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@menu
* Introduction to interpol::
* Functions and Variables for interpol::
@end menu
@node Introduction to interpol, Functions and Variables for interpol, interpol, interpol
@section Introduction to interpol
Package @code{interpol} defines the Lagrangian, the linear and the cubic
splines methods for polynomial interpolation.
For comments, bugs or suggestions, please contact me at @var{'mario AT edu DOT xunta DOT es'}.
@opencatbox
@category{Numerical methods} @category{Share packages} @category{Package interpol}
@closecatbox
@node Functions and Variables for interpol, , Introduction to interpol, interpol
@section Functions and Variables for interpol
@deffn {Function} lagrange (@var{points})
@deffnx {Function} lagrange (@var{points}, @var{option})
Computes the polynomial interpolation by the Lagrangian method. Argument @var{points} must be either:
@itemize @bullet
@item
a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
@item
a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
@item
a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
@end itemize
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
With the @var{option} argument it is possible to select the name for the independent variable, which is @code{'x} by default; to define another one, write something like @code{varname='z}.
Note that when working with high degree polynomials, floating point evaluations are unstable.
Examples:
@example
(%i1) load(interpol)$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) lagrange(p);
(x - 7) (x - 6) (x - 3) (x - 1)
(%o3) -------------------------------
35
(x - 8) (x - 6) (x - 3) (x - 1)
- -------------------------------
12
7 (x - 8) (x - 7) (x - 3) (x - 1)
+ ---------------------------------
30
(x - 8) (x - 7) (x - 6) (x - 1)
- -------------------------------
60
(x - 8) (x - 7) (x - 6) (x - 3)
+ -------------------------------
84
(%i4) f(x):=''%;
(x - 7) (x - 6) (x - 3) (x - 1)
(%o4) f(x) := -------------------------------
35
(x - 8) (x - 6) (x - 3) (x - 1)
- -------------------------------
12
7 (x - 8) (x - 7) (x - 3) (x - 1)
+ ---------------------------------
30
(x - 8) (x - 7) (x - 6) (x - 1)
- -------------------------------
60
(x - 8) (x - 7) (x - 6) (x - 3)
+ -------------------------------
84
(%i5) /* Evaluate the polynomial at some points */
expand(map(f,[2.3,5/7,%pi]));
4 3 2
919062 73 %pi 701 %pi 8957 %pi
(%o5) [- 1.567535, ------, ------- - -------- + ---------
84035 420 210 420
5288 %pi 186
- -------- + ---]
105 5
(%i6) %,numer;
(%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
(%i7) load(draw)$ /* load draw package */
(%i8) /* Plot the polynomial together with points */
draw2d(
color = red,
key = "Lagrange polynomial",
explicit(f(x),x,0,10),
point_size = 3,
color = blue,
key = "Sample points",
points(p))$
(%i9) /* Change variable name */
lagrange(p, varname=w);
(w - 7) (w - 6) (w - 3) (w - 1)
(%o9) -------------------------------
35
(w - 8) (w - 6) (w - 3) (w - 1)
- -------------------------------
12
7 (w - 8) (w - 7) (w - 3) (w - 1)
+ ---------------------------------
30
(w - 8) (w - 7) (w - 6) (w - 1)
- -------------------------------
60
(w - 8) (w - 7) (w - 6) (w - 3)
+ -------------------------------
84
@end example
@opencatbox
@category{Package interpol}
@closecatbox
@end deffn
@deffn {Function} charfun2 (@var{x}, @var{a}, @var{b})
Returns @code{true} if number @var{x} belongs to the interval @math{[a, b)}, and @code{false} otherwise.
@opencatbox
@category{Package interpol}
@closecatbox
@end deffn
@deffn {Function} linearinterpol (@var{points})
@deffnx {Function} linearinterpol (@var{points}, @var{option})
Computes the polynomial interpolation by the linear method. Argument @var{points} must be either:
@itemize @bullet
@item
a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
@item
a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
@item
a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
@end itemize
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
With the @var{option} argument it is possible to select the name for the independent variable, which is @code{'x} by default; to define another one, write something like @code{varname='z}.
