ratinterp - Maple Programming Help

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ratinterp

rational interpolation

 Calling Sequence ratinterp(xpoints, ypoints, z, mn)

Parameters

 xpoints - list of independent values, x0,..xk ypoints - list of dependent values, y0,..yk z - variable to be used in polynomial mn - list of numerator and denominator degrees, [m, n]

Description

 • Important: The ratinterp function has been deprecated.  Use the function CurveFitting[RationalInterpolation] instead.  A call to ratinterp automatically generates a call to CurveFitting[RationalInterpolation].
 • The ratinterp function computes a rational function u/v that satisfies

$\mathrm{yi}v\left(\mathrm{xi}\right)-u\left(\mathrm{xi}\right)=0\mathrm{for}i=0\mathrm{..}m+n.$

 with u and v being polynomials in z having degrees at most m and n. In most (but not all) cases, this is equivalent to finding a rational function in z having certain degree constraints that interpolate the points (x0,y0), (x1,y1), ..., (xk,yk). Here $\mathrm{mn}=\left[m,n\right]$. At least m+n+1 points are required, with all the independent points distinct.
 • Sometimes the linearized equation gives u and v satisfying the linearized equation, but with at least one of the xi a common zero. In this case, the resulting rational function does not interpolate the point xi (called an unattainable'' point), since common factors are removed from the rational function. In such cases ratinterp finds the rational function that interpolates a maximum number of points with the given degree constraints. The case where there are unattainable points is recognizable by the fact that the degrees of both the numerator and the denominator are less than the degree constraints.
 • Two procedures are available in the case of exact input: one that avoids singularities in the corresponding table of interpolants by a look-around'' procedure, and another that jumps singularities by a look-ahead'' procedure. Both algorithms are fraction-free. The look-ahead procedure uses a recusion similar to that used for subresultant computation. The second procedure can be specified by using a fifth argument, subresultant. The procedure produces an error message in the case of inexact data.

Examples

Important: The ratinterp function has been deprecated.  Use the function CurveFitting[RationalInterpolation] instead.  A call to ratinterp automatically generates a call to CurveFitting[RationalInterpolation].

 > $\mathrm{xpoints}≔\left[0,1,2,3\right]:$$\mathrm{ypoints}≔\left[0,3,1,3\right]:$
 > $\mathrm{f1}≔\mathrm{ratinterp}\left(\mathrm{xpoints},\mathrm{ypoints},x,\left[1,2\right]\right)$
 ${\mathrm{f1}}{:=}{-}\frac{{3}{}{x}}{{4}{}{{x}}^{{2}}{-}{17}{}{x}{+}{12}}$ (1)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}4\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{normal}\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\phantom{x={\mathrm{xpoints}}_{i}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{f1}}}{x={\mathrm{xpoints}}_{i}}-{\mathrm{ypoints}}_{i}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}$
 ${0}$ (2)
 > $\mathrm{xpoints}≔\left[0,1,2,3,4,-1\right]:$$\mathrm{ypoints}≔\left[0,3,1,3,a,\frac{1}{11}\right]:$
 > $\mathrm{f2}≔\mathrm{ratinterp}\left(\mathrm{xpoints},\mathrm{ypoints},x,\left[2,3\right]\right)$
 ${\mathrm{f2}}{:=}{-}\frac{{3}{}{x}}{{4}{}{{x}}^{{2}}{-}{17}{}{x}{+}{12}}$ (3)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}6\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{normal}\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\phantom{x={\mathrm{xpoints}}_{i}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{f1}}}{x={\mathrm{xpoints}}_{i}}-{\mathrm{ypoints}}_{i}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}$
 ${0}$ (4)

Implies that x=4 is an unattainable point.

References

 Beckermann, B., and Labahn, G. Fraction-free Computation of Matrix GCD's and Rational Interpolants. CS Tech Report, University of Waterloo, 1997.
 See also: SIAM Journal on Matrix Analysis and Applications, 2000.
 Beckermann, B., and Labahn, G. Numeric and Symbolic Computation of Problems Defined by Structured Linear Systems. CS Tech Report, University of Waterloo, 1999.