PLplot 5.15.0
dspline.c
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1//
2// Copyright (C) 2009 Alan W. Irwin
3//
4// This file is part of PLplot.
5//
6// PLplot is free software; you can redistribute it and/or modify
7// it under the terms of the GNU Library General Public License as published
8// by the Free Software Foundation; either version 2 of the License, or
9// (at your option) any later version.
10//
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12// but WITHOUT ANY WARRANTY; without even the implied warranty of
13// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14// GNU Library General Public License for more details.
15//
16// You should have received a copy of the GNU Library General Public License
17// along with PLplot; if not, write to the Free Software
18// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
19//
20// Provenance: This code was originally developed under the GPL as part of
21// the FreeEOS project (revision 121). This code has been converted from
22// Fortran to C with the aid of f2c and relicensed for PLplot under the LGPL
23// with the permission of the FreeEOS copyright holder (Alan W. Irwin).
24//
25
26#include "dspline.h"
27
28int dspline( double *x, double *y, int n,
29 int if1, double cond1, int ifn, double condn, double *y2 )
30{
31 int i__1, i__, k;
32 double p, u[2000], qn, un, sig;
33
34// input parameters:
35// x(n) are the spline knot points
36// y(n) are the function values at the knot points
37// if1 = 1 specifies cond1 is the first derivative at the
38// first knot point.
39// if1 = 2 specifies cond1 is the second derivative at the
40// first knot point.
41// ifn = 1 specifies condn is the first derivative at the
42// nth knot point.
43// ifn = 2 specifies condn is the second derivative at the
44// nth knot point.
45// output values:
46// y2(n) is the second derivative of the spline evaluated at
47// the knot points.
48 // Parameter adjustments
49 --y2;
50 --y;
51 --x;
52
53 // Function Body
54 if ( n > 2000 )
55 {
56 return 1;
57 }
58// y2(i) = u(i) + d(i)*y2(i+1), where
59// d(i) is temporarily stored in y2(i) (see below).
60 if ( if1 == 2 )
61 {
62// cond1 is second derivative at first point.
63// these two values assure that for above equation with d(i) temporarily
64// stored in y2(i)
65 y2[1] = 0.;
66 u[0] = cond1;
67 }
68 else if ( if1 == 1 )
69 {
70// cond1 is first derivative at first point.
71// special case (Press et al 3.3.5 with A = 1, and B=0)
72// of equations below where
73// a_j = 0
74// b_j = -(x_j+1 - x_j)/3
75// c_j = -(x_j+1 - x_j)/6
76// r_j = cond1 - (y_j+1 - y_j)/(x_j+1 - x_j)
77// u(i) = r(i)/b(i)
78// d(i) = -c(i)/b(i)
79// N.B. d(i) is temporarily stored in y2.
80 y2[1] = -.5;
81 u[0] = 3. / ( x[2] - x[1] ) * ( ( y[2] - y[1] ) / ( x[2] - x[1] ) - cond1 );
82 }
83 else
84 {
85 return 2;
86 }
87// if original tri-diagonal system is characterized as
88// a_j y2_j-1 + b_j y2_j + c_j y2_j+1 = r_j
89// Then from Press et al. 3.3.7, we have the unscaled result:
90// a_j = (x_j - x_j-1)/6
91// b_j = (x_j+1 - x_j-1)/3
92// c_j = (x_j+1 - x_j)/6
93// r_j = (y_j+1 - y_j)/(x_j+1 - x_j) - (y_j - y_j-1)/(x_j - x_j-1)
94// In practice, all these values are divided through by b_j/2 to scale
95// them, and from now on we will use these scaled values.
96
97// forward elimination step: assume y2(i-1) = u(i-1) + d(i-1)*y2(i).
98// When this is substituted into above tridiagonal equation ==>
99// y2(i) = u(i) + d(i)*y2(i+1), where
100// u(i) = [r(i) - a(i) u(i-1)]/[b(i) + a(i) d(i-1)]
101// d(i) = -c(i)/[b(i) + a(i) d(i-1)]
102// N.B. d(i) is temporarily stored in y2.
103 i__1 = n - 1;
104 for ( i__ = 2; i__ <= i__1; ++i__ )
105 {
106// sig is scaled a(i)
107 sig = ( x[i__] - x[i__ - 1] ) / ( x[i__ + 1] - x[i__ - 1] );
108// p is denominator = scaled a(i) d(i-1) + scaled b(i), where scaled
109// b(i) is 2.
110 p = sig * y2[i__ - 1] + 2.;
111// propagate d(i) equation above. Note sig-1 = -c(i)
112 y2[i__] = ( sig - 1. ) / p;
113// propagate scaled u(i) equation above
114 u[i__ - 1] = ( ( ( y[i__ + 1] - y[i__] ) / ( x[i__ + 1] - x[i__] ) - ( y[i__]
115 - y[i__ - 1] ) / ( x[i__] - x[i__ - 1] ) ) * 6. / ( x[i__ + 1] -
116 x[i__ - 1] ) - sig * u[i__ - 2] ) / p;
117 }
118 if ( ifn == 2 )
119 {
120// condn is second derivative at nth point.
121// These two values assure that in the equation below.
122 qn = 0.;
123 un = condn;
124 }
125 else if ( ifn == 1 )
126 {
127// specify condn is first derivative at nth point.
128// special case (Press et al 3.3.5 with A = 0, and B=1)
129// implies a_n y2(n-1) + b_n y2(n) = r_n, where
130// a_n = (x_n - x_n-1)/6
131// b_n = (x_n - x_n-1)/3
132// r_n = cond1 - (y_n - y_n-1)/(x_n - x_n-1)
133// use same propagation equation as above, only with c_n = 0
134// ==> d_n = 0 ==> y2(n) = u(n) =>
135// y(n) = [r(n) - a(n) u(n-1)]/[b(n) + a(n) d(n-1)]
136// qn is scaled a_n
137 qn = .5;
138// un is scaled r_n (N.B. un is not u(n))! Sorry for the mixed notation.
139 un = 3. / ( x[n] - x[n - 1] ) * ( condn - ( y[n] - y[n - 1] ) / ( x[n]
140 - x[n - 1] ) );
141 }
142 else
143 {
144 return 3;
145 }
146// N.B. d(i) is temporarily stored in y2, and everything is
147// scaled by b_n.
148// qn is scaled a_n, 1.d0 is scaled b_n, and un is scaled r_n.
149 y2[n] = ( un - qn * u[n - 2] ) / ( qn * y2[n - 1] + 1. );
150// back substitution.
151 for ( k = n - 1; k >= 1; --k )
152 {
153 y2[k] = y2[k] * y2[k + 1] + u[k - 1];
154 }
155 return 0;
156}
157
int dspline(double *x, double *y, int n, int if1, double cond1, int ifn, double condn, double *y2)
Definition: dspline.c:28