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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from
../../gmp/doc/gmp.texi.
This manual describes how to install and use the GNU multiple precision
arithmetic library, version 6.0.0.
Copyright 1991, 1993-2014 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License, Version
1.3 or any later version published by the Free Software Foundation;
with no Invariant Sections, with the Front-Cover Texts being "A GNU
Manual", and with the Back-Cover Texts being "You have freedom to copy
and modify this GNU Manual, like GNU software". A copy of the license
is included in *note GNU Free Documentation License::.
INFO-DIR-SECTION GNU libraries
START-INFO-DIR-ENTRY
* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
END-INFO-DIR-ENTRY
File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
15.2.6 Exact Remainder
----------------------
If the exact division algorithm is done with a full subtraction at each
stage and the dividend isn't a multiple of the divisor, then low zero
limbs are produced but with a remainder in the high limbs. For
dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this
remainder r is of the form
a = q*d + r*b^n
n represents the number of zero limbs produced by the subtractions,
that being the number of limbs produced for q. r will be in the range
0<=r<d and can be viewed as a remainder, but one shifted up by a factor
of b^n.
Carrying out full subtractions at each stage means the same number
of cross products must be done as a normal division, but there's still
some single limb divisions saved. When d is a single limb some
simplifications arise, providing good speedups on a number of
processors.
The functions `mpn_divexact_by3', `mpn_modexact_1_odd' and the
internal `mpn_redc_X' functions differ subtly in how they return r,
leading to some negations in the above formula, but all are essentially
the same.
Clearly r is zero when a is a multiple of d, and this leads to
divisibility or congruence tests which are potentially more efficient
than a normal division.
The factor of b^n on r can be ignored in a GCD when d is odd, hence
the use of `mpn_modexact_1_odd' by `mpn_gcd_1' and `mpz_kronecker_ui'
etc (*note Greatest Common Divisor Algorithms::).
Montgomery's REDC method for modular multiplications uses operands
of the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n)
uses the factor of b^n in the exact remainder to reach a product in the
same form (x*y)*b^-n (*note Modular Powering Algorithm::).
Notice that r generally gives no useful information about the
ordinary remainder a mod d since b^n mod d could be anything. If
however b^n == 1 mod d, then r is the negative of the ordinary
remainder. This occurs whenever d is a factor of b^n-1, as for example
with 3 in `mpn_divexact_by3'. For a 32 or 64 bit limb other such
factors include 5, 17 and 257, but no particular use has been found for
this.
File: gmp.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
15.2.7 Small Quotient Division
------------------------------
An NxM division where the number of quotient limbs Q=N-M is small can
be optimized somewhat.
An ordinary basecase division normalizes the divisor by shifting it
to make the high bit set, shifting the dividend accordingly, and
shifting the remainder back down at the end of the calculation. This
is wasteful if only a few quotient limbs are to be formed. Instead a
division of just the top 2*Q limbs of the dividend by the top Q limbs
of the divisor can be used to form a trial quotient. This requires
only those limbs normalized, not the whole of the divisor and dividend.
A multiply and subtract then applies the trial quotient to the M-Q
unused limbs of the divisor and N-Q dividend limbs (which includes Q
limbs remaining from the trial quotient division). The starting trial
quotient can be 1 or 2 too big, but all cases of 2 too big and most
cases of 1 too big are detected by first comparing the most significant
limbs that will arise from the subtraction. An addback is done if the
quotient still turns out to be 1 too big.
This whole procedure is essentially the same as one step of the
basecase algorithm done in a Q limb base, though with the trial
quotient test done only with the high limbs, not an entire Q limb
"digit" product. The correctness of this weaker test can be
established by following the argument of Knuth section 4.3.1 exercise
20 but with the v2*q>b*r+u2 condition appropriately relaxed.
File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
15.3 Greatest Common Divisor
============================
* Menu:
* Binary GCD::
* Lehmer's Algorithm::
* Subquadratic GCD::
* Extended GCD::
* Jacobi Symbol::
File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
15.3.1 Binary GCD
-----------------
At small sizes GMP uses an O(N^2) binary style GCD. This is described
in many textbooks, for example Knuth section 4.5.2 algorithm B. It
simply consists of successively reducing odd operands a and b using
a,b = abs(a-b),min(a,b)
strip factors of 2 from a
The Euclidean GCD algorithm, as per Knuth algorithms E and A,
repeatedly computes the quotient q = floor(a/b) and replaces a,b by v,
u - q v. The binary algorithm has so far been found to be faster than
the Euclidean algorithm everywhere. One reason the binary method does
well is that the implied quotient at each step is usually small, so
often only one or two subtractions are needed to get the same effect as
a division. Quotients 1, 2 and 3 for example occur 67.7% of the time,
see Knuth section 4.5.3 Theorem E.
When the implied quotient is large, meaning b is much smaller than
a, then a division is worthwhile. This is the basis for the initial a
mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
and 1x1 cases). But after that initial reduction, big quotients occur
too rarely to make it worth checking for them.
The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as
described above. For two N-bit operands, the algorithm takes about
0.68 iterations per bit. For optimum performance some attention needs
to be paid to the way the factors of 2 are stripped from a.
Firstly it may be noted that in twos complement the number of low
zero bits on a-b is the same as b-a, so counting or testing can begin on
a-b without waiting for abs(a-b) to be determined.
A loop stripping low zero bits tends not to branch predict well,
since the condition is data dependent. But on average there's only a
few low zeros, so an option is to strip one or two bits arithmetically
then loop for more (as done for AMD K6). Or use a lookup table to get
a count for several bits then loop for more (as done for AMD K7). An
alternative approach is to keep just one of a or b odd and iterate
a,b = abs(a-b), min(a,b)
a = a/2 if even
b = b/2 if even
This requires about 1.25 iterations per bit, but stripping of a
single bit at each step avoids any branching. Repeating the bit strip
reduces to about 0.9 iterations per bit, which may be a worthwhile
tradeoff.
Generally with the above approaches a speed of perhaps 6 cycles per
bit can be achieved, which is still not terribly fast with for instance
a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
means it's not usually advantageous to combine a set of divisibility
tests into a GCD.
Currently, the binary algorithm is used for GCD only when N < 3.
File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
15.3.2 Lehmer's algorithm
-------------------------
Lehmer's improvement of the Euclidean algorithms is based on the
observation that the initial part of the quotient sequence depends only
on the most significant parts of the inputs. The variant of Lehmer's
algorithm used in GMP splits off the most significant two limbs, as
suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by
Jebelean (*note References::). The quotients of two double-limb inputs
are collected as a 2 by 2 matrix with single-limb elements. This is
done by the function `mpn_hgcd2'. The resulting matrix is applied to
the inputs using `mpn_mul_1' and `mpn_submul_1'. Each iteration usually
reduces the inputs by almost one limb. In the rare case of a large
quotient, no progress can be made by examining just the most
significant two limbs, and the quotient is computed using plain
division.
The resulting algorithm is asymptotically O(N^2), just as the
Euclidean algorithm and the binary algorithm. The quadratic part of the
work are the calls to `mpn_mul_1' and `mpn_submul_1'. For small sizes,
the linear work is also significant. There are roughly N calls to the
`mpn_hgcd2' function. This function uses a couple of important
optimizations:
* It uses the same relaxed notion of correctness as `mpn_hgcd' (see
next section). This means that when called with the most
significant two limbs of two large numbers, the returned matrix
does not always correspond exactly to the initial quotient
sequence for the two large numbers; the final quotient may
sometimes be one off.
* It takes advantage of the fact the quotients are usually small.
The division operator is not used, since the corresponding
assembler instruction is very slow on most architectures. (This
code could probably be improved further, it uses many branches
that are unfriendly to prediction).
* It switches from double-limb calculations to single-limb
calculations half-way through, when the input numbers have been
reduced in size from two limbs to one and a half.
File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms
15.3.3 Subquadratic GCD
-----------------------
For inputs larger than `GCD_DC_THRESHOLD', GCD is computed via the HGCD
(Half GCD) function, as a generalization to Lehmer's algorithm.
Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1.
Then HGCD(a,b) returns a transformation matrix T with non-negative
elements, and reduced numbers (c;d) = T^-1 (a;b). The reduced numbers
c,d must be larger than S limbs, while their difference abs(c-d) must
fit in S limbs. The matrix elements will also be of size roughly N/2.
The HGCD base case uses Lehmer's algorithm, but with the above stop
condition that returns reduced numbers and the corresponding
transformation matrix half-way through. For inputs larger than
`HGCD_THRESHOLD', HGCD is computed recursively, using the divide and
conquer algorithm in "On Sch�nhage's algorithm and subquadratic integer
GCD computation" by M�ller (*note References::). The recursive
algorithm consists of these main steps.
* Call HGCD recursively, on the most significant N/2 limbs. Apply the
resulting matrix T_1 to the full numbers, reducing them to a size
just above 3N/2.
* Perform a small number of division or subtraction steps to reduce
the numbers to size below 3N/2. This is essential mainly for the
unlikely case of large quotients.
* Call HGCD recursively, on the most significant N/2 limbs of the
reduced numbers. Apply the resulting matrix T_2 to the full
numbers, reducing them to a size just above N/2.
* Compute T = T_1 T_2.
* Perform a small number of division and subtraction steps to
satisfy the requirements, and return.
GCD is then implemented as a loop around HGCD, similarly to Lehmer's
algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
`mpn_hgcd2', and applies the resulting matrix to the full numbers, the
subquadratic GCD chops off the most significant third of the limbs (the
proportion is a tuning parameter, and 1/3 seems to be more efficient
than, e.g, 1/2), calls `mpn_hgcd', and applies the resulting matrix.
Once the input numbers are reduced to size below `GCD_DC_THRESHOLD',
Lehmer's algorithm is used for the rest of the work.
The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)),
where M(N) is the time for multiplying two N-limb numbers.
File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms
15.3.4 Extended GCD
-------------------
The extended GCD function, or GCDEXT, calculates gcd(a,b) and also
cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used
for plain GCD are extended to handle this case. The binary algorithm is
used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes
up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is
implemented as a loop around HGCD, but with more book-keeping to keep
track of the cofactors. This gives the same asymptotic running time as
for GCD and HGCD, O(M(N)*log(N))
One difference to plain GCD is that while the inputs a and b are
reduced as the algorithm proceeds, the cofactors x and y grow in size.
This makes the tuning of the chopping-point more difficult. The current
code chops off the most significant half of the inputs for the call to
HGCD in the first iteration, and the most significant two thirds for
the remaining calls. This strategy could surely be improved. Also the
stop condition for the loop, where Lehmer's algorithm is invoked once
the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be
improved by taking into account the current size of the cofactors.
File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
15.3.5 Jacobi Symbol
--------------------
[This section is obsolete. The current Jacobi code actually uses a very
efficient algorithm.]
`mpz_jacobi' and `mpz_kronecker' are currently implemented with a
simple binary algorithm similar to that described for the GCDs (*note
Binary GCD::). They're not very fast when both inputs are large.
Lehmer's multi-step improvement or a binary based multi-step algorithm
is likely to be better.
When one operand fits a single limb, and that includes
`mpz_kronecker_ui' and friends, an initial reduction is done with
either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
algorithm on a single limb. The binary algorithm is well suited to a
single limb, and the whole calculation in this case is quite efficient.
In all the routines sign changes for the result are accumulated
using some bit twiddling, avoiding table lookups or conditional jumps.
File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
15.4 Powering Algorithms
========================
* Menu:
* Normal Powering Algorithm::
* Modular Powering Algorithm::
File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
15.4.1 Normal Powering
----------------------
Normal `mpz' or `mpf' powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is
seen in the exponent, as per Knuth section 4.6.3. The "left to right"
variant described there is used rather than algorithm A, since it's
just as easy and can be done with somewhat less temporary memory.
File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
15.4.2 Modular Powering
-----------------------
Modular powering is implemented using a 2^k-ary sliding window
algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
(*note References::). k is chosen according to the size of the
exponent. Larger exponents use larger values of k, the choice being
made to minimize the average number of multiplications that must
supplement the squaring.
The modular multiplies and squarings use either a simple division or
the REDC method by Montgomery (*note References::). REDC is a little
faster, essentially saving N single limb divisions in a fashion similar
to an exact remainder (*note Exact Remainder::).
File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
15.5 Root Extraction Algorithms
===============================
* Menu:
* Square Root Algorithm::
* Nth Root Algorithm::
* Perfect Square Algorithm::
* Perfect Power Algorithm::
File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
15.5.1 Square Root
------------------
Square roots are taken using the "Karatsuba Square Root" algorithm by
Paul Zimmermann (*note References::).
An input n is split into four parts of k bits each, so with b=2^k we
have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
that either the high or second highest bit is set. In GMP, k is kept
on a limb boundary and the input is left shifted (by an even number of
bits) to normalize.
The square root of the high two parts is taken, by recursive
application of the algorithm (bottoming out in a one-limb Newton's
method),
s1,r1 = sqrtrem (a3*b + a2)
This is an approximation to the desired root and is extended by a
division to give s,r,
q,u = divrem (r1*b + a1, 2*s1)
s = s1*b + q
r = u*b + a0 - q^2
The normalization requirement on a3 means at this point s is either
correct or 1 too big. r is negative in the latter case, so
if r < 0 then
r = r + 2*s - 1
s = s - 1
The algorithm is expressed in a divide and conquer form, but as
noted in the paper it can also be viewed as a discrete variant of
Newton's method, or as a variation on the schoolboy method (no longer
taught) for square roots two digits at a time.
If the remainder r is not required then usually only a few high limbs
of r and u need to be calculated to determine whether an adjustment to
s is required. This optimization is not currently implemented.
In the Karatsuba multiplication range this algorithm is
O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
limbs. In the FFT multiplication range this grows to a bound of
O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the
Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
The algorithm does all its calculations in integers and the resulting
`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.
File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
15.5.2 Nth Root
---------------
Integer Nth roots are taken using Newton's method with the following
iteration, where A is the input and n is the root to be taken.
1 A
a[i+1] = - * ( --------- + (n-1)*a[i] )
n a[i]^(n-1)
The initial approximation a[1] is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just
above the true root. The iteration converges quadratically when
started from a good approximation. When n is large more initial bits
are needed to get good convergence. The current implementation is not
particularly well optimized.
File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
15.5.3 Perfect Square
---------------------
A significant fraction of non-squares can be quickly identified by
checking whether the input is a quadratic residue modulo small integers.
`mpz_perfect_square_p' first tests the input mod 256, which means
just examining the low byte. Only 44 different values occur for
squares mod 256, so 82.8% of inputs can be immediately identified as
non-squares.
On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
for a total 99.25% of inputs identified as non-squares. On a 64-bit
system 97 is tested too, for a total 99.62%.
These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
for 64-bits), and such a remainder can be quickly taken just using
additions (see `mpn_mod_34lsub1').
When nails are in use moduli are instead selected by the `gen-psqr.c'
program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1
could be done with nails using some extra bit shifts, but this is not
currently implemented.
In any case each modulus is applied to the `mpn_mod_34lsub1' or
`mpn_mod_1' remainder and a table lookup identifies non-squares. By
using a "modexact" style calculation, and suitably permuted tables,
just one multiply each is required, see the code for details. Moduli
are also combined to save operations, so long as the lookup tables
don't become too big. `gen-psqr.c' does all the pre-calculations.
A square root must still be taken for any value that passes these
tests, to verify it's really a square and not one of the small fraction
of non-squares that get through (i.e. a pseudo-square to all the tested
bases).
Clearly more residue tests could be done, `mpz_perfect_square_p' only
uses a compact and efficient set. Big inputs would probably benefit
from more residue testing, small inputs might be better off with less.
The assumed distribution of squares versus non-squares in the input
would affect such considerations.
File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
15.5.4 Perfect Power
--------------------
Detecting perfect powers is required by some factorization algorithms.
Currently `mpz_perfect_power_p' is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(*Note Nth Root Algorithm::.)
If a prime divisor p with multiplicity e can be found, then only
roots which are divisors of e need to be considered, much reducing the
work necessary. To this end divisibility by a set of small primes is
checked.
File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
15.6 Radix Conversion
=====================
Radix conversions are less important than other algorithms. A program
dominated by conversions should probably use a different data
representation.
* Menu:
* Binary to Radix::
* Radix to Binary::
File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms
15.6.1 Binary to Radix
----------------------
Conversions from binary to a power-of-2 radix use a simple and fast
O(N) bit extraction algorithm.