Examples:
@example
(%i1) load(interpol)$
(%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
(%i3) linearinterpol(p);
13 3 x
(%o3) (-- - ---) charfun2(x, minf, 3)
2 2
+ (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
5 x
+ (--- - 3) charfun2(x, 3, 6)
3
(%i4) f(x):=''%;
13 3 x
(%o4) f(x) := (-- - ---) charfun2(x, minf, 3)
2 2
+ (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
5 x
+ (--- - 3) charfun2(x, 3, 6)
3
(%i5) /* Evaluate the polynomial at some points */
map(f,[7.3,25/7,%pi]);
62 5 %pi
(%o5) [2.3, --, ----- - 3]
21 3
(%i6) %,numer;
(%o6) [2.3, 2.952380952380953, 2.235987755982989]
(%i7) load(draw)$ /* load draw package */
(%i8) /* Plot the polynomial together with points */
draw2d(
color = red,
key = "Linear interpolator",
explicit(f(x),x,-5,20),
point_size = 3,
color = blue,
key = "Sample points",
points(args(p)))$
(%i9) /* Change variable name */
linearinterpol(p, varname='s);
13 3 s
(%o9) (-- - ---) charfun2(s, minf, 3)
2 2
+ (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
5 s
+ (--- - 3) charfun2(s, 3, 6)
3
@end example
@opencatbox
@category{Package interpol}
@closecatbox
@end deffn
@deffn {Function} cspline (@var{points})
@deffnx {Function} cspline (@var{points}, @var{option1}, @var{option2}, ...)
Computes the polynomial interpolation by the cubic splines method. Argument @var{points} must be either:
@itemize @bullet
@item
a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
@item
a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
@item
a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
@end itemize
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
There are three options to fit specific needs:
@itemize @bullet
@item
@code{'d1}, default @code{'unknown}, is the first derivative at @math{x_1}; if it is @code{'unknown}, the second derivative at @math{x_1} is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
@item
@code{'dn}, default @code{'unknown}, is the first derivative at @math{x_n}; if it is @code{'unknown}, the second derivative at @math{x_n} is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
@item
@code{'varname}, default @code{'x}, is the name of the independent variable.
@end itemize
Examples:
@example
(%i1) load(interpol)$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) /* Unknown first derivatives at the extremes
is equivalent to natural cubic splines */
cspline(p);
3 2
1159 x 1159 x 6091 x 8283
(%o3) (------- - ------- - ------ + ----) charfun2(x, minf, 3)
3288 1096 3288 1096
3 2
2587 x 5174 x 494117 x 108928
+ (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
1644 137 1644 137
3 2
4715 x 15209 x 579277 x 199575
+ (------- - -------- + -------- - ------) charfun2(x, 6, 7)
1644 274 1644 274
3 2
3287 x 2223 x 48275 x 9609
+ (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
4932 274 1644 274
(%i4) f(x):=''%$
(%i5) /* Some evaluations */
map(f,[2.3,5/7,%pi]), numer;
(%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
(%i6) load(draw)$ /* load draw package */
(%i7) /* Plotting interpolating function */
draw2d(
color = red,
key = "Cubic splines",
explicit(f(x),x,0,10),
point_size = 3,
color = blue,
key = "Sample points",
points(p))$
(%i8) /* New call, but giving values at the derivatives */
cspline(p,d1=0,dn=0);
3 2
1949 x 11437 x 17027 x 1247
(%o8) (------- - -------- + ------- + ----) charfun2(x, minf, 3)
2256 2256 2256 752
3 2
1547 x 35581 x 68068 x 173546
+ (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
564 564 141 141
3 2
607 x 35147 x 55706 x 38420
+ (------ - -------- + ------- - -----) charfun2(x, 6, 7)
188 564 141 47
3 2
3895 x 1807 x 5146 x 2148
+ (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
5076 188 141 47
(%i8) /* Defining new interpolating function */
g(x):=''%$
(%i9) /* Plotting both functions together */
draw2d(
color = black,
key = "Cubic splines (default)",
explicit(f(x),x,0,10),
color = red,
key = "Cubic splines (d1=0,dn=0)",
explicit(g(x),x,0,10),
point_size = 3,
color = blue,
key = "Sample points",
points(p))$
@end example
@opencatbox
@category{Package interpol}
@closecatbox
@end deffn
@deffn {Function} ratinterpol (@var{points}, @var{numdeg})
@deffnx {Function} ratinterpol (@var{points}, @var{numdeg}, @var{option1}, @var{option2}, ...)
Generates a rational interpolator for data given by @var{points} and the degree of the numerator
being equal to @var{numdeg}; the degree of the denominator is calculated
automatically. Argument @var{points} must be either:
@itemize @bullet
@item
a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
@item
a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
@item
a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
@end itemize
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
There are two options to fit specific needs:
@itemize @bullet
@item
@code{'denterm}, default @code{1}, is the independent term of the polynomial in the denominator.
@item
@code{'varname}, default @code{'x}, is the name of the independent variable.
@end itemize
Examples:
@example
(%i1) load(interpol)$
(%i2) load(draw)$
(%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
(%i4) for k:0 thru length(p)-1 do
draw2d(
explicit(ratinterpol(p,k),x,0,9),
point_size = 3,
points(p),
title = concat("Degree of numerator = ",k),
yrange=[0,10])$
@end example
@opencatbox
@category{Package interpol}
@closecatbox
@end deffn
|
