interpol
charfun2 (x, a, b) — Function
The characteristic or indicator function on the half-open interval $[a, b)$, that is, including a and excluding b.
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.
maxima
(%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.
maxima
(%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
See also: charfun.
cspline (points) — Function
Computes the polynomial interpolation by the cubic splines method. Argument points must be either:
a two column matrix, p:matrix([2,4],[5,6],[9,3]),
a list of pairs, p: [[2,4],[5,6],[9,3]],
a list of numbers, 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:
maxima
(%i1) load("interpol")$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) cspline(p);
3 2
1159 x 1159 x 6091 x 8283
(%o3) (------- - ------- - ------ + ----) charfun(x < 3)
3288 1096 3288 1096
+ charfun((6 <= x) and (x < 7))
3 2
4715 x 15209 x 579277 x 199575
(------- - -------- + -------- - ------)
1644 274 1644 274
+ charfun((3 <= x) and (x < 6))
3 2
3287 x 2223 x 48275 x 9609
(- ------- + ------- - ------- + ----)
4932 274 1644 274
3 2
2587 x 5174 x 494117 x 108928
+ charfun(7 <= x) (- ------- + ------- - -------- + ------)
1644 137 1644 137
(%i4) define (f(x),%)$
(%i5) float (map (f, [2.3,5/7,%pi]));
(%o5) [1.9914607664233568, 5.823200187269903, 2.2274053124295072]
(%i6) plot2d ([f,[discrete,p]], [x,0,10], [y,-8,13], [style,lines,points],
[legend,"Cubic splines","Points"])$
(%i7) cspline(p,d1=0,dn=0);
3 2
1949 x 11437 x 17027 x 1247
(%o7) (------- - -------- + ------- + ----) charfun(x < 3)
2256 2256 2256 752
+ charfun((6 <= x) and (x < 7))
3 2
607 x 35147 x 55706 x 38420
(------ - -------- + ------- - -----)
188 564 141 47
+ charfun((3 <= x) and (x < 6))
3 2
3895 x 1807 x 5146 x 2148
(- ------- + ------- - ------ + ----)
5076 188 141 47
3 2
1547 x 35581 x 68068 x 173546
+ charfun(7 <= x) (- ------- + -------- - ------- + ------)
564 564 141 141
(%i8) define (g(x),%)$
(%i9) plot2d ([f,g,[discrete,p]], [x,0,10], [y,-15,15], [style,lines,lines,points],
[legend,"Cubic splines","Splines with derivatives","Points"])$
See also: lagrange, linearinterpol, ratinterpol.
lagrange (points) — Function
Computes the polynomial interpolation by the Lagrangian method. Argument points must be either:
a two column matrix, p:matrix([2,4],[5,6],[9,3]),
a list of pairs, p: [[2,4],[5,6],[9,3]],
a list of numbers, 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:
maxima
(%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) define(f(x),%)$
(%i5) 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) plot2d ([f,[discrete,p]], [x,0,10], [style,lines,points],
[legend,"Polynomial","Points"])$
(%i7) lagrange(p, varname=w);
(w - 7) (w - 6) (w - 3) (w - 1)
(%o7) -------------------------------
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
See also: linearinterpol, cspline, ratinterpol.
linearinterpol (points) — Function
Computes the polynomial interpolation by the linear method. Argument points must be either:
a two column matrix, p:matrix([2,4],[5,6],[9,3]),
a list of pairs, p: [[2,4],[5,6],[9,3]],
a list of numbers, 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:
maxima
(%i1) load ("interpol") $
(%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
(%i3) linearinterpol(p);
13 3 x
(%o3) (-- - ---) charfun(x < 3) + charfun((3 <= x) and (x < 6))
2 2
5 x
(--- - 3) + charfun(7 <= x) (x - 5)
3
+ charfun((6 <= x) and (x < 7)) (37 - 5 x)
(%i4) define(f(x),%)$
(%i5) map(f, [7.3,25/7,%pi]);
62 5 %pi
(%o5) [2.3, --, ----- - 3]
21 3
(%i6) float(%);
(%o6) [2.3, 2.9523809523809526, 2.235987755982989]
(%i7) plot2d ([f,[discrete,args(p)]], [x,-5,20], [style,lines,points],
[legend,"Interpolation line","Points"])$
(%i8) lagrange(p, varname=w);
3 (w - 7) (w - 6) (w - 3) (w - 1)
(%o8) ---------------------------------
70
(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
See also: lagrange, cspline, ratinterpol.
ratinterpol (points, numdeg) — Function
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:
a two column matrix, p:matrix([2,4],[5,6],[9,3]),
a list of pairs, p: [[2,4],[5,6],[9,3]],
a list of numbers, 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:
maxima
(%i1) load("interpol")$
(%i2) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
(%i3) for k:0 thru length(p)-1 do
(%i4) plot2d([ratinterpol(p,k),[discrete,p]], [x,0,9], [y,0,10], [style,lines,points],
[title,concat("Degree of numerator = ",k)], nolegend, gnuplot)$
See also: lagrange, linearinterpol, cspline, minpack_lsquares, Package-lbfgs.