Conversions from binary to other radices use one of two algorithms.
Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
Repeated divisions by b^n are made, where b is the radix and n is the
biggest power that fits in a limb. But instead of simply using the
remainder r from such divisions, an extra divide step is done to give a
fractional limb representing r/b^n. The digits of r can then be
extracted using multiplications by b rather than divisions. Special
case code is provided for decimal, allowing multiplications by 10 to
optimize to shifts and adds.
Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used. For an input t, powers b^(n*2^i) of the radix are calculated,
until a power between t and sqrt(t) is reached. t is then divided by
that largest power, giving a quotient which is the digits above that
power, and a remainder which is those below. These two parts are in
turn divided by the second highest power, and so on recursively. When
a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
limbs, the basecase algorithm described above is used.
The advantage of this algorithm is that big divisions can make use
of the sub-quadratic divide and conquer division (*note Divide and
Conquer Division::), and big divisions tend to have less overheads than
lots of separate single limb divisions anyway. But in any case the
cost of calculating the powers b^(n*2^i) must first be overcome.
`GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
the same basic thing, the point where it becomes worth doing a big
division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD'
includes the cost of calculating the radix power required, whereas
`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
case when recursing.
Since the base case produces digits from least to most significant
but they want to be stored from most to least, it's necessary to
calculate in advance how many digits there will be, or at least be sure
not to underestimate that. For GMP the number of input bits is
multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
The result is either correct or one too big.
Examining some of the high bits of the input could increase the
chance of getting the exact number of digits, but an exact result every
time would not be practical, since in general the difference between
numbers 100... and 99... is only in the last few bits and the work to
identify 99... might well be almost as much as a full conversion.
`mpf_get_str' doesn't currently use the algorithm described here, it
multiplies or divides by a power of b to move the radix point to the
just above the highest non-zero digit (or at worst one above that
location), then multiplies by b^n to bring out digits. This is O(N^2)
and is certainly not optimal.
The r/b^n scheme described above for using multiplications to bring
out digits might be useful for more than a single limb. Some brief
experiments with it on the base case when recursing didn't give a
noticeable improvement, but perhaps that was only due to the
implementation. Something similar would work for the sub-quadratic
divisions too, though there would be the cost of calculating a bigger
radix power.
Another possible improvement for the sub-quadratic part would be to
arrange for radix powers that balanced the sizes of quotient and
remainder produced, i.e. the highest power would be an b^(n*k)
approximately equal to sqrt(t), not restricted to a 2^i factor. That
ought to smooth out a graph of times against sizes, but may or may not
be a net speedup.
File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms
15.6.2 Radix to Binary
----------------------
*This section needs to be rewritten, it currently describes the
algorithms used before GMP 4.3.*
Conversions from a power-of-2 radix into binary use a simple and fast
O(N) bitwise concatenation algorithm.
Conversions from other radices use one of two algorithms. Sizes
below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups
of n digits are converted to limbs, where n is the biggest power of the
base b which will fit in a limb, then those groups are accumulated into
the result by multiplying by b^n and adding. This saves
multi-precision operations, as per Knuth section 4.4 part E (*note
References::). Some special case code is provided for decimal, giving
the compiler a chance to optimize multiplications by 10.
Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used. First groups of n digits are converted into limbs. Then adjacent
limbs are combined into limb pairs with x*b^n+y, where x and y are the
limbs. Adjacent limb pairs are combined into quads similarly with
x*b^(2n)+y. This continues until a single block remains, that being
the result.
The advantage of this method is that the multiplications for each x
are big blocks, allowing Karatsuba and higher algorithms to be used.
But the cost of calculating the powers b^(n*2^i) must be overcome.
`SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000
digits, and on some processors much bigger still.
`SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and
tuned for decimal), though it might be better based on a limb count, so
as to be independent of the base. But that sort of count isn't used by
the base case and so would need some sort of initial calculation or
estimate.
The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger
than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that
`mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5,
or more).
File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms
15.7 Other Algorithms
=====================
* Menu:
* Prime Testing Algorithm::
* Factorial Algorithm::
* Binomial Coefficients Algorithm::
* Fibonacci Numbers Algorithm::
* Lucas Numbers Algorithm::
* Random Number Algorithms::
File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
15.7.1 Prime Testing
--------------------
The primality testing in `mpz_probab_prime_p' (*note Number Theoretic
Functions::) first does some trial division by small factors and then
uses the Miller-Rabin probabilistic primality testing algorithm, as
described in Knuth section 4.5.4 algorithm P (*note References::).
For an odd input n, and with n = q*2^k+1 where q is odd, this
algorithm selects a random base x and tests whether x^q mod n is 1 or
-1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably
prime, if not then n is definitely composite.
Any prime n will pass the test, but some composites do too. Such
composites are known as strong pseudoprimes to base x. No n is a
strong pseudoprime to more than 1/4 of all bases (see Knuth exercise
22), hence with x chosen at random there's no more than a 1/4 chance a
"probable prime" will in fact be composite.
In fact strong pseudoprimes are quite rare, making the test much more
powerful than this analysis would suggest, but 1/4 is all that's proven
for an arbitrary n.
File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
15.7.2 Factorial
----------------
Factorials are calculated by a combination of two algorithms. An idea is
shared among them: to compute the odd part of the factorial; a final
step takes account of the power of 2 term, by shifting.
For small n, the odd factor of n! is computed with the simple
observation that it is equal to the product of all positive odd numbers
smaller than n times the odd factor of [n/2]!, where [x] is the integer
part of x, and so on recursively. The procedure can be best illustrated
with an example,
23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^19
Current code collects all the factors in a single list, with a loop
and no recursion, and compute the product, with no special care for
repeated chunks.
When n is larger, computation pass trough prime sieving. An helper
function is used, as suggested by Peter Luschny:
n
-----
n! | | L(p,n)
msf(n) = -------------- = | | p
[n/2]!^2.2^k p=3
Where p ranges on odd prime numbers. The exponent k is chosen to
obtain an odd integer number: k is the number of 1 bits in the binary
representation of [n/2]. The function L(p,n) can be defined as zero
when p is composite, and, for any prime p, it is computed with:
---
\ n
L(p,n) = / [---] mod 2 <= log (n) .
--- p^i p
i>0
With this helper function, we are able to compute the odd part of n!
using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion
stops using the small-n algorithm on some [n/2^i].
Both the above algorithms use binary splitting to compute the
product of many small factors. At first as many products as possible
are accumulated in a single register, generating a list of factors that
fit in a machine word. This list is then split into halves, and the
product is computed recursively.
Such splitting is more efficient than repeated Nx1 multiplies since
it forms big multiplies, allowing Karatsuba and higher algorithms to be
used. And even below the Karatsuba threshold a big block of work can
be more efficient for the basecase algorithm.
File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
15.7.3 Binomial Coefficients
----------------------------
Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
using C(n,k) = C(n,n-k) if necessary, and then evaluating the following
product simply from i=2 to i=k.
k (n-k+i)
C(n,k) = (n-k+1) * prod -------
i=2 i
It's easy to show that each denominator i will divide the product so
far, so the exact division algorithm is used (*note Exact Division::).
The numerators n-k+i and denominators i are first accumulated into
as many fit a limb, to save multi-precision operations, though for
`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
and n-k+i in general won't fit in a limb at all.
File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms
15.7.4 Fibonacci Numbers
------------------------
The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
calculating isolated F[n] or F[n],F[n-1] values efficiently.
For small n, a table of single limb values in `__gmp_fib_table' is
used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
to F[93]. For convenience the table starts at F[-1].
Beyond the table, values are generated with a binary powering
algorithm, calculating a pair F[n] and F[n-1] working from high to low
across the bits of n. The formulas used are
F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
F[2k-1] = F[k]^2 + F[k-1]^2
F[2k] = F[2k+1] - F[2k-1]
At each step, k is the high b bits of n. If the next bit of n is 0
then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
and the process repeated until all bits of n are incorporated. Notice
these formulas require just two squares per bit of n.
It'd be possible to handle the first few n above the single limb
table with simple additions, using the defining Fibonacci recurrence
F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
be faster for only about 10 or 20 values of n, and including a block of
code for just those doesn't seem worthwhile. If they really mattered
it'd be better to extend the data table.
Using a table avoids lots of calculations on small numbers, and
makes small n go fast. A bigger table would make more small n go fast,
it's just a question of balancing size against desired speed. For GMP
the code is kept compact, with the emphasis primarily on a good
powering algorithm.
`mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
interested in F[n]. In this case the last step of the algorithm can
become one multiply instead of two squares. One of the following two
formulas is used, according as n is odd or even.
F[2k] = F[k]*(F[k]+2F[k-1])
F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
F[2k+1] here is the same as above, just rearranged to be a multiply.
For interest, the 2*(-1)^k term both here and above can be applied just
to the low limb of the calculation, without a carry or borrow into
further limbs, which saves some code size. See comments with
`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.
File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms
15.7.5 Lucas Numbers
--------------------
`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
Fibonacci numbers with the following simple formulas.
L[k] = F[k] + 2*F[k-1]
L[k-1] = 2*F[k] - F[k-1]
`mpz_lucnum_ui' is only interested in L[n], and some work can be
saved. Trailing zero bits on n can be handled with a single square
each.
L[2k] = L[k]^2 - 2*(-1)^k
And the lowest 1 bit can be handled with one multiply of a pair of
Fibonacci numbers, similar to what `mpz_fib_ui' does.
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms
15.7.6 Random Numbers
---------------------
For the `urandomb' functions, random numbers are generated simply by
concatenating bits produced by the generator. As long as the generator
has good randomness properties this will produce well-distributed N bit
numbers.
For the `urandomm' functions, random numbers in a range 0<=R<N are
generated by taking values R of ceil(log2(N)) bits each until one
satisfies R<N. This will normally require only one or two attempts,
but the attempts are limited in case the generator is somehow
degenerate and produces only 1 bits or similar.
The Mersenne Twister generator is by Matsumoto and Nishimura (*note
References::). It has a non-repeating period of 2^19937-1, which is a
Mersenne prime, hence the name of the generator. The state is 624
words of 32-bits each, which is iterated with one XOR and shift for each
32-bit word generated, making the algorithm very fast. Randomness
properties are also very good and this is the default algorithm used by
GMP.
Linear congruential generators are described in many text books, for
instance Knuth volume 2 (*note References::). With a modulus M and
parameters A and C, an integer state S is iterated by the formula S <-
A*S+C mod M. At each step the new state is a linear function of the
previous, mod M, hence the name of the generator.
In GMP only moduli of the form 2^N are supported, and the current
implementation is not as well optimized as it could be. Overheads are
significant when N is small, and when N is large clearly the multiply
at each step will become slow. This is not a big concern, since the
Mersenne Twister generator is better in every respect and is therefore
recommended for all normal applications.
For both generators the current state can be deduced by observing
enough output and applying some linear algebra (over GF(2) in the case
of the Mersenne Twister). This generally means raw output is
unsuitable for cryptographic applications without further hashing or
the like.
File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms
15.8 Assembly Coding
====================
The assembly subroutines in GMP are the most significant source of
speed at small to moderate sizes. At larger sizes algorithm selection
becomes more important, but of course speedups in low level routines
will still speed up everything proportionally.
Carry handling and widening multiplies that are important for GMP
can't be easily expressed in C. GCC `asm' blocks help a lot and are
provided in `longlong.h', but hand coding low level routines invariably
offers a speedup over generic C by a factor of anything from 2 to 10.
* Menu:
* Assembly Code Organisation::
* Assembly Basics::
* Assembly Carry Propagation::
* Assembly Cache Handling::
* Assembly Functional Units::
* Assembly Floating Point::
* Assembly SIMD Instructions::
* Assembly Software Pipelining::
* Assembly Loop Unrolling::
* Assembly Writing Guide::
File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding
15.8.1 Code Organisation
------------------------
The various `mpn' subdirectories contain machine-dependent code, written
in C or assembly. The `mpn/generic' subdirectory contains default code,
used when there's no machine-specific version of a particular file.
Each `mpn' subdirectory is for an ISA family. Generally 32-bit and
64-bit variants in a family cannot share code and have separate
directories. Within a family further subdirectories may exist for CPU
variants.
In each directory a `nails' subdirectory may exist, holding code with
nails support for that CPU variant. A `NAILS_SUPPORT' directive in each
file indicates the nails values the code handles. Nails code only
exists where it's faster, or promises to be faster, than plain code.
There's no effort put into nails if they're not going to enhance a
given CPU.
File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding
15.8.2 Assembly Basics
----------------------
`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
overall GMP performance. All multiplications and divisions come down to
repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and
`mpn_rshift' are next most important.
On some CPUs assembly versions of the internal functions
`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
mainly through avoiding function call overheads. They can also
potentially make better use of a wide superscalar processor, as can
bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'.
The restrictions on overlaps between sources and destinations (*note
Low-level Functions::) are designed to facilitate a variety of
implementations. For example, knowing `mpn_add_n' won't have partly
overlapping sources and destination means reading can be done far ahead
of writing on superscalar processors, and loops can be vectorized on a
vector processor, depending on the carry handling.
File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding
15.8.3 Carry Propagation
------------------------
The problem that presents most challenges in GMP is propagating carries
from one limb to the next. In functions like `mpn_addmul_1' and
`mpn_add_n', carries are the only dependencies between limb operations.
On processors with carry flags, a straightforward CISC style `adc' is
generally best. AMD K6 `mpn_addmul_1' however is an example of an
unusual set of circumstances where a branch works out better.
On RISC processors generally an add and compare for overflow is
used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some
carry propagation schemes require 4 instructions, meaning at least 4
cycles per limb, but other schemes may use just 1 or 2. On wide
superscalar processors performance may be completely determined by the
number of dependent instructions between carry-in and carry-out for
each limb.
On vector processors good use can be made of the fact that a carry
bit only very rarely propagates more than one limb. When adding a
single bit to a limb, there's only a carry out if that limb was
`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
`mpn/cray/add_n.c' is an example of this, it adds all limbs in
parallel, adds one set of carry bits in parallel and then only rarely
needs to fall through to a loop propagating further carries.
On the x86s, GCC (as of version 2.95.2) doesn't generate
particularly good code for the RISC style idioms that are necessary to
handle carry bits in C. Often conditional jumps are generated where
`adc' or `sbb' forms would be better. And so unfortunately almost any
loop involving carry bits needs to be coded in assembly for best
results.
File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding
15.8.4 Cache Handling
---------------------
GMP aims to perform well both on operands that fit entirely in L1 cache
and those which don't.
Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
large operands, so L2 and main memory performance is important for them.
`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
basecases, so L1 performance matters most for them, unless assembly
versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
case the remaining uses are mostly for larger operands.
For L2 or main memory operands, memory access times will almost
certainly be more than the calculation time. The aim therefore is to
maximize memory throughput, by starting a load of the next cache line
while processing the contents of the previous one. Clearly this is
only possible if the chip has a lock-up free cache or some sort of
prefetch instruction. Most current chips have both these features.
Prefetching sources combines well with loop unrolling, since a
prefetch can be initiated once per unrolled loop (or more than once if
the loop covers more than one cache line).
On CPUs without write-allocate caches, prefetching destinations will
ensure individual stores don't go further down the cache hierarchy,
limiting bandwidth. Of course for calculations which are slow anyway,
like `mpn_divrem_1', write-throughs might be fine.
The distance ahead to prefetch will be determined by memory latency
versus throughput. The aim of course is to have data arriving
continuously, at peak throughput. Some CPUs have limits on the number
of fetches or prefetches in progress.
If a special prefetch instruction doesn't exist then a plain load
can be used, but in that case care must be taken not to attempt to read
past the end of an operand, since that might produce a segmentation
violation.
Some CPUs or systems have hardware that detects sequential memory
accesses and initiates suitable cache movements automatically, making
life easy.
File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding
15.8.5 Functional Units
-----------------------
When choosing an approach for an assembly loop, consideration is given
to what operations can execute simultaneously and what throughput can
thereby be achieved. In some cases an algorithm can be tweaked to
accommodate available resources.
Loop control will generally require a counter and pointer updates,
costing as much as 5 instructions, plus any delays a branch introduces.
CPU addressing modes might reduce pointer updates, perhaps by allowing
just one updating pointer and others expressed as offsets from it, or
on CISC chips with all addressing done with the loop counter as a
scaled index.
