Video Introduction

(Note: These are notes for the friam.org mail list.)

There are some really useful introductiory videos to Sage by Jose Unpingco. This is a worksheet for the first one.

Note that the workbook allows you to enter HTML wherever you'd like: Move your mouse where a blue line appears and click-shift. Use the help link above for lots more information

1+1

2

for i in range(3):
    print i

0
1
2

# Imports work as usual 
import numpy
dir(numpy)

['ALLOW_THREADS', 'BUFSIZE', 'CLIP', 'ComplexWarning', 'DataSource', 'ERR_CALL', 'ERR_DEFAULT', 'ERR_DEFAULT2', 'ERR_IGNORE', 'ERR_LOG', 'ERR_PRINT', 'ERR_RAISE', 'ERR_WARN', 'FLOATING_POINT_SUPPORT', 'FPE_DIVIDEBYZERO', 'FPE_INVALID', 'FPE_OVERFLOW', 'FPE_UNDERFLOW', 'False_', 'Inf', 'Infinity', 'MAXDIMS', 'MachAr', 'NAN', 'NINF', 'NZERO', 'NaN', 'PINF', 'PZERO', 'PackageLoader', 'RAISE', 'RankWarning', 'SHIFT_DIVIDEBYZERO', 'SHIFT_INVALID', 'SHIFT_OVERFLOW', 'SHIFT_UNDERFLOW', 'ScalarType', 'Tester', 'True_', 'UFUNC_BUFSIZE_DEFAULT', 'UFUNC_PYVALS_NAME', 'WRAP', '__NUMPY_SETUP__', '__all__', '__builtins__', '__config__', '__doc__', '__file__', '__name__', '__package__', '__path__', '__version__', '_import_tools', '_mat', 'abs', 'absolute', 'add', 'add_docstring', 'add_newdoc', 'add_newdocs', 'alen', 'all', 'allclose', 'alltrue', 'alterdot', 'amax', 'amin', 'angle', 'any', 'append', 'apply_along_axis', 'apply_over_axes', 'arange', 'arccos', 'arccosh', 'arcsin', 'arcsinh', 'arctan', 'arctan2', 'arctanh', 'argmax', 'argmin', 'argsort', 'argwhere', 'around', 'array', 'array2string', 'array_equal', 'array_equiv', 'array_repr', 'array_split', 'array_str', 'asanyarray', 'asarray', 'asarray_chkfinite', 'ascontiguousarray', 'asfarray', 'asfortranarray', 'asmatrix', 'asscalar', 'atleast_1d', 'atleast_2d', 'atleast_3d', 'average', 'bartlett', 'base_repr', 'bench', 'binary_repr', 'bincount', 'bitwise_and', 'bitwise_not', 'bitwise_or', 'bitwise_xor', 'blackman', 'bmat', 'bool', 'bool8', 'bool_', 'broadcast', 'broadcast_arrays', 'byte', 'byte_bounds', 'bytes_', 'c_', 'can_cast', 'cast', 'cdouble', 'ceil', 'cfloat', 'char', 'character', 'chararray', 'choose', 'clip', 'clongdouble', 'clongfloat', 'column_stack', 'common_type', 'compare_chararrays', 'compat', 'complex', 'complex128', 'complex256', 'complex64', 'complex_', 'complexfloating', 'compress', 'concatenate', 'conj', 'conjugate', 'convolve', 'copy', 'copysign', 'core', 'corrcoef', 'correlate', 'cos', 'cosh', 'cov', 'cross', 'csingle', 'ctypeslib', 'cumprod', 'cumproduct', 'cumsum', 'datetime_data', 'deg2rad', 'degrees', 'delete', 'deprecate', 'deprecate_with_doc', 'diag', 'diag_indices', 'diag_indices_from', 'diagflat', 'diagonal', 'diff', 'digitize', 'disp', 'divide', 'dot', 'double', 'dsplit', 'dstack', 'dtype', 'e', 'ediff1d', 'emath', 'empty', 'empty_like', 'equal', 'errstate', 'exp', 'exp2', 'expand_dims', 'expm1', 'extract', 'eye', 'fabs', 'fastCopyAndTranspose', 'fft', 'fill_diagonal', 'find_common_type', 'finfo', 'fix', 'flatiter', 'flatnonzero', 'flexible', 'fliplr', 'flipud', 'float', 'float128', 'float32', 'float64', 'float_', 'floating', 'floor', 'floor_divide', 'fmax', 'fmin', 'fmod', 'format_parser', 'frexp', 'frombuffer', 'fromfile', 'fromfunction', 'fromiter', 'frompyfunc', 'fromregex', 'fromstring', 'fv', 'generic', 'genfromtxt', 'get_array_wrap', 'get_include', 'get_numarray_include', 'get_numpy_include', 'get_printoptions', 'getbuffer', 