︠b96482d5-8c81-4c9e-81cc-b56888bfcd6fi︠
%hide
# by Mao Ziyang (mao-ziyang@yeah.net)

var('x')
@interact
def interpolate(pts = input_box(default=[[-2,2],[0,1],[1,2],[2,7]], label='Points'), 
    interpolate_rule=selector(['Lagrange', 'Newton'], nrows=1, label='Interpolation rule'),
    show_basis=checkbox(False, "Show basis functions")):
    n = len(pts)-1    #插值多项式次数
    if interpolate_rule == 'Lagrange':
        l = []   #插值基函数
        for i in range(n+1):
            xvalues = [pt[0] for pt in pts]
            deleted_x = xvalues.pop(i)
            numerator = prod((x - j) for j in xvalues)
            denominator = numerator(x = deleted_x)
            l.append(numerator / denominator)
        p(x) = sum([l[i] * pts[i][1] for i in range(n+1)])
        html(r'''
          <h3>Lagrange Interpolation</h3>
          <div class="math">
          \begin{align*}
          %s = & %s\\
             = & %s
          \end{align*}
          </div>
          ''' % ('L_%s(x)'%n,latex(p(x)),latex(expand(p(x))))
          )
    elif interpolate_rule =='Newton':
        l = []    #插值基函数
        a = [pt[1] for pt in pts]
        xvalues = [pt[0] for pt in pts]
        
        for k in range(1, n+1):
            for i in range(n, k-1, -1):
                a[i] = (a[i] - a[i-1])/(xvalues[i] - xvalues[i-k])
        p(x) = a[0]
        l.append(1)
        for i in range(1, n+1):
            l.append(l[i-1]*(x-xvalues[i-1]))
            p(x) = p(x) + a[i]*l[i]
            
        #另一种计算方法，计算量更小
        #for k in range(1,n+1):
        #    for i in range(k, n+1):
        #        a[i] = (a[i] - a[k-1])/(xvalues[i] - xvalues[k-1])
        #p(x) = a[n]
        #l.append(1)
        #for k in range(1,n+1):
        #    p(x) = a[n-k] + (x - xvalues[n-k]) * p(x)
        #    l.append((x-xvalues[n-k])*l[k-1])
               
        html(r'''
          <h3>Newton Interpolation</h3>
          <div class="math">
          \begin{align*}
          %s = & %s\\
             = & %s
          \end{align*}
          </div>
          ''' % ('N_%s(x)'%n,latex(p(x)),latex(expand(p(x))))
          )
    xvalues = [pt[0] for pt in pts]
    fig = scatter_plot(pts)+plot(p(x), [min(xvalues),max(xvalues)])
    fig.show()
    if show_basis:
        fig2 = plot([l[i] for i in range(n+1)], [min(xvalues), max(xvalues)], fill=true)
        if interpolate_rule == 'Lagrange':
            fig2 += scatter_plot([(pt[0],1) for pt in pts])
        fig2.show()
︡b5873ba1-e22a-47e2-b8d7-7f2b15fbacc6︡{"html": "<div padding=6 id=\"div-interact-2\"> <table width=800px height=20px bgcolor=\"#c5c5c5\"\n                 cellpadding=15><tr><td bgcolor=\"#f9f9f9\" valign=top align=left><table>\n<tr><td colspan=3><table><tr><td align=right><font color=\"black\">Points&nbsp;</font></td><td><input type=\"text\" value=\"[[-2, 2], [0, 1], [1, 2], [2, 7]]\" size=80 onchange=\"interact(2, '_interact_.update(\\'2\\', \\'pts\\', 1, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'2\\');')\"></input></td>\n</tr><tr><td align=right><font color=\"black\">Interpolation rule&nbsp;</font></td><td><table style=\"border:1px solid #dfdfdf; background-color:#efefef;\">\n<tr><td><button style=\"border-style:inset;\" value=\"0\" onclick=\"$('BUTTON', this.parentNode).css('border-style', 'outset'); $(this).css('border-style', 'inset'); interact(2, '_interact_.update(\\'2\\', \\'interpolate_rule\\', 2, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'2\\');')\">Lagrange</button>\n<button style=\"border-style:outset;\" value=\"1\" onclick=\"$('BUTTON', this.parentNode).css('border-style', 'outset'); $(this).css('border-style', 'inset'); interact(2, '_interact_.update(\\'2\\', \\'interpolate_rule\\', 2, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'2\\');')\">Newton</button>\n</td></tr></table></td>\n</tr><tr><td align=right><font color=\"black\">Show basis functions&nbsp;</font></td><td><input type=\"checkbox\"  width=200px onchange=\"interact(2, '_interact_.update(\\'2\\', \\'show_basis\\', 3, _interact_.standard_b64decode(\\''+encode64(this.checked)+'\\'), globals()); _interact_.recompute(\\'2\\');')\"></input></td>\n</tr></table></td></tr>\n<tr><td></td><td style='width: 100%;'><div id=\"cell-interact-2\"><?__SAGE__START>\n        <table border=0 bgcolor=\"white\" width=100%>\n        <tr><td bgcolor=\"white\" align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>\n        <tr><td  align=left valign=top><?__SAGE__HTML></td></tr>\n        </table><?__SAGE__END></div></td><td></td></tr>\n<tr><td colspan=3></td></tr>\n</table></td>\n                 </tr></table></div>\n                 "}︡

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