The final loop control cost can be amortised by processing several
limbs in each iteration (*note Assembly Loop Unrolling::). This at
least ensures loop control isn't a big fraction the work done.
Memory throughput is always a limit. If perhaps only one load or
one store can be done per cycle then 3 cycles/limb will the top speed
for "binary" operations like `mpn_add_n', and any code achieving that
is optimal.
Integer resources can be freed up by having the loop counter in a
float register, or by pressing the float units into use for some
multiplying, perhaps doing every second limb on the float side (*note
Assembly Floating Point::).
Float resources can be freed up by doing carry propagation on the
integer side, or even by doing integer to float conversions in integers
using bit twiddling.
File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding
15.8.6 Floating Point
---------------------
Floating point arithmetic is used in GMP for multiplications on CPUs
with poor integer multipliers. It's mostly useful for `mpn_mul_1',
`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
`mpn_mul_basecase' on both 32-bit and 64-bit machines.
With IEEE 53-bit double precision floats, integer multiplications
producing up to 53 bits will give exact results. Breaking a 64x64
multiplication into eight 16x32->48 bit pieces is convenient. With
some care though six 21x32->53 bit products can be used, if one of the
lower two 21-bit pieces also uses the sign bit.
For the `mpn_mul_1' family of functions on a 64-bit machine, the
invariant single limb is split at the start, into 3 or 4 pieces.
Inside the loop, the bignum operand is split into 32-bit pieces. Fast
conversion of these unsigned 32-bit pieces to floating point is highly
machine-dependent. In some cases, reading the data into the integer
unit, zero-extending to 64-bits, then transferring to the floating
point unit back via memory is the only option.
Converting partial products back to 64-bit limbs is usually best
done as a signed conversion. Since all values are smaller than 2^53,
signed and unsigned are the same, but most processors lack unsigned
conversions.
Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or
`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
into four 16-bit parts. The multi-limb operand U is split in the loop
into two 32-bit parts.
+---+---+---+---+
|v48|v32|v16|v00| V operand
+---+---+---+---+
+-------+---+---+
x | u32 | u00 | U operand (one limb)
+---------------+
---------------------------------
+-----------+
| u00 x v00 | p00 48-bit products
+-----------+
+-----------+
| u00 x v16 | p16
+-----------+
+-----------+
| u00 x v32 | p32
+-----------+
+-----------+
| u00 x v48 | p48
+-----------+
+-----------+
| u32 x v00 | r32
+-----------+
+-----------+
| u32 x v16 | r48
+-----------+
+-----------+
| u32 x v32 | r64
+-----------+
+-----------+
| u32 x v48 | r80
+-----------+
p32 and r32 can be summed using floating-point addition, and
likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from
the previous iteration.
For each loop then, four 49-bit quantities are transferred to the
integer unit, aligned as follows,
|-----64bits----|-----64bits----|
+------------+
| p00 + r64' | i00
+------------+
+------------+
| p16 + r80' | i16
+------------+
+------------+
| p32 + r32 | i32
+------------+
+------------+
| p48 + r48 | i48
+------------+
The challenge then is to sum these efficiently and add in a carry
limb, generating a low 64-bit result limb and a high 33-bit carry limb
(i48 extends 33 bits into the high half).
File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding
15.8.7 SIMD Instructions
------------------------
The single-instruction multiple-data support in current microprocessors
is aimed at signal processing algorithms where each data point can be
treated more or less independently. There's generally not much support
for propagating the sort of carries that arise in GMP.
SIMD multiplications of say four 16x16 bit multiplies only do as much
work as one 32x32 from GMP's point of view, and need some shifts and
adds besides. But of course if say the SIMD form is fully pipelined
and uses less instruction decoding then it may still be worthwhile.
On the x86 chips, MMX has so far found a use in `mpn_rshift' and
`mpn_lshift', and is used in a special case for 16-bit multipliers in
the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1',
`mpn_addmul_1', and `mpn_submul_1'.
File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding
15.8.8 Software Pipelining
--------------------------
Software pipelining consists of scheduling instructions around the
branch point in a loop. For example a loop might issue a load not for
use in the present iteration but the next, thereby allowing extra
cycles for the data to arrive from memory.
Naturally this is wanted only when doing things like loads or
multiplies that take several cycles to complete, and only where a CPU
has multiple functional units so that other work can be done in the
meantime.
A pipeline with several stages will have a data value in progress at
each stage and each loop iteration moves them along one stage. This is
like juggling.
If the latency of some instruction is greater than the loop time
then it will be necessary to unroll, so one register has a result ready
to use while another (or multiple others) are still in progress.
(*note Assembly Loop Unrolling::).
File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding
15.8.9 Loop Unrolling
---------------------
Loop unrolling consists of replicating code so that several limbs are
processed in each loop. At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for
example alternately using one register combination and then another.
Judicious use of `m4' macros can help avoid lots of duplication in the
source code.
Any amount of unrolling can be handled with a loop counter that's
decremented by N each time, stopping when the remaining count is less
than the further N the loop will process. Or by subtracting N at the
start, the termination condition becomes when the counter C is less
than 0 (and the count of remaining limbs is C+N).
Alternately for a power of 2 unroll the loop count and remainder can
be established with a shift and mask. This is convenient if also
making a computed jump into the middle of a large loop.
The limbs not a multiple of the unrolling can be handled in various
ways, for example
* A simple loop at the end (or the start) to process the excess.
Care will be wanted that it isn't too much slower than the
unrolled part.
* A set of binary tests, for example after an 8-limb unrolling, test
for 4 more limbs to process, then a further 2 more or not, and
finally 1 more or not. This will probably take more code space
than a simple loop.
* A `switch' statement, providing separate code for each possible
excess, for example an 8-limb unrolling would have separate code
for 0 remaining, 1 remaining, etc, up to 7 remaining. This might
take a lot of code, but may be the best way to optimize all cases
in combination with a deep pipelined loop.
* A computed jump into the middle of the loop, thus making the first
iteration handle the excess. This should make times smoothly
increase with size, which is attractive, but setups for the jump
and adjustments for pointers can be tricky and could become quite
difficult in combination with deep pipelining.
File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding
15.8.10 Writing Guide
---------------------
This is a guide to writing software pipelined loops for processing limb
vectors in assembly.
First determine the algorithm and which instructions are needed.
Code it without unrolling or scheduling, to make sure it works. On a
3-operand CPU try to write each new value to a new register, this will
greatly simplify later steps.
Then note for each instruction the functional unit and/or issue port
requirements. If an instruction can use either of two units, like U0
or U1 then make a category "U0/U1". Count the total using each unit
(or combined unit), and count all instructions.
Figure out from those counts the best possible loop time. The goal
will be to find a perfect schedule where instruction latencies are
completely hidden. The total instruction count might be the limiting
factor, or perhaps a particular functional unit. It might be possible
to tweak the instructions to help the limiting factor.
Suppose the loop time is N, then make N issue buckets, with the
final loop branch at the end of the last. Now fill the buckets with
dummy instructions using the functional units desired. Run this to
make sure the intended speed is reached.
Now replace the dummy instructions with the real instructions from
the slow but correct loop you started with. The first will typically
be a load instruction. Then the instruction using that value is placed
in a bucket an appropriate distance down. Run the loop again, to check
it still runs at target speed.
Keep placing instructions, frequently measuring the loop. After a
few you will need to wrap around from the last bucket back to the top
of the loop. If you used the new-register for new-value strategy above
then there will be no register conflicts. If not then take care not to
clobber something already in use. Changing registers at this time is
very error prone.
The loop will overlap two or more of the original loop iterations,
and the computation of one vector element result will be started in one
iteration of the new loop, and completed one or several iterations
later.
The final step is to create feed-in and wind-down code for the loop.
A good way to do this is to make a copy (or copies) of the loop at the
start and delete those instructions which don't have valid antecedents,
and at the end replicate and delete those whose results are unwanted
(including any further loads).
The loop will have a minimum number of limbs loaded and processed,
so the feed-in code must test if the request size is smaller and skip
either to a suitable part of the wind-down or to special code for small
sizes.
File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top
16 Internals
************
*This chapter is provided only for informational purposes and the
various internals described here may change in future GMP releases.
Applications expecting to be compatible with future releases should use
only the documented interfaces described in previous chapters.*
* Menu:
* Integer Internals::
* Rational Internals::
* Float Internals::
* Raw Output Internals::
* C++ Interface Internals::
File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals
16.1 Integer Internals
======================
`mpz_t' variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated. The fields are as follows.
`_mp_size'
The number of limbs, or the negative of that when representing a
negative integer. Zero is represented by `_mp_size' set to zero,
in which case the `_mp_d' data is unused.
`_mp_d'
A pointer to an array of limbs which is the magnitude. These are
stored "little endian" as per the `mpn' functions, so `_mp_d[0]'
is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the
most significant. Whenever `_mp_size' is non-zero, the most
significant limb is non-zero.
Currently there's always at least one limb allocated, so for
instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui'
can fetch `_mp_d[0]' unconditionally (though its value is then
only wanted if `_mp_size' is non-zero).
`_mp_alloc'
`_mp_alloc' is the number of limbs currently allocated at `_mp_d',
and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine
is about to (or might be about to) increase `_mp_size', it checks
`_mp_alloc' to see whether there's enough space, and reallocates
if not. `MPZ_REALLOC' is generally used for this.
The various bitwise logical functions like `mpz_and' behave as if
negative values were twos complement. But sign and magnitude is always
used internally, and necessary adjustments are made during the
calculations. Sometimes this isn't pretty, but sign and magnitude are
best for other routines.
Some internal temporary variables are setup with `MPZ_TMP_INIT' and
these have `_mp_d' space obtained from `TMP_ALLOC' rather than the
memory allocation functions. Care is taken to ensure that these are
big enough that no reallocation is necessary (since it would have
unpredictable consequences).
`_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is
usually a `long'. This is done to make the fields just 32 bits on some
64 bits systems, thereby saving a few bytes of data space but still
providing plenty of range.
File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals
16.2 Rational Internals
=======================
`mpq_t' variables represent rationals using an `mpz_t' numerator and
denominator (*note Integer Internals::).
The canonical form adopted is denominator positive (and non-zero),
no common factors between numerator and denominator, and zero uniquely
represented as 0/1.
It's believed that casting out common factors at each stage of a
calculation is best in general. A GCD is an O(N^2) operation so it's
better to do a few small ones immediately than to delay and have to do
a big one later. Knowing the numerator and denominator have no common
factors can be used for example in `mpq_mul' to make only two cross
GCDs necessary, not four.
This general approach to common factors is badly sub-optimal in the
presence of simple factorizations or little prospect for cancellation,
but GMP has no way to know when this will occur. As per *note
Efficiency::, that's left to applications. The `mpq_t' framework might
still suit, with `mpq_numref' and `mpq_denref' for direct access to the
numerator and denominator, or of course `mpz_t' variables can be used
directly.
File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals
16.3 Float Internals
====================
Efficient calculation is the primary aim of GMP floats and the use of
whole limbs and simple rounding facilitates this.
`mpf_t' floats have a variable precision mantissa and a single
machine word signed exponent. The mantissa is represented using sign
and magnitude.
most least
significant significant
limb limb
_mp_d
|---- _mp_exp ---> |
_____ _____ _____ _____ _____
|_____|_____|_____|_____|_____|
. <------------ radix point
<-------- _mp_size --------->
The fields are as follows.
`_mp_size'
The number of limbs currently in use, or the negative of that when
representing a negative value. Zero is represented by `_mp_size'
and `_mp_exp' both set to zero, and in that case the `_mp_d' data
is unused. (In the future `_mp_exp' might be undefined when
representing zero.)
`_mp_prec'
The precision of the mantissa, in limbs. In any calculation the
aim is to produce `_mp_prec' limbs of result (the most significant
being non-zero).
`_mp_d'
A pointer to the array of limbs which is the absolute value of the
mantissa. These are stored "little endian" as per the `mpn'
functions, so `_mp_d[0]' is the least significant limb and
`_mp_d[ABS(_mp_size)-1]' the most significant.
The most significant limb is always non-zero, but there are no
other restrictions on its value, in particular the highest 1 bit
can be anywhere within the limb.
`_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being
for convenience (see below). There are no reallocations during a
calculation, only in a change of precision with `mpf_set_prec'.
`_mp_exp'
The exponent, in limbs, determining the location of the implied
radix point. Zero means the radix point is just above the most
significant limb. Positive values mean a radix point offset
towards the lower limbs and hence a value >= 1, as for example in
the diagram above. Negative exponents mean a radix point further
above the highest limb.
Naturally the exponent can be any value, it doesn't have to fall
within the limbs as the diagram shows, it can be a long way above
or a long way below. Limbs other than those included in the
`{_mp_d,_mp_size}' data are treated as zero.
The `_mp_size' and `_mp_prec' fields are `int', although the
`mp_size_t' type is usually a `long'. The `_mp_exp' field is usually
`long'. This is done to make some fields just 32 bits on some 64 bits
systems, thereby saving a few bytes of data space but still providing
plenty of precision and a very large range.
The following various points should be noted.
Low Zeros
The least significant limbs `_mp_d[0]' etc can be zero, though
such low zeros can always be ignored. Routines likely to produce
low zeros check and avoid them to save time in subsequent
calculations, but for most routines they're quite unlikely and
aren't checked.
Mantissa Size Range
The `_mp_size' count of limbs in use can be less than `_mp_prec' if
the value can be represented in less. This means low precision
values or small integers stored in a high precision `mpf_t' can
still be operated on efficiently.
`_mp_size' can also be greater than `_mp_prec'. Firstly a value is
allowed to use all of the `_mp_prec+1' limbs available at `_mp_d',
and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves
`_mp_size' unchanged and so the size can be arbitrarily bigger than
`_mp_prec'.
Rounding
All rounding is done on limb boundaries. Calculating `_mp_prec'
limbs with the high non-zero will ensure the application requested
minimum precision is obtained.
The use of simple "trunc" rounding towards zero is efficient,
since there's no need to examine extra limbs and increment or
decrement.
Bit Shifts
Since the exponent is in limbs, there are no bit shifts in basic
operations like `mpf_add' and `mpf_mul'. When differing exponents
are encountered all that's needed is to adjust pointers to line up
the relevant limbs.
Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit
shifts, but the choice is between an exponent in limbs which
requires shifts there, or one in bits which requires them almost
everywhere else.
Use of `_mp_prec+1' Limbs
The extra limb on `_mp_d' (`_mp_prec+1' rather than just
`_mp_prec') helps when an `mpf' routine might get a carry from its
operation. `mpf_add' for instance will do an `mpn_add' of
`_mp_prec' limbs. If there's no carry then that's the result, but
if there is a carry then it's stored in the extra limb of space and
`_mp_size' becomes `_mp_prec+1'.
Whenever `_mp_prec+1' limbs are held in a variable, the low limb
is not needed for the intended precision, only the `_mp_prec' high
limbs. But zeroing it out or moving the rest down is unnecessary.
Subsequent routines reading the value will simply take the high
limbs they need, and this will be `_mp_prec' if their target has
that same precision. This is no more than a pointer adjustment,
and must be checked anyway since the destination precision can be
different from the sources.
Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs
if available. This ensures that a variable which has `_mp_size'
equal to `_mp_prec+1' will get its full exact value copied.
Strictly speaking this is unnecessary since only `_mp_prec' limbs
are needed for the application's requested precision, but it's
considered that an `mpf_set' from one variable into another of the
same precision ought to produce an exact copy.
Application Precisions
`__GMPF_BITS_TO_PREC' converts an application requested precision
to an `_mp_prec'. The value in bits is rounded up to a whole limb
then an extra limb is added since the most significant limb of
`_mp_d' is only non-zero and therefore might contain only one bit.
`__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
extra limb from `_mp_prec' before converting to bits. The net
effect of reading back with `mpf_get_prec' is simply the precision
rounded up to a multiple of `mp_bits_per_limb'.
Note that the extra limb added here for the high only being
non-zero is in addition to the extra limb allocated to `_mp_d'.
For example with a 32-bit limb, an application request for 250
bits will be rounded up to 8 limbs, then an extra added for the
high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then
gets 10 limbs allocated. Reading back with `mpf_get_prec' will
take `_mp_prec' subtract 1 limb and multiply by 32, giving 256
bits.
Strictly speaking, the fact the high limb has at least one bit
means that a float with, say, 3 limbs of 32-bits each will be
holding at least 65 bits, but for the purposes of `mpf_t' it's
considered simply to be 64 bits, a nice multiple of the limb size.