'getbufsize', 'geterr', 'geterrcall', 'geterrobj', 'gradient', 'greater', 'greater_equal', 'hamming', 'hanning', 'histogram', 'histogram2d', 'histogramdd', 'hsplit', 'hstack', 'hypot', 'i0', 'identity', 'iinfo', 'imag', 'in1d', 'index_exp', 'indices', 'inexact', 'inf', 'info', 'infty', 'inner', 'insert', 'int', 'int0', 'int16', 'int32', 'int64', 'int8', 'int_', 'int_asbuffer', 'intc', 'integer', 'interp', 'intersect1d', 'intersect1d_nu', 'intp', 'invert', 'ipmt', 'irr', 'iscomplex', 'iscomplexobj', 'isfinite', 'isfortran', 'isinf', 'isnan', 'isneginf', 'isposinf', 'isreal', 'isrealobj', 'isscalar', 'issctype', 'issubclass_', 'issubdtype', 'issubsctype', 'iterable', 'ix_', 'kaiser', 'kron', 'ldexp', 'left_shift', 'less', 'less_equal', 'lexsort', 'lib', 'linalg', 'linspace', 'little_endian', 'load', 'loads', 'loadtxt', 'log', 'log10', 'log1p', 'log2', 'logaddexp', 'logaddexp2', 'logical_and', 'logical_not', 'logical_or', 'logical_xor', 'logspace', 'long', 'longcomplex', 'longdouble', 'longfloat', 'longlong', 'lookfor', 'ma', 'mafromtxt', 'mask_indices', 'mat', 'math', 'matrix', 'matrixlib', 'max', 'maximum', 'maximum_sctype', 'may_share_memory', 'mean', 'median', 'memmap', 'meshgrid', 'mgrid', 'min', 'minimum', 'mintypecode', 'mirr', 'mod', 'modf', 'msort', 'multiply', 'nan', 'nan_to_num', 'nanargmax', 'nanargmin', 'nanmax', 'nanmin', 'nansum', 'nbytes', 'ndarray', 'ndenumerate', 'ndfromtxt', 'ndim', 'ndindex', 'negative', 'newaxis', 'newbuffer', 'nextafter', 'nonzero', 'not_equal', 'nper', 'npv', 'number', 'obj2sctype', 'object', 'object0', 'object_', 'ogrid', 'ones', 'ones_like', 'outer', 'packbits', 'percentile', 'pi', 'piecewise', 'pkgload', 'place', 'pmt', 'poly', 'poly1d', 'polyadd', 'polyder', 'polydiv', 'polyfit', 'polyint', 'polymul', 'polynomial', 'polysub', 'polyval', 'power', 'ppmt', 'prod', 'product', 'ptp', 'put', 'putmask', 'pv', 'r_', 'rad2deg', 'radians', 'random', 'rank', 'rate', 'ravel', 'real', 'real_if_close', 'rec', 'recarray', 'recfromcsv', 'recfromtxt', 'reciprocal', 'record', 'remainder', 'repeat', 'require', 'reshape', 'resize', 'restoredot', 'right_shift', 'rint', 'roll', 'rollaxis', 'roots', 'rot90', 'round', 'round_', 'row_stack', 's_', 'safe_eval', 'save', 'savetxt', 'savez', 'sctype2char', 'sctypeDict', 'sctypeNA', 'sctypes', 'searchsorted', 'select', 'set_numeric_ops', 'set_printoptions', 'set_string_function', 'setbufsize', 'setdiff1d', 'seterr', 'seterrcall', 'seterrobj', 'setmember1d', 'setxor1d', 'shape', 'short', 'show_config', 'sign', 'signbit', 'signedinteger', 'sin', 'sinc', 'single', 'singlecomplex', 'sinh', 'size', 'sometrue', 'sort', 'sort_complex', 'source', 'spacing', 'split', 'sqrt', 'square', 'squeeze', 'std', 'str', 'str_', 'string0', 'string_', 'subtract', 'sum', 'swapaxes', 'take', 'tan', 'tanh', 'tensordot', 'test', 'testing', 'tile', 'trace', 'transpose', 'trapz', 'tri', 'tril', 'tril_indices', 'tril_indices_from', 'trim_zeros', 'triu', 'triu_indices', 'triu_indices_from', 'true_divide', 'trunc', 'typeDict', 'typeNA', 'typecodes', 'typename', 'ubyte', 'ufunc', 'uint', 'uint0', 'uint16', 'uint32', 'uint64', 'uint8', 'uintc', 'uintp', 'ulonglong', 'unicode', 'unicode0', 'unicode_', 'union1d', 'unique', 'unique1d', 'unpackbits', 'unravel_index', 'unsignedinteger', 'unwrap', 'ushort', 'vander', 'var', 'vdot', 'vectorize', 'version', 'void', 'void0', 'vsplit', 'vstack', 'where', 'who', 'zeros', 'zeros_like']

