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@menu
* Introduction to interpol::
* Definitions for interpol::
@end menu
@node Introduction to interpol, Definitions for interpol, interpol, interpol
@section Introduction to interpol
Package @code{interpol} defines de 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'}.
@node Definitions for interpol, , Introduction to interpol, interpol
@section Definitions 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}.
Examples:
@example
(%i1) load("interpol")$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) lagrange(p);
4 3 2
73 x - 1402 x + 8957 x - 21152 x + 15624
(%o3) -------------------------------------------
420
(%i4) f(x):=''%;
4 3 2
73 x - 1402 x + 8957 x - 21152 x + 15624
(%o4) f(x) := -------------------------------------------
420
(%i5) /* Evaluate the polynomial at some points */
map(f,[2.3,5/7,%pi]);
919062
(%o5) [- 1.567535000000005, ------,
84035
4 3 2
73 %pi - 1402 %pi + 8957 %pi - 21152 %pi + 15624
---------------------------------------------------]
420
(%i6) %,numer;
(%o6) [- 1.567535000000005, 10.9366573451538,
2.89319655125692]
(%i7) /* Plot the polynomial together with points */
plot2d([f(x),[discrete,p]],[x,0,10],
[gnuplot_curve_styles,
["with lines","with points pointsize 3"]])$
(%i8) /* Change variable name */
lagrange(p, varname=w);
4 3 2
73 w - 1402 w + 8957 w - 21152 w + 15624
(%o8) -------------------------------------------
420
@end example
@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.
@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);
(%o3) - ((9 x - 39) charfun2(x, minf, 3)
+ (30 - 6 x) charfun2(x, 7, inf)
+ (30 x - 222) charfun2(x, 6, 7)
+ (18 - 10 x) charfun2(x, 3, 6))/6
(%i4) f(x):=''%;
(%o4) f(x) := - ((9 x - 39) charfun2(x, minf, 3)
+ (30 - 6 x) charfun2(x, 7, inf)
+ (30 x - 222) charfun2(x, 6, 7)
+ (18 - 10 x) charfun2(x, 3, 6))/6
(%i5) /* Evaluate the polynomial at some points */
map(f,[7.3,25/7,%pi]);
62 18 - 10 %pi
(%o5) [2.3, --, - -----------]
21 6
(%i6) %,numer;
(%o6) [2.3, 2.952380952380953, 2.235987755982988]
(%i7) /* Plot the polynomial together with points */
plot2d(['(f(x)),[discrete,args(p)]],[x,-5,20],
[gnuplot_curve_styles,
["with lines","with points pointsize 3"]])$
(%i8) /* Change variable name */
linearinterpol(p, varname='s);
(%o8) - ((9 s - 39) charfun2(s, minf, 3)
+ (30 - 6 s) charfun2(s, 7, inf)
+ (30 s - 222) charfun2(s, 6, 7)
+ (18 - 10 s) charfun2(s, 3, 6))/6
@end example
@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
(%o3) ((3477 x - 10431 x - 18273 x + 74547)
3 2
charfun2(x, minf, 3) + (- 15522 x + 372528 x - 2964702 x
+ 7842816) charfun2(x, 7, inf)
3 2
+ (28290 x - 547524 x + 3475662 x - 7184700)
3 2
charfun2(x, 6, 7) + (- 6574 x + 80028 x - 289650 x
+ 345924) charfun2(x, 3, 6))/9864
(%i4) f(x):=''%$
(%i5) /* Some evaluations */
map(f,[2.3,5/7,%pi]), numer;
(%o5) [1.991460766423358, 5.823200187269904,
2.227405312429501]
(%i6) /* Plotting interpolating function */
plot2d(['(f(x)),[discrete,p]],[x,0,10],
[gnuplot_curve_styles,
["with lines","with points pointsize 3"]])$
(%i7) /* New call, but giving values at the derivatives */
cspline(p,d1=0,dn=0);
3 2
(%o7) ((17541 x - 102933 x + 153243 x + 33669)
3 2
charfun2(x, minf, 3) + (- 55692 x + 1280916 x - 9801792 x
+ 24990624) charfun2(x, 7, inf)
3 2
+ (65556 x - 1265292 x + 8021664 x - 16597440)
3 2
charfun2(x, 6, 7) + (- 15580 x + 195156 x - 741024 x
+ 927936) charfun2(x, 3, 6))/20304
(%i8) /* Defining new interpolating function */
g(x):=''%$
(%i9) /* Plotting both functions together */
plot2d(['(f(x)),'(g(x)),[discrete,p]],[x,0,10],
[gnuplot_curve_styles,
["with lines","with lines","with points pointsize 3"]])$
@end example
@end deffn
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