File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals
16.4 Raw Output Internals
=========================
`mpz_out_raw' uses the following format.
+------+------------------------+
| size | data bytes |
+------+------------------------+
The size is 4 bytes written most significant byte first, being the
number of subsequent data bytes, or the twos complement negative of
that when a negative integer is represented. The data bytes are the
absolute value of the integer, written most significant byte first.
The most significant data byte is always non-zero, so the output is
the same on all systems, irrespective of limb size.
In GMP 1, leading zero bytes were written to pad the data bytes to a
multiple of the limb size. `mpz_inp_raw' will still accept this, for
compatibility.
The use of "big endian" for both the size and data fields is
deliberate, it makes the data easy to read in a hex dump of a file.
Unfortunately it also means that the limb data must be reversed when
reading or writing, so neither a big endian nor little endian system
can just read and write `_mp_d'.
File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals
16.5 C++ Interface Internals
============================
A system of expression templates is used to ensure something like
`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the
scheme also ensures the precision of the final destination is used for
any temporaries within a statement like `f=w*x+y*z'. These are
important features which a naive implementation cannot provide.
A simplified description of the scheme follows. The true scheme is
complicated by the fact that expressions have different return types.
For detailed information, refer to the source code.
To perform an operation, say, addition, we first define a "function
object" evaluating it,
struct __gmp_binary_plus
{
static void eval(mpf_t f, const mpf_t g, const mpf_t h)
{
mpf_add(f, g, h);
}
};
And an "additive expression" object,
__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
operator+(const mpf_class &f, const mpf_class &g)
{
return __gmp_expr
<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
}
The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
to encapsulate any possible kind of expression into a single template
type. In fact even `mpf_class' etc are `typedef' specializations of
`__gmp_expr'.
Next we define assignment of `__gmp_expr' to `mpf_class'.
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
{
expr.eval(this->get_mpf_t(), this->precision());
return *this;
}
template <class Op>
void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
(mpf_t f, mp_bitcnt_t precision)
{
Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
}
where `expr.val1' and `expr.val2' are references to the expression's
operands (here `expr' is the `__gmp_binary_expr' stored within the
`__gmp_expr').
This way, the expression is actually evaluated only at the time of
assignment, when the required precision (that of `f') is known.
Furthermore the target `mpf_t' is now available, thus we can call
`mpf_add' directly with `f' as the output argument.
Compound expressions are handled by defining operators taking
subexpressions as their arguments, like this:
template <class T, class U>
__gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
{
return __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
(expr1, expr2);
}
And the corresponding specializations of `__gmp_expr::eval':
template <class T, class U, class Op>
void __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
(mpf_t f, mp_bitcnt_t precision)
{
// declare two temporaries
mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
}
The expression is thus recursively evaluated to any level of
complexity and all subexpressions are evaluated to the precision of `f'.
File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top
Appendix A Contributors
***********************
Torbj�rn Granlund wrote the original GMP library and is still the main
developer. Code not explicitly attributed to others, was contributed by
Torbj�rn. Several other individuals and organizations have contributed
GMP. Here is a list in chronological order on first contribution:
Gunnar Sj�din and Hans Riesel helped with mathematical problems in
early versions of the library.
Richard Stallman helped with the interface design and revised the
first version of this manual.
Brian Beuning and Doug Lea helped with testing of early versions of
the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
`mpz_probab_prime_p'.
Paul Zimmermann wrote the REDC-based mpz_powm code, the
Sch�nhage-Strassen FFT multiply code, and the Karatsuba square root
code. He also improved the Toom3 code for GMP 4.2. Paul sparked the
development of GMP 2, with his comparisons between bignum packages.
The ECMNET project Paul is organizing was a driving force behind many
of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth
root code (with Torbj�rn).
Ken Weber (Kent State University, Universidade Federal do Rio Grande
do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact',
`mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil)
grant 301314194-2.
Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
configure. He has also made valuable suggestions and tested numerous
intermediary releases.
Joachim Hollman was involved in the design of the `mpf' interface,
and in the `mpz' design revisions for version 2.
Bennet Yee contributed the initial versions of `mpz_jacobi' and
`mpz_legendre'.
Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
`mpn/m68k/rshift.S' (now in `.asm' form).
Robert Harley of Inria, France and David Seal of ARM, England,
suggested clever improvements for population count. Robert also wrote
highly optimized Karatsuba and 3-way Toom multiplication functions for
GMP 3, and contributed the ARM assembly code.
Torsten Ekedahl of the Mathematical department of Stockholm
University provided significant inspiration during several phases of
the GMP development. His mathematical expertise helped improve several
algorithms.
Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.
Kevin Ryde worked on a large number of things: optimized x86 code,
m4 asm macros, parameter tuning, speed measuring, the configure system,
function inlining, divisibility tests, bit scanning, Jacobi symbols,
Fibonacci and Lucas number functions, printf and scanf functions, perl
interface, demo expression parser, the algorithms chapter in the
manual, `gmpasm-mode.el', and various miscellaneous improvements
elsewhere.
Kent Boortz made the Mac OS 9 port.
Steve Root helped write the optimized alpha 21264 assembly code.
Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
`istream' input routines.
Jason Moxham rewrote `mpz_fac_ui'.
Pedro Gimeno implemented the Mersenne Twister and made other random
number improvements.
Niels M�ller wrote the sub-quadratic GCD, extended GCD and jacobi
code, the quadratic Hensel division code, and (with Torbj�rn) the new
divide and conquer division code for GMP 4.3. Niels also helped
implement the new Toom multiply code for GMP 4.3 and implemented helper
functions to simplify Toom evaluations for GMP 5.0. He wrote the
original version of mpn_mulmod_bnm1, and he is the main author of the
mini-gmp package used for gmp bootstrapping.
Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply
strategy, and found the optimal strategies for evaluation and
interpolation in Toom multiplication.
Marco Bodrato helped implement the new Toom multiply code for GMP
4.3 and implemented most of the new Toom multiply and squaring code for
5.0. He is the main author of the current mpn_mulmod_bnm1 and
mpn_mullo_n. Marco also wrote the functions mpn_invert and
mpn_invertappr. He is the author of the current combinatorial
functions: binomial, factorial, multifactorial, primorial.
David Harvey suggested the internal function `mpn_bdiv_dbm1',
implementing division relevant to Toom multiplication. He also worked
on fast assembly sequences, in particular on a fast AMD64
`mpn_mul_basecase'. He wrote the internal middle product functions
`mpn_mulmid_basecase', `mpn_toom42_mulmid', `mpn_mulmid_n' and related
helper routines.
Martin Boij wrote `mpn_perfect_power_p'.
Marc Glisse improved `gmpxx.h': use fewer temporaries (faster),
specializations of `numeric_limits' and `common_type', C++11 features
(move constructors, explicit bool conversion, UDL), make the conversion
from `mpq_class' to `mpz_class' explicit, optimize operations where one
argument is a small compile-time constant, replace some heap
allocations by stack allocations. He also fixed the eofbit handling of
C++ streams, and removed one division from `mpq/aors.c'.
David S Miller wrote assembly code for SPARC T3 and T4.
Mark Sofroniou cleaned up the types of mul_fft.c, letting it work
for huge operands.
Ulrich Weigand ported GMP to the powerpc64le ABI.
(This list is chronological, not ordered after significance. If you
have contributed to GMP but are not listed above, please tell
<[email protected]> about the omission!)
The development of floating point functions of GNU MP 2, were
supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
project POSSO (POlynomial System SOlving).
The development of GMP 2, 3, and 4.0 was supported in part by the
IDA Center for Computing Sciences.
The development of GMP 4.3, 5.0, and 5.1 was supported in part by
the Swedish Foundation for Strategic Research.
Thanks go to Hans Thorsen for donating an SGI system for the GMP
test system environment.
File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
Appendix B References
*********************
B.1 Books
=========
* Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study
in Analytic Number Theory and Computational Complexity", Wiley,
1998.
* Richard Crandall and Carl Pomerance, "Prime Numbers: A
Computational Perspective", 2nd edition, Springer-Verlag, 2005.
`http://www.math.dartmouth.edu/~carlp/'
* Henri Cohen, "A Course in Computational Algebraic Number Theory",
Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
`http://www.math.u-bordeaux.fr/~cohen/'
* Donald E. Knuth, "The Art of Computer Programming", volume 2,
"Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
`http://www-cs-faculty.stanford.edu/~knuth/taocp.html'
* John D. Lipson, "Elements of Algebra and Algebraic Computing", The
Benjamin Cummings Publishing Company Inc, 1981.
* Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
"Handbook of Applied Cryptography",
`http://www.cacr.math.uwaterloo.ca/hac/'
* Richard M. Stallman and the GCC Developer Community, "Using the
GNU Compiler Collection", Free Software Foundation, 2008,
available online `https://gcc.gnu.org/onlinedocs/', and in the GCC
package `https://ftp.gnu.org/gnu/gcc/'
B.2 Papers
==========
* Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP
Square Root", Journal of Automated Reasoning, volume 29, 2002, pp.
225-252. Also available online as INRIA Research Report 4475,
June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf'
* Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
`http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022'
* Torbj�rn Granlund and Peter L. Montgomery, "Division by Invariant
Integers using Multiplication", in Proceedings of the SIGPLAN
PLDI'94 Conference, June 1994. Also available
`https://gmplib.org/~tege/divcnst-pldi94.pdf'.
* Niels M�ller and Torbj�rn Granlund, "Improved division by invariant
integers", IEEE Transactions on Computers, 11 June 2010.
`https://gmplib.org/~tege/division-paper.pdf'
* Torbj�rn Granlund and Niels M�ller, "Division of integers large and
small", to appear.
* Tudor Jebelean, "An algorithm for exact division", Journal of
Symbolic Computation, volume 15, 1993, pp. 169-180. Research
report version available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'
* Tudor Jebelean, "Exact Division with Karatsuba Complexity -
Extended Abstract", RISC-Linz technical report 96-31,
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz'
* Tudor Jebelean, "Practical Integer Division with Karatsuba
Complexity", ISSAC 97, pp. 339-341. Technical report available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz'
* Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
ISSAC 93, pp. 111-116. Technical report version available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz'
* Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for
Finding the GCD of Long Integers", Journal of Symbolic
Computation, volume 19, 1995, pp. 145-157. Technical report
version also available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz'
* Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
Division", Journal of Symbolic Computation, volume 21, 1996, pp.
441-455. Early technical report version also available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz'
* Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
623-dimensionally equidistributed uniform pseudorandom number
generator", ACM Transactions on Modelling and Computer Simulation,
volume 8, January 1998, pp. 3-30. Available online
`http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz'
(or .pdf)
* R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
Proceedings of the 13th Annual IEEE Symposium on Switching and
Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast
Modular Transforms", Journal of Computer and System Sciences,
volume 8, number 3, June 1974, pp. 366-386.
* Niels M�ller, "On Sch�nhage's algorithm and subquadratic integer
GCD computation", in Mathematics of Computation, volume 77,
January 2008, pp. 589-607.
* Peter L. Montgomery, "Modular Multiplication Without Trial
Division", in Mathematics of Computation, volume 44, number 170,
April 1985.
* Arnold Sch�nhage and Volker Strassen, "Schnelle Multiplikation
grosser Zahlen", Computing 7, 1971, pp. 281-292.
* Kenneth Weber, "The accelerated integer GCD algorithm", ACM
Transactions on Mathematical Software, volume 21, number 1, March
1995, pp. 111-122.
* Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
3805, November 1999,
`http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf'
* Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
Implementations",
`http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'
* Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
Reprinted as "More on Multiplying and Squaring Large Integers",
IEEE Transactions on Computers, volume 43, number 8, August 1994,
pp. 899-908.
File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
Appendix C GNU Free Documentation License
*****************************************
Version 1.3, 3 November 2008
Copyright (C) 2000-2002, 2007, 2008 Free Software Foundation, Inc.
`http://fsf.org/'
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
0. PREAMBLE
The purpose of this License is to make a manual, textbook, or other
functional and useful document "free" in the sense of freedom: to
assure everyone the effective freedom to copy and redistribute it,
with or without modifying it, either commercially or
noncommercially. Secondarily, this License preserves for the
author and publisher a way to get credit for their work, while not
being considered responsible for modifications made by others.
This License is a kind of "copyleft", which means that derivative
works of the document must themselves be free in the same sense.
It complements the GNU General Public License, which is a copyleft
license designed for free software.
We have designed this License in order to use it for manuals for
free software, because free software needs free documentation: a
free program should come with manuals providing the same freedoms
that the software does. But this License is not limited to
software manuals; it can be used for any textual work, regardless
of subject matter or whether it is published as a printed book.
We recommend this License principally for works whose purpose is
instruction or reference.
1. APPLICABILITY AND DEFINITIONS
This License applies to any manual or other work, in any medium,
that contains a notice placed by the copyright holder saying it
can be distributed under the terms of this License. Such a notice
grants a world-wide, royalty-free license, unlimited in duration,
to use that work under the conditions stated herein. The
"Document", below, refers to any such manual or work. Any member
of the public is a licensee, and is addressed as "you". You
accept the license if you copy, modify or distribute the work in a
way requiring permission under copyright law.
A "Modified Version" of the Document means any work containing the
Document or a portion of it, either copied verbatim, or with
modifications and/or translated into another language.
A "Secondary Section" is a named appendix or a front-matter section
of the Document that deals exclusively with the relationship of the
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The "Invariant Sections" are certain Secondary Sections whose
titles are designated, as being those of Invariant Sections, in
the notice that says that the Document is released under this
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Secondary then it is not allowed to be designated as Invariant.
The Document may contain zero Invariant Sections. If the Document
does not identify any Invariant Sections then there are none.
The "Cover Texts" are certain short passages of text that are
listed, as Front-Cover Texts or Back-Cover Texts, in the notice
that says that the Document is released under this License. A
Front-Cover Text may be at most 5 words, and a Back-Cover Text may
be at most 25 words.
A "Transparent" copy of the Document means a machine-readable copy,
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copy that is not "Transparent" is called "Opaque".
Examples of suitable formats for Transparent copies include plain
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can be read and edited only by proprietary word processors, SGML or
XML for which the DTD and/or processing tools are not generally
available, and the machine-generated HTML, PostScript or PDF
produced by some word processors for output purposes only.
The "Title Page" means, for a printed book, the title page itself,
plus such following pages as are needed to hold, legibly, the
material this License requires to appear in the title page. For
works in formats which do not have any title page as such, "Title
Page" means the text near the most prominent appearance of the
work's title, preceding the beginning of the body of the text.
The "publisher" means any person or entity that distributes copies
of the Document to the public.
A section "Entitled XYZ" means a named subunit of the Document
whose title either is precisely XYZ or contains XYZ in parentheses
following text that translates XYZ in another language. (Here XYZ
stands for a specific section name mentioned below, such as
"Acknowledgements", "Dedications", "Endorsements", or "History".)
To "Preserve the Title" of such a section when you modify the
Document means that it remains a section "Entitled XYZ" according
to this definition.
The Document may include Warranty Disclaimers next to the notice
which states that this License applies to the Document. These
Warranty Disclaimers are considered to be included by reference in
this License, but only as regards disclaiming warranties: any other
implication that these Warranty Disclaimers may have is void and
has no effect on the meaning of this License.
2. VERBATIM COPYING
You may copy and distribute the Document in any medium, either
commercially or noncommercially, provided that this License, the
copyright notices, and the license notice saying this License
applies to the Document are reproduced in all copies, and that you
add no other conditions whatsoever to those of this License. You
may not use technical measures to obstruct or control the reading
or further copying of the copies you make or distribute. However,
you may accept compensation in exchange for copies. If you
distribute a large enough number of copies you must also follow
the conditions in section 3.
You may also lend copies, under the same conditions stated above,
and you may publicly display copies.
3. COPYING IN QUANTITY
If you publish printed copies (or copies in media that commonly
have printed covers) of the Document, numbering more than 100, and
the Document's license notice requires Cover Texts, you must
enclose the copies in covers that carry, clearly and legibly, all
these Cover Texts: Front-Cover Texts on the front cover, and
Back-Cover Texts on the back cover. Both covers must also clearly
and legibly identify you as the publisher of these copies. The
front cover must present the full title with all words of the
title equally prominent and visible. You may add other material
on the covers in addition. Copying with changes limited to the
covers, as long as they preserve the title of the Document and
satisfy these conditions, can be treated as verbatim copying in
other respects.
