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Next: Functions and Variables for interpol, Previous: Package interpol, Up: Package interpol [Contents][Index]
Package interpol defines the Lagrangian, the linear and the cubic
splines methods for polynomial interpolation.
For comments, bugs or suggestions, please contact me at ’mario AT edu DOT xunta DOT es’.
Previous: Introduction to interpol, Up: Package interpol [Contents][Index]
Computes the polynomial interpolation by the Lagrangian method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
With the option argument it is possible to select the name for the independent variable, which is 'x by default; to define another one, write something like varname='z.
Note that when working with high degree polynomials, floating point evaluations are unstable.
See also linearinterpol, cspline, and ratinterpol.
Examples:
(%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
The characteristic or indicator function on the half-open interval [a, b), that is, including a and excluding b.
When x >= a and x < b evaluates to true or false,
charfun2 returns 1 or 0, respectively.
Otherwise, charfun2 returns a partially-evaluated result in terms of charfun.
Package interpol loads this function.
See also charfun.
Examples:
When x >= a and x < b evaluates to true or false,
charfun2 returns 1 or 0, respectively.
(%i1) load ("interpol") $
(%i2) charfun2 (5, 0, 100);
(%o2) 1
(%i3) charfun2 (-5, 0, 100);
(%o3) 0
Otherwise, charfun2 returns a partially-evaluated result in terms of charfun.
(%i1) load ("interpol") $
(%i2) charfun2 (t, 0, 100);
(%o2) charfun((0 <= t) and (t < 100))
(%i3) charfun2 (5, u, v);
(%o3) charfun((u <= 5) and (5 < v))
(%i4) assume (v > u, u > 5);
(%o4) [v > u, u > 5]
(%i5) charfun2 (5, u, v);
(%o5) 0
Computes the polynomial interpolation by the linear method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
With the option argument it is possible to select the name for the independent variable, which is 'x by default; to define another one, write something like varname='z.
See also lagrange, cspline, and ratinterpol.
Examples:
(%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
Computes the polynomial interpolation by the cubic splines method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
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:
'd1, default 'unknown, is the first derivative at x_1; if it is 'unknown, the second derivative at 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.
'dn, default 'unknown, is the first derivative at x_n; if it is 'unknown, the second derivative at 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.
'varname, default 'x, is the name of the independent variable.
See also lagrange, linearinterpol, and ratinterpol.
Examples:
(%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))$
Generates a rational interpolator for data given by points and the degree of the numerator being equal to numdeg; the degree of the denominator is calculated automatically. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
There is one option to fit specific needs:
'varname, default 'x, is the name of the independent variable.
See also lagrange, linearinterpol, cspline, minpack_lsquares, and Package lbfgs
Examples:
(%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])$
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