[sqrt(i) for i in srange(1,10,.1)]

[1.00000000000000, 1.04880884817015, 1.09544511501033, 1.14017542509914, 1.18321595661992, 1.22474487139159, 1.26491106406735, 1.30384048104053, 1.34164078649987, 1.37840487520902, 1.41421356237310, 1.44913767461894, 1.48323969741913, 1.51657508881031, 1.54919333848297, 1.58113883008419, 1.61245154965971, 1.64316767251550, 1.67332005306815, 1.70293863659264, 1.73205080756888, 1.76068168616590, 1.78885438199983, 1.81659021245850, 1.84390889145858, 1.87082869338697, 1.89736659610103, 1.92353840616714, 1.94935886896179, 1.97484176581315, 2.00000000000000, 2.02484567313166, 2.04939015319192, 2.07364413533277, 2.09761769634030, 2.12132034355964, 2.14476105895272, 2.16794833886788, 2.19089023002066, 2.21359436211787, 2.23606797749979, 2.25831795812724, 2.28035085019828, 2.30217288664427, 2.32379000772445, 2.34520787991171, 2.36643191323985, 2.38746727726266, 2.40831891575846, 2.42899156029822, 2.44948974278318, 2.46981780704569, 2.48997991959775, 2.50998007960223, 2.52982212813470, 2.54950975679639, 2.56904651573302, 2.58843582110896, 2.60768096208106, 2.62678510731274, 2.64575131106459, 2.66458251889484, 2.68328157299975, 2.70185121722126, 2.72029410174709, 2.73861278752583, 2.75680975041804, 2.77488738510232, 2.79284800875379, 2.81069386451104, 2.82842712474619, 2.84604989415154, 2.86356421265527, 2.88097205817758, 2.89827534923789, 2.91547594742265, 2.93257565972303, 2.94957624075052, 2.96647939483826, 2.98328677803526, 3.00000000000000, 3.01662062579967, 3.03315017762062, 3.04959013639538, 3.06594194335118, 3.08220700148449, 3.09838667696593, 3.11448230047948, 3.13049516849970, 3.14642654451045]

var('x')

x

diff(sin(x))

cos(x)

b=matrix(2,2,[diff(sin(x),i) for i in range(4)]);b

[ sin(x)  cos(x)]
[-sin(x) -cos(x)]

b.eigenvalues()

[sin(x) - cos(x), 0]

b=matrix(2,2,[diff(sin(x)^2+cos(x),i) for i in range(4)]);b

[                sin(x)^2 + cos(x)          2*sin(x)*cos(x) - sin(x)]
[-2*sin(x)^2 + 2*cos(x)^2 - cos(x)         -8*sin(x)*cos(x) + sin(x)]

b.det()

(2*sin(x)*cos(x) - sin(x))*(2*sin(x)^2 - 2*cos(x)^2 + cos(x)) - (8*sin(x)*cos(x) - sin(x))*(sin(x)^2 + cos(x))

plot(b.det(),-3,3)

find_root(b.det(),1,3)

1.9106332362490221

plot(sin,0,2)+plot(cos,0,2,color='red')