If the required texts for either cover are too voluminous to fit
legibly, you should put the first ones listed (as many as fit
reasonably) on the actual cover, and continue the rest onto
adjacent pages.
If you publish or distribute Opaque copies of the Document
numbering more than 100, you must either include a
machine-readable Transparent copy along with each Opaque copy, or
state in or with each Opaque copy a computer-network location from
which the general network-using public has access to download
using public-standard network protocols a complete Transparent
copy of the Document, free of added material. If you use the
latter option, you must take reasonably prudent steps, when you
begin distribution of Opaque copies in quantity, to ensure that
this Transparent copy will remain thus accessible at the stated
location until at least one year after the last time you
distribute an Opaque copy (directly or through your agents or
retailers) of that edition to the public.
It is requested, but not required, that you contact the authors of
the Document well before redistributing any large number of
copies, to give them a chance to provide you with an updated
version of the Document.
4. MODIFICATIONS
You may copy and distribute a Modified Version of the Document
under the conditions of sections 2 and 3 above, provided that you
release the Modified Version under precisely this License, with
the Modified Version filling the role of the Document, thus
licensing distribution and modification of the Modified Version to
whoever possesses a copy of it. In addition, you must do these
things in the Modified Version:
A. Use in the Title Page (and on the covers, if any) a title
distinct from that of the Document, and from those of
previous versions (which should, if there were any, be listed
in the History section of the Document). You may use the
same title as a previous version if the original publisher of
that version gives permission.
B. List on the Title Page, as authors, one or more persons or
entities responsible for authorship of the modifications in
the Modified Version, together with at least five of the
principal authors of the Document (all of its principal
authors, if it has fewer than five), unless they release you
from this requirement.
C. State on the Title page the name of the publisher of the
Modified Version, as the publisher.
D. Preserve all the copyright notices of the Document.
E. Add an appropriate copyright notice for your modifications
adjacent to the other copyright notices.
F. Include, immediately after the copyright notices, a license
notice giving the public permission to use the Modified
Version under the terms of this License, in the form shown in
the Addendum below.
G. Preserve in that license notice the full lists of Invariant
Sections and required Cover Texts given in the Document's
license notice.
H. Include an unaltered copy of this License.
I. Preserve the section Entitled "History", Preserve its Title,
and add to it an item stating at least the title, year, new
authors, and publisher of the Modified Version as given on
the Title Page. If there is no section Entitled "History" in
the Document, create one stating the title, year, authors,
and publisher of the Document as given on its Title Page,
then add an item describing the Modified Version as stated in
the previous sentence.
J. Preserve the network location, if any, given in the Document
for public access to a Transparent copy of the Document, and
likewise the network locations given in the Document for
previous versions it was based on. These may be placed in
the "History" section. You may omit a network location for a
work that was published at least four years before the
Document itself, or if the original publisher of the version
it refers to gives permission.
K. For any section Entitled "Acknowledgements" or "Dedications",
Preserve the Title of the section, and preserve in the
section all the substance and tone of each of the contributor
acknowledgements and/or dedications given therein.
L. Preserve all the Invariant Sections of the Document,
unaltered in their text and in their titles. Section numbers
or the equivalent are not considered part of the section
titles.
M. Delete any section Entitled "Endorsements". Such a section
may not be included in the Modified Version.
N. Do not retitle any existing section to be Entitled
"Endorsements" or to conflict in title with any Invariant
Section.
O. Preserve any Warranty Disclaimers.
If the Modified Version includes new front-matter sections or
appendices that qualify as Secondary Sections and contain no
material copied from the Document, you may at your option
designate some or all of these sections as invariant. To do this,
add their titles to the list of Invariant Sections in the Modified
Version's license notice. These titles must be distinct from any
other section titles.
You may add a section Entitled "Endorsements", provided it contains
nothing but endorsements of your Modified Version by various
parties--for example, statements of peer review or that the text
has been approved by an organization as the authoritative
definition of a standard.
You may add a passage of up to five words as a Front-Cover Text,
and a passage of up to 25 words as a Back-Cover Text, to the end
of the list of Cover Texts in the Modified Version. Only one
passage of Front-Cover Text and one of Back-Cover Text may be
added by (or through arrangements made by) any one entity. If the
Document already includes a cover text for the same cover,
previously added by you or by arrangement made by the same entity
you are acting on behalf of, you may not add another; but you may
replace the old one, on explicit permission from the previous
publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this
License give permission to use their names for publicity for or to
assert or imply endorsement of any Modified Version.
5. COMBINING DOCUMENTS
You may combine the Document with other documents released under
this License, under the terms defined in section 4 above for
modified versions, provided that you include in the combination
all of the Invariant Sections of all of the original documents,
unmodified, and list them all as Invariant Sections of your
combined work in its license notice, and that you preserve all
their Warranty Disclaimers.
The combined work need only contain one copy of this License, and
multiple identical Invariant Sections may be replaced with a single
copy. If there are multiple Invariant Sections with the same name
but different contents, make the title of each such section unique
by adding at the end of it, in parentheses, the name of the
original author or publisher of that section if known, or else a
unique number. Make the same adjustment to the section titles in
the list of Invariant Sections in the license notice of the
combined work.
In the combination, you must combine any sections Entitled
"History" in the various original documents, forming one section
Entitled "History"; likewise combine any sections Entitled
"Acknowledgements", and any sections Entitled "Dedications". You
must delete all sections Entitled "Endorsements."
6. COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other
documents released under this License, and replace the individual
copies of this License in the various documents with a single copy
that is included in the collection, provided that you follow the
rules of this License for verbatim copying of each of the
documents in all other respects.
You may extract a single document from such a collection, and
distribute it individually under this License, provided you insert
a copy of this License into the extracted document, and follow
this License in all other respects regarding verbatim copying of
that document.
7. AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other
separate and independent documents or works, in or on a volume of
a storage or distribution medium, is called an "aggregate" if the
copyright resulting from the compilation is not used to limit the
legal rights of the compilation's users beyond what the individual
works permit. When the Document is included in an aggregate, this
License does not apply to the other works in the aggregate which
are not themselves derivative works of the Document.
If the Cover Text requirement of section 3 is applicable to these
copies of the Document, then if the Document is less than one half
of the entire aggregate, the Document's Cover Texts may be placed
on covers that bracket the Document within the aggregate, or the
electronic equivalent of covers if the Document is in electronic
form. Otherwise they must appear on printed covers that bracket
the whole aggregate.
8. TRANSLATION
Translation is considered a kind of modification, so you may
distribute translations of the Document under the terms of section
4. Replacing Invariant Sections with translations requires special
permission from their copyright holders, but you may include
translations of some or all Invariant Sections in addition to the
original versions of these Invariant Sections. You may include a
translation of this License, and all the license notices in the
Document, and any Warranty Disclaimers, provided that you also
include the original English version of this License and the
original versions of those notices and disclaimers. In case of a
disagreement between the translation and the original version of
this License or a notice or disclaimer, the original version will
prevail.
If a section in the Document is Entitled "Acknowledgements",
"Dedications", or "History", the requirement (section 4) to
Preserve its Title (section 1) will typically require changing the
actual title.
9. TERMINATION
You may not copy, modify, sublicense, or distribute the Document
except as expressly provided under this License. Any attempt
otherwise to copy, modify, sublicense, or distribute it is void,
and will automatically terminate your rights under this License.
However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
provisionally, unless and until the copyright holder explicitly
and finally terminates your license, and (b) permanently, if the
copyright holder fails to notify you of the violation by some
reasonable means prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from
that copyright holder, and you cure the violation prior to 30 days
after your receipt of the notice.
Termination of your rights under this section does not terminate
the licenses of parties who have received copies or rights from
you under this License. If your rights have been terminated and
not permanently reinstated, receipt of a copy of some or all of
the same material does not give you any rights to use it.
10. FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of
the GNU Free Documentation License from time to time. Such new
versions will be similar in spirit to the present version, but may
differ in detail to address new problems or concerns. See
`https://www.gnu.org/copyleft/'.
Each version of the License is given a distinguishing version
number. If the Document specifies that a particular numbered
version of this License "or any later version" applies to it, you
have the option of following the terms and conditions either of
that specified version or of any later version that has been
published (not as a draft) by the Free Software Foundation. If
the Document does not specify a version number of this License,
you may choose any version ever published (not as a draft) by the
Free Software Foundation. If the Document specifies that a proxy
can decide which future versions of this License can be used, that
proxy's public statement of acceptance of a version permanently
authorizes you to choose that version for the Document.
11. RELICENSING
"Massive Multiauthor Collaboration Site" (or "MMC Site") means any
World Wide Web server that publishes copyrightable works and also
provides prominent facilities for anybody to edit those works. A
public wiki that anybody can edit is an example of such a server.
A "Massive Multiauthor Collaboration" (or "MMC") contained in the
site means any set of copyrightable works thus published on the MMC
site.
"CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
license published by Creative Commons Corporation, a not-for-profit
corporation with a principal place of business in San Francisco,
California, as well as future copyleft versions of that license
published by that same organization.
"Incorporate" means to publish or republish a Document, in whole or
in part, as part of another Document.
An MMC is "eligible for relicensing" if it is licensed under this
License, and if all works that were first published under this
License somewhere other than this MMC, and subsequently
incorporated in whole or in part into the MMC, (1) had no cover
texts or invariant sections, and (2) were thus incorporated prior
to November 1, 2008.
The operator of an MMC Site may republish an MMC contained in the
site under CC-BY-SA on the same site at any time before August 1,
2009, provided the MMC is eligible for relicensing.
ADDENDUM: How to use this License for your documents
====================================================
To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and license
notices just after the title page:
Copyright (C) YEAR YOUR NAME.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
Texts. A copy of the license is included in the section entitled ``GNU
Free Documentation License''.
If you have Invariant Sections, Front-Cover Texts and Back-Cover
Texts, replace the "with...Texts." line with this:
with the Invariant Sections being LIST THEIR TITLES, with
the Front-Cover Texts being LIST, and with the Back-Cover Texts
being LIST.
If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.
If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License, to
permit their use in free software.
File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top
Concept Index
*************
[index ]
* Menu:
* #include: Headers and Libraries.
(line 6)
* --build: Build Options. (line 52)
* --disable-fft: Build Options. (line 313)
* --disable-shared: Build Options. (line 45)
* --disable-static: Build Options. (line 45)
* --enable-alloca: Build Options. (line 274)
* --enable-assert: Build Options. (line 319)
* --enable-cxx: Build Options. (line 226)
* --enable-fat: Build Options. (line 161)
* --enable-profiling <1>: Build Options. (line 323)
* --enable-profiling: Profiling. (line 6)
* --exec-prefix: Build Options. (line 32)
* --host: Build Options. (line 66)
* --prefix: Build Options. (line 32)
* -finstrument-functions: Profiling. (line 66)
* 2exp functions: Efficiency. (line 43)
* 68000: Notes for Particular Systems.
(line 94)
* 80x86: Notes for Particular Systems.
(line 150)
* ABI <1>: Build Options. (line 168)
* ABI: ABI and ISA. (line 6)
* About this manual: Introduction to GMP. (line 57)
* AC_CHECK_LIB: Autoconf. (line 11)
* AIX <1>: Notes for Particular Systems.
(line 7)
* AIX: ABI and ISA. (line 178)
* Algorithms: Algorithms. (line 6)
* alloca: Build Options. (line 274)
* Allocation of memory: Custom Allocation. (line 6)
* AMD64: ABI and ISA. (line 44)
* Anonymous FTP of latest version: Introduction to GMP. (line 37)
* Application Binary Interface: ABI and ISA. (line 6)
* Arithmetic functions <1>: Rational Arithmetic. (line 6)
* Arithmetic functions <2>: Float Arithmetic. (line 6)
* Arithmetic functions: Integer Arithmetic. (line 6)
* ARM: Notes for Particular Systems.
(line 20)
* Assembly cache handling: Assembly Cache Handling.
(line 6)
* Assembly carry propagation: Assembly Carry Propagation.
(line 6)
* Assembly code organisation: Assembly Code Organisation.
(line 6)
* Assembly coding: Assembly Coding. (line 6)
* Assembly floating Point: Assembly Floating Point.
(line 6)
* Assembly loop unrolling: Assembly Loop Unrolling.
(line 6)
* Assembly SIMD: Assembly SIMD Instructions.
(line 6)
* Assembly software pipelining: Assembly Software Pipelining.
(line 6)
* Assembly writing guide: Assembly Writing Guide.
(line 6)
* Assertion checking <1>: Debugging. (line 79)
* Assertion checking: Build Options. (line 319)
* Assignment functions <1>: Assigning Integers. (line 6)
* Assignment functions <2>: Initializing Rationals.
(line 6)
* Assignment functions <3>: Simultaneous Float Init & Assign.
(line 6)
* Assignment functions <4>: Simultaneous Integer Init & Assign.
(line 6)
* Assignment functions: Assigning Floats. (line 6)
* Autoconf: Autoconf. (line 6)
* Basics: GMP Basics. (line 6)
* Binomial coefficient algorithm: Binomial Coefficients Algorithm.
(line 6)
* Binomial coefficient functions: Number Theoretic Functions.
(line 128)
* Binutils strip: Known Build Problems.
(line 28)
* Bit manipulation functions: Integer Logic and Bit Fiddling.
(line 6)
* Bit scanning functions: Integer Logic and Bit Fiddling.
(line 40)
* Bit shift left: Integer Arithmetic. (line 38)
* Bit shift right: Integer Division. (line 62)
* Bits per limb: Useful Macros and Constants.
(line 7)
* Bug reporting: Reporting Bugs. (line 6)
* Build directory: Build Options. (line 19)
* Build notes for binary packaging: Notes for Package Builds.
(line 6)
* Build notes for particular systems: Notes for Particular Systems.
(line 6)
* Build options: Build Options. (line 6)
* Build problems known: Known Build Problems.
(line 6)
* Build system: Build Options. (line 52)
* Building GMP: Installing GMP. (line 6)
* Bus error: Debugging. (line 7)
* C compiler: Build Options. (line 179)
* C++ compiler: Build Options. (line 250)
* C++ interface: C++ Class Interface. (line 6)
* C++ interface internals: C++ Interface Internals.
(line 6)
* C++ istream input: C++ Formatted Input. (line 6)
* C++ ostream output: C++ Formatted Output.
(line 6)
* C++ support: Build Options. (line 226)
* CC: Build Options. (line 179)
* CC_FOR_BUILD: Build Options. (line 213)
* CFLAGS: Build Options. (line 179)
* Checker: Debugging. (line 115)
* checkergcc: Debugging. (line 122)
* Code organisation: Assembly Code Organisation.
(line 6)
* Compaq C++: Notes for Particular Systems.
(line 25)
* Comparison functions <1>: Integer Comparisons. (line 6)
* Comparison functions <2>: Float Comparison. (line 6)
* Comparison functions: Comparing Rationals. (line 6)
* Compatibility with older versions: Compatibility with older versions.
(line 6)
* Conditions for copying GNU MP: Copying. (line 6)
* Configuring GMP: Installing GMP. (line 6)
* Congruence algorithm: Exact Remainder. (line 30)
* Congruence functions: Integer Division. (line 137)
* Constants: Useful Macros and Constants.
(line 6)
* Contributors: Contributors. (line 6)
* Conventions for parameters: Parameter Conventions.
(line 6)
* Conventions for variables: Variable Conventions.
(line 6)
* Conversion functions <1>: Converting Integers. (line 6)
* Conversion functions <2>: Converting Floats. (line 6)
* Conversion functions: Rational Conversions.
(line 6)
* Copying conditions: Copying. (line 6)
* CPPFLAGS: Build Options. (line 205)
* CPU types <1>: Introduction to GMP. (line 24)
* CPU types: Build Options. (line 108)
* Cross compiling: Build Options. (line 66)
* Cryptography functions, low-level: Low-level Functions. (line 495)
* Custom allocation: Custom Allocation. (line 6)
* CXX: Build Options. (line 250)
* CXXFLAGS: Build Options. (line 250)
* Cygwin: Notes for Particular Systems.
(line 57)
* Darwin: Known Build Problems.
(line 51)
* Debugging: Debugging. (line 6)
* Demonstration programs: Demonstration Programs.
(line 6)
* Digits in an integer: Miscellaneous Integer Functions.
(line 23)
* Divisibility algorithm: Exact Remainder. (line 30)
* Divisibility functions: Integer Division. (line 137)
* Divisibility testing: Efficiency. (line 91)
* Division algorithms: Division Algorithms. (line 6)
* Division functions <1>: Rational Arithmetic. (line 24)
* Division functions <2>: Integer Division. (line 6)
* Division functions: Float Arithmetic. (line 33)
* DJGPP <1>: Notes for Particular Systems.
(line 57)
* DJGPP: Known Build Problems.
(line 18)
* DLLs: Notes for Particular Systems.
(line 70)
* DocBook: Build Options. (line 346)
* Documentation formats: Build Options. (line 339)
* Documentation license: GNU Free Documentation License.
(line 6)
* DVI: Build Options. (line 342)
* Efficiency: Efficiency. (line 6)
* Emacs: Emacs. (line 6)
* Exact division functions: Integer Division. (line 112)
* Exact remainder: Exact Remainder. (line 6)
* Example programs: Demonstration Programs.
(line 6)
* Exec prefix: Build Options. (line 32)
* Execution profiling <1>: Build Options. (line 323)
* Execution profiling: Profiling. (line 6)
* Exponentiation functions <1>: Float Arithmetic. (line 41)
* Exponentiation functions: Integer Exponentiation.
(line 6)
* Export: Integer Import and Export.
(line 45)
* Expression parsing demo: Demonstration Programs.
(line 15)
* Extended GCD: Number Theoretic Functions.
(line 49)
* Factor removal functions: Number Theoretic Functions.
(line 108)
* Factorial algorithm: Factorial Algorithm. (line 6)
* Factorial functions: Number Theoretic Functions.
(line 116)
* Factorization demo: Demonstration Programs.
(line 25)
* Fast Fourier Transform: FFT Multiplication. (line 6)
* Fat binary: Build Options. (line 161)
* FFT multiplication <1>: FFT Multiplication. (line 6)
* FFT multiplication: Build Options. (line 313)
* Fibonacci number algorithm: Fibonacci Numbers Algorithm.
(line 6)
* Fibonacci sequence functions: Number Theoretic Functions.
(line 136)
* Float arithmetic functions: Float Arithmetic. (line 6)
* Float assignment functions <1>: Simultaneous Float Init & Assign.
(line 6)
* Float assignment functions: Assigning Floats. (line 6)
* Float comparison functions: Float Comparison. (line 6)
* Float conversion functions: Converting Floats. (line 6)
* Float functions: Floating-point Functions.
(line 6)
* Float initialization functions <1>: Simultaneous Float Init & Assign.
(line 6)
* Float initialization functions: Initializing Floats. (line 6)
* Float input and output functions: I/O of Floats. (line 6)
* Float internals: Float Internals. (line 6)
* Float miscellaneous functions: Miscellaneous Float Functions.
(line 6)
* Float random number functions: Miscellaneous Float Functions.
(line 27)
* Float rounding functions: Miscellaneous Float Functions.
(line 9)
* Float sign tests: Float Comparison. (line 35)
* Floating point mode: Notes for Particular Systems.
(line 34)
* Floating-point functions: Floating-point Functions.
(line 6)
* Floating-point number: Nomenclature and Types.
(line 21)
* fnccheck: Profiling. (line 77)
* Formatted input: Formatted Input. (line 6)
* Formatted output: Formatted Output. (line 6)
* Free Documentation License: GNU Free Documentation License.
(line 6)
* FreeBSD: Notes for Particular Systems.
(line 43)
* frexp <1>: Converting Integers. (line 43)
* frexp: Converting Floats. (line 24)
* FTP of latest version: Introduction to GMP. (line 37)
* Function classes: Function Classes. (line 6)
* FunctionCheck: Profiling. (line 77)
* GCC Checker: Debugging. (line 115)
* GCD algorithms: Greatest Common Divisor Algorithms.
(line 6)
* GCD extended: Number Theoretic Functions.
(line 49)
* GCD functions: Number Theoretic Functions.
(line 32)
* GDB: Debugging. (line 58)
* Generic C: Build Options. (line 152)
* GMP Perl module: Demonstration Programs.
(line 35)
* GMP version number: Useful Macros and Constants.
(line 12)
* gmp.h: Headers and Libraries.
(line 6)
* gmpxx.h: C++ Interface General.
(line 8)
* GNU Debugger: Debugging. (line 58)
* GNU Free Documentation License: GNU Free Documentation License.
(line 6)
* GNU strip: Known Build Problems.
(line 28)
* gprof: Profiling. (line 41)
* Greatest common divisor algorithms: Greatest Common Divisor Algorithms.
(line 6)
* Greatest common divisor functions: Number Theoretic Functions.
(line 32)
* Hardware floating point mode: Notes for Particular Systems.
(line 34)
* Headers: Headers and Libraries.
(line 6)
* Heap problems: Debugging. (line 24)
* Home page: Introduction to GMP. (line 33)
* Host system: Build Options. (line 66)
* HP-UX: ABI and ISA. (line 77)
* HPPA: ABI and ISA. (line 77)
* I/O functions <1>: I/O of Integers. (line 6)
* I/O functions <2>: I/O of Floats. (line 6)
* I/O functions: I/O of Rationals. (line 6)
* i386: Notes for Particular Systems.
(line 150)
* IA-64: ABI and ISA. (line 116)
* Import: Integer Import and Export.
(line 11)
* In-place operations: Efficiency. (line 57)
* Include files: Headers and Libraries.
(line 6)
* info-lookup-symbol: Emacs. (line 6)
* Initialization functions <1>: Initializing Floats. (line 6)
* Initialization functions <2>: Random State Initialization.
(line 6)
* Initialization functions <3>: Simultaneous Float Init & Assign.
(line 6)
* Initialization functions <4>: Simultaneous Integer Init & Assign.
(line 6)
* Initialization functions <5>: Initializing Rationals.
(line 6)
* Initialization functions: Initializing Integers.
(line 6)
* Initializing and clearing: Efficiency. (line 21)
* Input functions <1>: Formatted Input Functions.
(line 6)
* Input functions <2>: I/O of Rationals. (line 6)
* Input functions <3>: I/O of Floats. (line 6)
* Input functions: I/O of Integers. (line 6)
* Install prefix: Build Options. (line 32)
* Installing GMP: Installing GMP. (line 6)
* Instruction Set Architecture: ABI and ISA. (line 6)
* instrument-functions: Profiling. (line 66)
* Integer: Nomenclature and Types.
(line 6)
* Integer arithmetic functions: Integer Arithmetic. (line 6)
* Integer assignment functions <1>: Assigning Integers. (line 6)
* Integer assignment functions: Simultaneous Integer Init & Assign.
(line 6)
* Integer bit manipulation functions: Integer Logic and Bit Fiddling.
(line 6)
* Integer comparison functions: Integer Comparisons. (line 6)
* Integer conversion functions: Converting Integers. (line 6)
* Integer division functions: Integer Division. (line 6)
* Integer exponentiation functions: Integer Exponentiation.
(line 6)
* Integer export: Integer Import and Export.
(line 45)
* Integer functions: Integer Functions. (line 6)
* Integer import: Integer Import and Export.
(line 11)
* Integer initialization functions <1>: Initializing Integers.
(line 6)
* Integer initialization functions: Simultaneous Integer Init & Assign.
(line 6)
* Integer input and output functions: I/O of Integers. (line 6)
* Integer internals: Integer Internals. (line 6)
* Integer logical functions: Integer Logic and Bit Fiddling.
(line 6)
* Integer miscellaneous functions: Miscellaneous Integer Functions.
(line 6)
* Integer random number functions: Integer Random Numbers.
(line 6)
* Integer root functions: Integer Roots. (line 6)
* Integer sign tests: Integer Comparisons. (line 28)
* Integer special functions: Integer Special Functions.
(line 6)
* Interix: Notes for Particular Systems.
(line 65)
* Internals: Internals. (line 6)
* Introduction: Introduction to GMP. (line 6)
* Inverse modulo functions: Number Theoretic Functions.
(line 76)
* IRIX <1>: ABI and ISA. (line 141)
* IRIX: Known Build Problems.
(line 38)
* ISA: ABI and ISA. (line 6)
* istream input: C++ Formatted Input. (line 6)
* Jacobi symbol algorithm: Jacobi Symbol. (line 6)
* Jacobi symbol functions: Number Theoretic Functions.
(line 83)
* Karatsuba multiplication: Karatsuba Multiplication.
(line 6)
* Karatsuba square root algorithm: Square Root Algorithm.
(line 6)
* Kronecker symbol functions: Number Theoretic Functions.
(line 95)
* Language bindings: Language Bindings. (line 6)
* Latest version of GMP: Introduction to GMP. (line 37)
* LCM functions: Number Theoretic Functions.
(line 70)
* Least common multiple functions: Number Theoretic Functions.
(line 70)
* Legendre symbol functions: Number Theoretic Functions.
(line 86)
* libgmp: Headers and Libraries.
(line 22)
* libgmpxx: Headers and Libraries.
(line 27)
* Libraries: Headers and Libraries.
(line 22)
* Libtool: Headers and Libraries.
(line 33)
* Libtool versioning: Notes for Package Builds.
(line 9)
* License conditions: Copying. (line 6)
* Limb: Nomenclature and Types.
(line 31)
* Limb size: Useful Macros and Constants.
(line 7)
* Linear congruential algorithm: Random Number Algorithms.
(line 25)
* Linear congruential random numbers: Random State Initialization.
(line 32)
* Linking: Headers and Libraries.
(line 22)
* Logical functions: Integer Logic and Bit Fiddling.
(line 6)
* Low-level functions: Low-level Functions. (line 6)
* Low-level functions for cryptography: Low-level Functions. (line 495)
* Lucas number algorithm: Lucas Numbers Algorithm.
(line 6)
* Lucas number functions: Number Theoretic Functions.
(line 147)
* MacOS X: Known Build Problems.
(line 51)
* Mailing lists: Introduction to GMP. (line 44)
* Malloc debugger: Debugging. (line 30)
* Malloc problems: Debugging. (line 24)
* Memory allocation: Custom Allocation. (line 6)
* Memory management: Memory Management. (line 6)
* Mersenne twister algorithm: Random Number Algorithms.
(line 17)
* Mersenne twister random numbers: Random State Initialization.
(line 13)
* MINGW: Notes for Particular Systems.
(line 57)
* MIPS: ABI and ISA. (line 141)
* Miscellaneous float functions: Miscellaneous Float Functions.
(line 6)
* Miscellaneous integer functions: Miscellaneous Integer Functions.
(line 6)
* MMX: Notes for Particular Systems.
(line 156)
* Modular inverse functions: Number Theoretic Functions.
(line 76)
* Most significant bit: Miscellaneous Integer Functions.
(line 34)
* MPN_PATH: Build Options. (line 327)
* MS Windows: Notes for Particular Systems.
(line 57)
* MS-DOS: Notes for Particular Systems.
(line 57)
* Multi-threading: Reentrancy. (line 6)
* Multiplication algorithms: Multiplication Algorithms.
(line 6)
* Nails: Low-level Functions. (line 664)
* Native compilation: Build Options. (line 52)
* NetBSD: Notes for Particular Systems.
(line 100)
* NeXT: Known Build Problems.
(line 57)
* Next prime function: Number Theoretic Functions.
(line 25)
* Nomenclature: Nomenclature and Types.
(line 6)
* Non-Unix systems: Build Options. (line 11)
* Nth root algorithm: Nth Root Algorithm. (line 6)
* Number sequences: Efficiency. (line 147)
* Number theoretic functions: Number Theoretic Functions.
(line 6)
* Numerator and denominator: Applying Integer Functions.
(line 6)
* obstack output: Formatted Output Functions.
(line 81)
* OpenBSD: Notes for Particular Systems.
(line 109)
* Optimizing performance: Performance optimization.
(line 6)
* ostream output: C++ Formatted Output.
(line 6)
* Other languages: Language Bindings. (line 6)
* Output functions <1>: Formatted Output Functions.
(line 6)
* Output functions <2>: I/O of Rationals. (line 6)
* Output functions <3>: I/O of Integers. (line 6)
* Output functions: I/O of Floats. (line 6)
* Packaged builds: Notes for Package Builds.
(line 6)
* Parameter conventions: Parameter Conventions.
(line 6)
* Parsing expressions demo: Demonstration Programs.
(line 21)
* Particular systems: Notes for Particular Systems.
(line 6)
* Past GMP versions: Compatibility with older versions.
(line 6)
* PDF: Build Options. (line 342)
* Perfect power algorithm: Perfect Power Algorithm.
(line 6)
* Perfect power functions: Integer Roots. (line 28)
* Perfect square algorithm: Perfect Square Algorithm.
(line 6)
* Perfect square functions: Integer Roots. (line 37)
* perl: Demonstration Programs.
(line 35)
* Perl module: Demonstration Programs.
(line 35)
* Postscript: Build Options. (line 342)
* Power/PowerPC <1>: Known Build Problems.
(line 63)
* Power/PowerPC: Notes for Particular Systems.
(line 115)
* Powering algorithms: Powering Algorithms. (line 6)
* Powering functions <1>: Float Arithmetic. (line 41)
* Powering functions: Integer Exponentiation.
(line 6)
* PowerPC: ABI and ISA. (line 176)
* Precision of floats: Floating-point Functions.
(line 6)
* Precision of hardware floating point: Notes for Particular Systems.
(line 34)
* Prefix: Build Options. (line 32)
* Prime testing algorithms: Prime Testing Algorithm.
(line 6)
* Prime testing functions: Number Theoretic Functions.
(line 7)
* Primorial functions: Number Theoretic Functions.
(line 121)
* printf formatted output: Formatted Output. (line 6)
* Probable prime testing functions: Number Theoretic Functions.
(line 7)
* prof: Profiling. (line 24)
* Profiling: Profiling. (line 6)
* Radix conversion algorithms: Radix Conversion Algorithms.
(line 6)
* Random number algorithms: Random Number Algorithms.
(line 6)
* Random number functions <1>: Integer Random Numbers.
(line 6)
* Random number functions <2>: Random Number Functions.
(line 6)
* Random number functions: Miscellaneous Float Functions.
(line 27)
* Random number seeding: Random State Seeding.
(line 6)
* Random number state: Random State Initialization.
(line 6)
* Random state: Nomenclature and Types.
(line 46)
* Rational arithmetic: Efficiency. (line 113)
* Rational arithmetic functions: Rational Arithmetic. (line 6)
* Rational assignment functions: Initializing Rationals.
(line 6)
* Rational comparison functions: Comparing Rationals. (line 6)
* Rational conversion functions: Rational Conversions.
(line 6)
* Rational initialization functions: Initializing Rationals.
(line 6)
* Rational input and output functions: I/O of Rationals. (line 6)
* Rational internals: Rational Internals. (line 6)
* Rational number: Nomenclature and Types.
(line 16)
* Rational number functions: Rational Number Functions.
(line 6)
* Rational numerator and denominator: Applying Integer Functions.
(line 6)
* Rational sign tests: Comparing Rationals. (line 27)
* Raw output internals: Raw Output Internals.
(line 6)
* Reallocations: Efficiency. (line 30)
* Reentrancy: Reentrancy. (line 6)
* References: References. (line 6)
* Remove factor functions: Number Theoretic Functions.
(line 108)
* Reporting bugs: Reporting Bugs. (line 6)
* Root extraction algorithm: Nth Root Algorithm. (line 6)
* Root extraction algorithms: Root Extraction Algorithms.
(line 6)
* Root extraction functions <1>: Float Arithmetic. (line 37)
* Root extraction functions: Integer Roots. (line 6)
* Root testing functions: Integer Roots. (line 28)
* Rounding functions: Miscellaneous Float Functions.
(line 9)
* Sample programs: Demonstration Programs.
(line 6)
* Scan bit functions: Integer Logic and Bit Fiddling.
(line 40)
* scanf formatted input: Formatted Input. (line 6)
* SCO: Known Build Problems.
(line 38)
* Seeding random numbers: Random State Seeding.
(line 6)
* Segmentation violation: Debugging. (line 7)
* Sequent Symmetry: Known Build Problems.
(line 68)
* Services for Unix: Notes for Particular Systems.
(line 65)
* Shared library versioning: Notes for Package Builds.
(line 9)
* Sign tests <1>: Comparing Rationals. (line 27)
* Sign tests <2>: Float Comparison. (line 35)
* Sign tests: Integer Comparisons. (line 28)
* Size in digits: Miscellaneous Integer Functions.
(line 23)
* Small operands: Efficiency. (line 7)
* Solaris <1>: ABI and ISA. (line 208)
* Solaris: Known Build Problems.
(line 72)
* Sparc: Notes for Particular Systems.
(line 127)
* Sparc V9: ABI and ISA. (line 208)
* Special integer functions: Integer Special Functions.
(line 6)
* Square root algorithm: Square Root Algorithm.
(line 6)
* SSE2: Notes for Particular Systems.
(line 156)
* Stack backtrace: Debugging. (line 50)
* Stack overflow <1>: Debugging. (line 7)
* Stack overflow: Build Options. (line 274)
* Static linking: Efficiency. (line 14)
* stdarg.h: Headers and Libraries.
(line 17)
* stdio.h: Headers and Libraries.
(line 11)
* Stripped libraries: Known Build Problems.
(line 28)
* Sun: ABI and ISA. (line 208)
* SunOS: Notes for Particular Systems.
(line 144)
* Systems: Notes for Particular Systems.
(line 6)
* Temporary memory: Build Options. (line 274)
* Texinfo: Build Options. (line 339)
* Text input/output: Efficiency. (line 153)
* Thread safety: Reentrancy. (line 6)
* Toom multiplication <1>: Higher degree Toom'n'half.
(line 6)
* Toom multiplication <2>: Other Multiplication.
(line 6)
* Toom multiplication <3>: Toom 4-Way Multiplication.
(line 6)
* Toom multiplication: Toom 3-Way Multiplication.
(line 6)
* Types: Nomenclature and Types.
(line 6)
* ui and si functions: Efficiency. (line 50)
* Unbalanced multiplication: Unbalanced Multiplication.
(line 6)
* Upward compatibility: Compatibility with older versions.
(line 6)
* Useful macros and constants: Useful Macros and Constants.
(line 6)
* User-defined precision: Floating-point Functions.
(line 6)
* Valgrind: Debugging. (line 130)
* Variable conventions: Variable Conventions.
(line 6)
* Version number: Useful Macros and Constants.
(line 12)
* Web page: Introduction to GMP. (line 33)
* Windows: Notes for Particular Systems.
(line 70)
* x86: Notes for Particular Systems.
(line 150)
* x87: Notes for Particular Systems.
(line 34)
* XML: Build Options. (line 346)
File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top
Function and Type Index
***********************
[index ]
* Menu:
* __GMP_CC: Useful Macros and Constants.
(line 23)
* __GMP_CFLAGS: Useful Macros and Constants.
(line 24)
* __GNU_MP_VERSION: Useful Macros and Constants.
(line 10)
* __GNU_MP_VERSION_MINOR: Useful Macros and Constants.
(line 11)
* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants.
(line 12)
* _mpz_realloc: Integer Special Functions.
(line 14)
* abs <1>: C++ Interface Rationals.
(line 49)
* abs <2>: C++ Interface Floats.
(line 83)
* abs: C++ Interface Integers.
(line 47)
* ceil: C++ Interface Floats.
(line 84)
* cmp <1>: C++ Interface Floats.
(line 86)
* cmp <2>: C++ Interface Rationals.
(line 51)
* cmp <3>: C++ Interface Floats.
(line 85)
* cmp <4>: C++ Interface Rationals.
(line 50)
* cmp: C++ Interface Integers.
(line 49)
* floor: C++ Interface Floats.
(line 93)
* gmp_asprintf: Formatted Output Functions.
(line 65)
* gmp_errno: Random State Initialization.
(line 55)
* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization.
(line 55)
* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization.
(line 55)
* gmp_fprintf: Formatted Output Functions.
(line 29)
* gmp_fscanf: Formatted Input Functions.
(line 25)
* GMP_LIMB_BITS: Low-level Functions. (line 694)
* GMP_NAIL_BITS: Low-level Functions. (line 692)
* GMP_NAIL_MASK: Low-level Functions. (line 702)
* GMP_NUMB_BITS: Low-level Functions. (line 693)
* GMP_NUMB_MASK: Low-level Functions. (line 703)
* GMP_NUMB_MAX: Low-level Functions. (line 711)
* gmp_obstack_printf: Formatted Output Functions.
(line 79)
* gmp_obstack_vprintf: Formatted Output Functions.
(line 81)
* gmp_printf: Formatted Output Functions.
(line 24)
* GMP_RAND_ALG_DEFAULT: Random State Initialization.
(line 49)
* GMP_RAND_ALG_LC: Random State Initialization.
(line 49)
* gmp_randclass: C++ Interface Random Numbers.
(line 7)
* gmp_randclass::get_f: C++ Interface Random Numbers.
(line 45)
* gmp_randclass::get_z_bits: C++ Interface Random Numbers.
(line 38)
* gmp_randclass::get_z_range: C++ Interface Random Numbers.
(line 42)
* gmp_randclass::gmp_randclass: C++ Interface Random Numbers.
(line 27)
* gmp_randclass::seed: C++ Interface Random Numbers.
(line 33)
* gmp_randclear: Random State Initialization.
(line 62)
* gmp_randinit: Random State Initialization.
(line 47)
* gmp_randinit_default: Random State Initialization.
(line 7)
* gmp_randinit_lc_2exp: Random State Initialization.
(line 18)
* gmp_randinit_lc_2exp_size: Random State Initialization.
(line 32)
* gmp_randinit_mt: Random State Initialization.
(line 13)
* gmp_randinit_set: Random State Initialization.
(line 43)
* gmp_randseed: Random State Seeding.
(line 8)
* gmp_randseed_ui: Random State Seeding.
(line 10)
* gmp_randstate_t: Nomenclature and Types.
(line 46)
* gmp_scanf: Formatted Input Functions.
(line 21)
* gmp_snprintf: Formatted Output Functions.
(line 46)
* gmp_sprintf: Formatted Output Functions.
(line 34)
* gmp_sscanf: Formatted Input Functions.
(line 29)
* gmp_urandomb_ui: Random State Miscellaneous.
(line 8)
* gmp_urandomm_ui: Random State Miscellaneous.
(line 14)
* gmp_vasprintf: Formatted Output Functions.
(line 66)
* gmp_version: Useful Macros and Constants.
(line 18)
* gmp_vfprintf: Formatted Output Functions.
(line 30)
* gmp_vfscanf: Formatted Input Functions.
(line 26)
* gmp_vprintf: Formatted Output Functions.
(line 25)
* gmp_vscanf: Formatted Input Functions.
(line 22)
* gmp_vsnprintf: Formatted Output Functions.
(line 48)
* gmp_vsprintf: Formatted Output Functions.
(line 35)
* gmp_vsscanf: Formatted Input Functions.
(line 31)
* hypot: C++ Interface Floats.
(line 94)
* mp_bitcnt_t: Nomenclature and Types.
(line 42)
* mp_bits_per_limb: Useful Macros and Constants.
(line 7)
* mp_exp_t: Nomenclature and Types.
(line 27)
* mp_get_memory_functions: Custom Allocation. (line 90)
* mp_limb_t: Nomenclature and Types.
(line 31)
* mp_set_memory_functions: Custom Allocation. (line 18)
* mp_size_t: Nomenclature and Types.
(line 37)
* mpf_abs: Float Arithmetic. (line 47)
* mpf_add: Float Arithmetic. (line 7)
* mpf_add_ui: Float Arithmetic. (line 9)
* mpf_ceil: Miscellaneous Float Functions.
(line 7)
* mpf_class: C++ Interface General.
(line 20)
* mpf_class::fits_sint_p: C++ Interface Floats.
(line 87)
* mpf_class::fits_slong_p: C++ Interface Floats.
(line 88)
* mpf_class::fits_sshort_p: C++ Interface Floats.
(line 89)
* mpf_class::fits_uint_p: C++ Interface Floats.
(line 90)
* mpf_class::fits_ulong_p: C++ Interface Floats.
(line 91)
* mpf_class::fits_ushort_p: C++ Interface Floats.
(line 92)
* mpf_class::get_d: C++ Interface Floats.
(line 95)
* mpf_class::get_mpf_t: C++ Interface General.
(line 66)
* mpf_class::get_prec: C++ Interface Floats.
(line 115)
* mpf_class::get_si: C++ Interface Floats.
(line 96)
* mpf_class::get_str: C++ Interface Floats.
(line 98)
* mpf_class::get_ui: C++ Interface Floats.
(line 99)
* mpf_class::mpf_class: C++ Interface Floats.
(line 12)
* mpf_class::operator=: C++ Interface Floats.
(line 60)
* mpf_class::set_prec: C++ Interface Floats.
(line 116)
* mpf_class::set_prec_raw: C++ Interface Floats.
(line 117)
* mpf_class::set_str: C++ Interface Floats.
(line 100)
* mpf_class::swap: C++ Interface Floats.
(line 104)
* mpf_clear: Initializing Floats. (line 37)
* mpf_clears: Initializing Floats. (line 41)
* mpf_cmp: Float Comparison. (line 7)
* mpf_cmp_d: Float Comparison. (line 8)
* mpf_cmp_si: Float Comparison. (line 10)
* mpf_cmp_ui: Float Comparison. (line 9)
* mpf_div: Float Arithmetic. (line 29)
* mpf_div_2exp: Float Arithmetic. (line 55)
* mpf_div_ui: Float Arithmetic. (line 33)
* mpf_eq: Float Comparison. (line 18)
* mpf_fits_sint_p: Miscellaneous Float Functions.
(line 20)
* mpf_fits_slong_p: Miscellaneous Float Functions.
(line 18)
* mpf_fits_sshort_p: Miscellaneous Float Functions.
(line 22)
* mpf_fits_uint_p: Miscellaneous Float Functions.
(line 19)
* mpf_fits_ulong_p: Miscellaneous Float Functions.
(line 17)
* mpf_fits_ushort_p: Miscellaneous Float Functions.
(line 21)
* mpf_floor: Miscellaneous Float Functions.
(line 8)
* mpf_get_d: Converting Floats. (line 7)
* mpf_get_d_2exp: Converting Floats. (line 17)
* mpf_get_default_prec: Initializing Floats. (line 12)
* mpf_get_prec: Initializing Floats. (line 62)
* mpf_get_si: Converting Floats. (line 28)
* mpf_get_str: Converting Floats. (line 38)
* mpf_get_ui: Converting Floats. (line 29)
* mpf_init: Initializing Floats. (line 19)
* mpf_init2: Initializing Floats. (line 26)
* mpf_init_set: Simultaneous Float Init & Assign.
(line 16)
* mpf_init_set_d: Simultaneous Float Init & Assign.
(line 19)
* mpf_init_set_si: Simultaneous Float Init & Assign.
(line 18)
* mpf_init_set_str: Simultaneous Float Init & Assign.
(line 26)
* mpf_init_set_ui: Simultaneous Float Init & Assign.
(line 17)
* mpf_inits: Initializing Floats. (line 31)
* mpf_inp_str: I/O of Floats. (line 39)
* mpf_integer_p: Miscellaneous Float Functions.
(line 14)
* mpf_mul: Float Arithmetic. (line 19)
* mpf_mul_2exp: Float Arithmetic. (line 51)
* mpf_mul_ui: Float Arithmetic. (line 21)
* mpf_neg: Float Arithmetic. (line 44)
* mpf_out_str: I/O of Floats. (line 19)
* mpf_pow_ui: Float Arithmetic. (line 41)
* mpf_random2: Miscellaneous Float Functions.
(line 37)
* mpf_reldiff: Float Comparison. (line 31)
* mpf_set: Assigning Floats. (line 10)
* mpf_set_d: Assigning Floats. (line 13)
* mpf_set_default_prec: Initializing Floats. (line 7)
* mpf_set_prec: Initializing Floats. (line 65)
* mpf_set_prec_raw: Initializing Floats. (line 72)
* mpf_set_q: Assigning Floats. (line 15)
* mpf_set_si: Assigning Floats. (line 12)
* mpf_set_str: Assigning Floats. (line 18)
* mpf_set_ui: Assigning Floats. (line 11)
* mpf_set_z: Assigning Floats. (line 14)
* mpf_sgn: Float Comparison. (line 35)
* mpf_sqrt: Float Arithmetic. (line 36)
* mpf_sqrt_ui: Float Arithmetic. (line 37)
* mpf_sub: Float Arithmetic. (line 12)
* mpf_sub_ui: Float Arithmetic. (line 16)
* mpf_swap: Assigning Floats. (line 52)
* mpf_t: Nomenclature and Types.
(line 21)
* mpf_trunc: Miscellaneous Float Functions.
(line 9)
* mpf_ui_div: Float Arithmetic. (line 31)
* mpf_ui_sub: Float Arithmetic. (line 14)
* mpf_urandomb: Miscellaneous Float Functions.
(line 27)
* mpn_add: Low-level Functions. (line 69)
* mpn_add_1: Low-level Functions. (line 64)
* mpn_add_n: Low-level Functions. (line 54)
* mpn_addmul_1: Low-level Functions. (line 150)
* mpn_and_n: Low-level Functions. (line 437)
* mpn_andn_n: Low-level Functions. (line 452)
* mpn_cmp: Low-level Functions. (line 286)
* mpn_cnd_add_n: Low-level Functions. (line 530)
* mpn_cnd_sub_n: Low-level Functions. (line 532)
* mpn_com: Low-level Functions. (line 477)
* mpn_copyd: Low-level Functions. (line 486)
* mpn_copyi: Low-level Functions. (line 482)
* mpn_divexact_by3: Low-level Functions. (line 231)
* mpn_divexact_by3c: Low-level Functions. (line 233)
* mpn_divmod: Low-level Functions. (line 226)
* mpn_divmod_1: Low-level Functions. (line 210)
* mpn_divrem: Low-level Functions. (line 184)
* mpn_divrem_1: Low-level Functions. (line 208)
* mpn_gcd: Low-level Functions. (line 291)
* mpn_gcd_1: Low-level Functions. (line 301)
* mpn_gcdext: Low-level Functions. (line 307)
* mpn_get_str: Low-level Functions. (line 361)
* mpn_hamdist: Low-level Functions. (line 426)
* mpn_ior_n: Low-level Functions. (line 442)
* mpn_iorn_n: Low-level Functions. (line 457)
* mpn_lshift: Low-level Functions. (line 262)
* mpn_mod_1: Low-level Functions. (line 257)
* mpn_mul: Low-level Functions. (line 116)
* mpn_mul_1: Low-level Functions. (line 135)
* mpn_mul_n: Low-level Functions. (line 105)
* mpn_nand_n: Low-level Functions. (line 462)
* mpn_neg: Low-level Functions. (line 98)
* mpn_nior_n: Low-level Functions. (line 467)
* mpn_perfect_square_p: Low-level Functions. (line 432)
* mpn_popcount: Low-level Functions. (line 422)
* mpn_random: Low-level Functions. (line 411)
* mpn_random2: Low-level Functions. (line 412)
* mpn_rshift: Low-level Functions. (line 274)
* mpn_scan0: Low-level Functions. (line 396)
* mpn_scan1: Low-level Functions. (line 404)
* mpn_sec_add_1: Low-level Functions. (line 543)
* mpn_sec_div_qr: Low-level Functions. (line 613)
* mpn_sec_div_qr_itch: Low-level Functions. (line 614)
* mpn_sec_div_r: Low-level Functions. (line 630)
* mpn_sec_div_r_itch: Low-level Functions. (line 631)
* mpn_sec_invert: Low-level Functions. (line 645)
* mpn_sec_invert_itch: Low-level Functions. (line 646)
* mpn_sec_mul: Low-level Functions. (line 558)
* mpn_sec_mul_itch: Low-level Functions. (line 559)
* mpn_sec_powm: Low-level Functions. (line 588)
* mpn_sec_powm_itch: Low-level Functions. (line 590)
* mpn_sec_sqr: Low-level Functions. (line 573)
* mpn_sec_sqr_itch: Low-level Functions. (line 574)
* mpn_sec_sub_1: Low-level Functions. (line 545)
* mpn_sec_tabselect: Low-level Functions. (line 604)
* mpn_set_str: Low-level Functions. (line 376)
* mpn_sizeinbase: Low-level Functions. (line 354)
* mpn_sqr: Low-level Functions. (line 127)
* mpn_sqrtrem: Low-level Functions. (line 336)
* mpn_sub: Low-level Functions. (line 90)
* mpn_sub_1: Low-level Functions. (line 85)
* mpn_sub_n: Low-level Functions. (line 76)
* mpn_submul_1: Low-level Functions. (line 161)
* mpn_tdiv_qr: Low-level Functions. (line 173)
* mpn_xnor_n: Low-level Functions. (line 472)
* mpn_xor_n: Low-level Functions. (line 447)
* mpn_zero: Low-level Functions. (line 489)
* mpq_abs: Rational Arithmetic. (line 34)
* mpq_add: Rational Arithmetic. (line 8)
* mpq_canonicalize: Rational Number Functions.
(line 22)
* mpq_class: C++ Interface General.
(line 19)
* mpq_class::canonicalize: C++ Interface Rationals.
(line 43)
* mpq_class::get_d: C++ Interface Rationals.
(line 52)
* mpq_class::get_den: C++ Interface Rationals.
(line 66)
* mpq_class::get_den_mpz_t: C++ Interface Rationals.
(line 76)
* mpq_class::get_mpq_t: C++ Interface General.
(line 65)
* mpq_class::get_num: C++ Interface Rationals.
(line 65)
* mpq_class::get_num_mpz_t: C++ Interface Rationals.
(line 75)
* mpq_class::get_str: C++ Interface Rationals.
(line 53)
* mpq_class::mpq_class: C++ Interface Rationals.
(line 12)
* mpq_class::set_str: C++ Interface Rationals.
(line 55)
* mpq_class::swap: C++ Interface Rationals.
(line 57)
* mpq_clear: Initializing Rationals.
(line 16)
* mpq_clears: Initializing Rationals.
(line 20)
* mpq_cmp: Comparing Rationals. (line 7)
* mpq_cmp_si: Comparing Rationals. (line 17)
* mpq_cmp_ui: Comparing Rationals. (line 15)
* mpq_denref: Applying Integer Functions.
(line 18)
* mpq_div: Rational Arithmetic. (line 24)
* mpq_div_2exp: Rational Arithmetic. (line 28)
* mpq_equal: Comparing Rationals. (line 33)
* mpq_get_d: Rational Conversions.
(line 7)
* mpq_get_den: Applying Integer Functions.
(line 24)
* mpq_get_num: Applying Integer Functions.
(line 23)
* mpq_get_str: Rational Conversions.
(line 22)
* mpq_init: Initializing Rationals.
(line 7)
* mpq_inits: Initializing Rationals.
(line 12)
* mpq_inp_str: I/O of Rationals. (line 27)
* mpq_inv: Rational Arithmetic. (line 37)
* mpq_mul: Rational Arithmetic. (line 16)
* mpq_mul_2exp: Rational Arithmetic. (line 20)
* mpq_neg: Rational Arithmetic. (line 31)
* mpq_numref: Applying Integer Functions.
(line 17)
* mpq_out_str: I/O of Rationals. (line 19)
* mpq_set: Initializing Rationals.
(line 24)
* mpq_set_d: Rational Conversions.
(line 17)
* mpq_set_den: Applying Integer Functions.
(line 26)
* mpq_set_f: Rational Conversions.
(line 18)
* mpq_set_num: Applying Integer Functions.
(line 25)
* mpq_set_si: Initializing Rationals.
(line 31)
* mpq_set_str: Initializing Rationals.
(line 36)
* mpq_set_ui: Initializing Rationals.
(line 29)
* mpq_set_z: Initializing Rationals.
(line 25)
* mpq_sgn: Comparing Rationals. (line 27)
* mpq_sub: Rational Arithmetic. (line 12)
* mpq_swap: Initializing Rationals.
(line 56)
* mpq_t: Nomenclature and Types.
(line 16)
* mpz_2fac_ui: Number Theoretic Functions.
(line 114)
* mpz_abs: Integer Arithmetic. (line 45)
* mpz_add: Integer Arithmetic. (line 7)
* mpz_add_ui: Integer Arithmetic. (line 9)
* mpz_addmul: Integer Arithmetic. (line 26)
* mpz_addmul_ui: Integer Arithmetic. (line 28)
* mpz_and: Integer Logic and Bit Fiddling.
(line 11)
* mpz_array_init: Integer Special Functions.
(line 11)
* mpz_bin_ui: Number Theoretic Functions.
(line 126)
* mpz_bin_uiui: Number Theoretic Functions.
(line 128)
* mpz_cdiv_q: Integer Division. (line 13)
* mpz_cdiv_q_2exp: Integer Division. (line 26)
* mpz_cdiv_q_ui: Integer Division. (line 18)
* mpz_cdiv_qr: Integer Division. (line 16)
* mpz_cdiv_qr_ui: Integer Division. (line 22)
* mpz_cdiv_r: Integer Division. (line 14)
* mpz_cdiv_r_2exp: Integer Division. (line 28)
* mpz_cdiv_r_ui: Integer Division. (line 20)
* mpz_cdiv_ui: Integer Division. (line 24)
* mpz_class: C++ Interface General.
(line 18)
* mpz_class::fits_sint_p: C++ Interface Integers.
(line 50)
* mpz_class::fits_slong_p: C++ Interface Integers.
(line 51)
* mpz_class::fits_sshort_p: C++ Interface Integers.
(line 52)
* mpz_class::fits_uint_p: C++ Interface Integers.
(line 53)
* mpz_class::fits_ulong_p: C++ Interface Integers.
(line 54)
* mpz_class::fits_ushort_p: C++ Interface Integers.
(line 55)
* mpz_class::get_d: C++ Interface Integers.
(line 56)
* mpz_class::get_mpz_t: C++ Interface General.
(line 64)
* mpz_class::get_si: C++ Interface Integers.
(line 57)
* mpz_class::get_str: C++ Interface Integers.
(line 58)
* mpz_class::get_ui: C++ Interface Integers.
(line 59)
* mpz_class::mpz_class: C++ Interface Integers.
(line 21)
* mpz_class::set_str: C++ Interface Integers.
(line 60)
* mpz_class::swap: C++ Interface Integers.
(line 64)
* mpz_clear: Initializing Integers.
(line 49)
* mpz_clears: Initializing Integers.
(line 53)
* mpz_clrbit: Integer Logic and Bit Fiddling.
(line 56)
* mpz_cmp: Integer Comparisons. (line 7)
* mpz_cmp_d: Integer Comparisons. (line 8)
* mpz_cmp_si: Integer Comparisons. (line 9)
* mpz_cmp_ui: Integer Comparisons. (line 10)
* mpz_cmpabs: Integer Comparisons. (line 18)
* mpz_cmpabs_d: Integer Comparisons. (line 19)
* mpz_cmpabs_ui: Integer Comparisons. (line 20)
* mpz_com: Integer Logic and Bit Fiddling.
(line 20)
* mpz_combit: Integer Logic and Bit Fiddling.
(line 59)
* mpz_congruent_2exp_p: Integer Division. (line 137)
* mpz_congruent_p: Integer Division. (line 133)
* mpz_congruent_ui_p: Integer Division. (line 135)
* mpz_divexact: Integer Division. (line 110)
* mpz_divexact_ui: Integer Division. (line 112)
* mpz_divisible_2exp_p: Integer Division. (line 123)
* mpz_divisible_p: Integer Division. (line 120)
* mpz_divisible_ui_p: Integer Division. (line 122)
* mpz_even_p: Miscellaneous Integer Functions.
(line 18)
* mpz_export: Integer Import and Export.
(line 45)
* mpz_fac_ui: Number Theoretic Functions.
(line 113)
* mpz_fdiv_q: Integer Division. (line 30)
* mpz_fdiv_q_2exp: Integer Division. (line 43)
* mpz_fdiv_q_ui: Integer Division. (line 35)
* mpz_fdiv_qr: Integer Division. (line 33)
* mpz_fdiv_qr_ui: Integer Division. (line 39)
* mpz_fdiv_r: Integer Division. (line 31)
* mpz_fdiv_r_2exp: Integer Division. (line 45)
* mpz_fdiv_r_ui: Integer Division. (line 37)
* mpz_fdiv_ui: Integer Division. (line 41)
* mpz_fib2_ui: Number Theoretic Functions.
(line 136)
* mpz_fib_ui: Number Theoretic Functions.
(line 134)
* mpz_fits_sint_p: Miscellaneous Integer Functions.
(line 10)
* mpz_fits_slong_p: Miscellaneous Integer Functions.
(line 8)
* mpz_fits_sshort_p: Miscellaneous Integer Functions.
(line 12)
* mpz_fits_uint_p: Miscellaneous Integer Functions.
(line 9)
* mpz_fits_ulong_p: Miscellaneous Integer Functions.
(line 7)
* mpz_fits_ushort_p: Miscellaneous Integer Functions.
(line 11)
* mpz_gcd: Number Theoretic Functions.
(line 32)
* mpz_gcd_ui: Number Theoretic Functions.
(line 39)
* mpz_gcdext: Number Theoretic Functions.
(line 49)
* mpz_get_d: Converting Integers. (line 27)
* mpz_get_d_2exp: Converting Integers. (line 36)
* mpz_get_si: Converting Integers. (line 18)
* mpz_get_str: Converting Integers. (line 47)
* mpz_get_ui: Converting Integers. (line 11)
* mpz_getlimbn: Integer Special Functions.
(line 23)
* mpz_hamdist: Integer Logic and Bit Fiddling.
(line 29)
* mpz_import: Integer Import and Export.
(line 11)
* mpz_init: Initializing Integers.
(line 26)
* mpz_init2: Initializing Integers.
(line 33)
* mpz_init_set: Simultaneous Integer Init & Assign.
(line 27)
* mpz_init_set_d: Simultaneous Integer Init & Assign.
(line 30)
* mpz_init_set_si: Simultaneous Integer Init & Assign.
(line 29)
* mpz_init_set_str: Simultaneous Integer Init & Assign.
(line 35)
* mpz_init_set_ui: Simultaneous Integer Init & Assign.
(line 28)
* mpz_inits: Initializing Integers.
(line 29)
* mpz_inp_raw: I/O of Integers. (line 62)
* mpz_inp_str: I/O of Integers. (line 31)
* mpz_invert: Number Theoretic Functions.
(line 76)
* mpz_ior: Integer Logic and Bit Fiddling.
(line 14)
* mpz_jacobi: Number Theoretic Functions.
(line 83)
* mpz_kronecker: Number Theoretic Functions.
(line 91)
* mpz_kronecker_si: Number Theoretic Functions.
(line 92)
* mpz_kronecker_ui: Number Theoretic Functions.
(line 93)
* mpz_lcm: Number Theoretic Functions.
(line 68)
* mpz_lcm_ui: Number Theoretic Functions.
(line 70)
* mpz_legendre: Number Theoretic Functions.
(line 86)
* mpz_limbs_finish: Integer Special Functions.
(line 48)
* mpz_limbs_modify: Integer Special Functions.
(line 41)
* mpz_limbs_read: Integer Special Functions.
(line 35)
* mpz_limbs_write: Integer Special Functions.
(line 40)
* mpz_lucnum2_ui: Number Theoretic Functions.
(line 147)
* mpz_lucnum_ui: Number Theoretic Functions.
(line 145)
* mpz_mfac_uiui: Number Theoretic Functions.
(line 116)
* mpz_mod: Integer Division. (line 100)
* mpz_mod_ui: Integer Division. (line 102)
* mpz_mul: Integer Arithmetic. (line 19)
* mpz_mul_2exp: Integer Arithmetic. (line 38)
* mpz_mul_si: Integer Arithmetic. (line 20)
* mpz_mul_ui: Integer Arithmetic. (line 22)
* mpz_neg: Integer Arithmetic. (line 42)
* mpz_nextprime: Number Theoretic Functions.
(line 25)
* mpz_odd_p: Miscellaneous Integer Functions.
(line 17)
* mpz_out_raw: I/O of Integers. (line 46)
* mpz_out_str: I/O of Integers. (line 19)
* mpz_perfect_power_p: Integer Roots. (line 28)
* mpz_perfect_square_p: Integer Roots. (line 37)
* mpz_popcount: Integer Logic and Bit Fiddling.
(line 23)
* mpz_pow_ui: Integer Exponentiation.
(line 31)
* mpz_powm: Integer Exponentiation.
(line 8)
* mpz_powm_sec: Integer Exponentiation.
(line 18)
* mpz_powm_ui: Integer Exponentiation.
(line 10)
* mpz_primorial_ui: Number Theoretic Functions.
(line 121)
* mpz_probab_prime_p: Number Theoretic Functions.
(line 7)
* mpz_random: Integer Random Numbers.
(line 42)
* mpz_random2: Integer Random Numbers.
(line 51)
* mpz_realloc2: Initializing Integers.
(line 57)
* mpz_remove: Number Theoretic Functions.
(line 108)
* mpz_roinit_n: Integer Special Functions.
(line 69)
* MPZ_ROINIT_N: Integer Special Functions.
(line 84)
* mpz_root: Integer Roots. (line 8)
* mpz_rootrem: Integer Roots. (line 14)
* mpz_rrandomb: Integer Random Numbers.
(line 31)
* mpz_scan0: Integer Logic and Bit Fiddling.
(line 38)
* mpz_scan1: Integer Logic and Bit Fiddling.
(line 40)
* mpz_set: Assigning Integers. (line 10)
* mpz_set_d: Assigning Integers. (line 13)
* mpz_set_f: Assigning Integers. (line 15)
* mpz_set_q: Assigning Integers. (line 14)
* mpz_set_si: Assigning Integers. (line 12)
* mpz_set_str: Assigning Integers. (line 21)
* mpz_set_ui: Assigning Integers. (line 11)
* mpz_setbit: Integer Logic and Bit Fiddling.
(line 53)
* mpz_sgn: Integer Comparisons. (line 28)
* mpz_si_kronecker: Number Theoretic Functions.
(line 94)
* mpz_size: Integer Special Functions.
(line 31)
* mpz_sizeinbase: Miscellaneous Integer Functions.
(line 23)
* mpz_sqrt: Integer Roots. (line 18)
* mpz_sqrtrem: Integer Roots. (line 21)
* mpz_sub: Integer Arithmetic. (line 12)
* mpz_sub_ui: Integer Arithmetic. (line 14)
* mpz_submul: Integer Arithmetic. (line 32)
* mpz_submul_ui: Integer Arithmetic. (line 34)
* mpz_swap: Assigning Integers. (line 37)
* mpz_t: Nomenclature and Types.
(line 6)
* mpz_tdiv_q: Integer Division. (line 47)
* mpz_tdiv_q_2exp: Integer Division. (line 60)
* mpz_tdiv_q_ui: Integer Division. (line 52)
* mpz_tdiv_qr: Integer Division. (line 50)
* mpz_tdiv_qr_ui: Integer Division. (line 56)
* mpz_tdiv_r: Integer Division. (line 48)
* mpz_tdiv_r_2exp: Integer Division. (line 62)
* mpz_tdiv_r_ui: Integer Division. (line 54)
* mpz_tdiv_ui: Integer Division. (line 58)
* mpz_tstbit: Integer Logic and Bit Fiddling.
(line 62)
* mpz_ui_kronecker: Number Theoretic Functions.
(line 95)
* mpz_ui_pow_ui: Integer Exponentiation.
(line 33)
* mpz_ui_sub: Integer Arithmetic. (line 16)
* mpz_urandomb: Integer Random Numbers.
(line 14)
* mpz_urandomm: Integer Random Numbers.
(line 23)
* mpz_xor: Integer Logic and Bit Fiddling.
(line 17)
* operator"" <1>: C++ Interface Floats.
(line 56)
* operator"" <2>: C++ Interface Integers.
(line 30)
* operator"": C++ Interface Rationals.
(line 38)
* operator%: C++ Interface Integers.
(line 35)
* operator/: C++ Interface Integers.
(line 34)
* operator<<: C++ Formatted Output.
(line 20)
* operator>> <1>: C++ Formatted Input. (line 14)
* operator>>: C++ Interface Rationals.
(line 85)
* sgn <1>: C++ Interface Floats.
(line 102)
* sgn <2>: C++ Interface Integers.
(line 62)
* sgn: C++ Interface Rationals.
(line 56)
* sqrt <1>: C++ Interface Integers.
(line 63)
* sqrt: C++ Interface Floats.
(line 103)
* swap <1>: C++ Interface Floats.
(line 105)
* swap <2>: C++ Interface Integers.
(line 65)
* swap: C++ Interface Rationals.
(line 58)
* trunc: C++ Interface Floats.
(line 106)
Local Variables:
coding: iso-8859-1
End: