PLplot  5.10.0
dspline.c
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00001 //
00002 // Copyright (C) 2009 Alan W. Irwin
00003 //
00004 // This file is part of PLplot.
00005 //
00006 // PLplot is free software; you can redistribute it and/or modify
00007 // it under the terms of the GNU Library General Public License as published
00008 // by the Free Software Foundation; either version 2 of the License, or
00009 // (at your option) any later version.
00010 //
00011 // PLplot is distributed in the hope that it will be useful,
00012 // but WITHOUT ANY WARRANTY; without even the implied warranty of
00013 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
00014 // GNU Library General Public License for more details.
00015 //
00016 // You should have received a copy of the GNU Library General Public License
00017 // along with PLplot; if not, write to the Free Software
00018 // Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
00019 //
00020 // Provenance: This code was originally developed under the GPL as part of
00021 // the FreeEOS project (revision 121).  This code has been converted from
00022 // Fortran to C with the aid of f2c and relicensed for PLplot under the LGPL
00023 // with the permission of the FreeEOS copyright holder (Alan W. Irwin).
00024 //
00025 
00026 #include "dspline.h"
00027 
00028 int dspline( double *x, double *y, int n,
00029              int if1, double cond1, int ifn, double condn, double *y2 )
00030 {
00031     int    i__1, i__, k;
00032     double p, u[2000], qn, un, sig;
00033 
00034 //      input parameters:
00035 //      x(n) are the spline knot points
00036 //      y(n) are the function values at the knot points
00037 //      if1 = 1 specifies cond1 is the first derivative at the
00038 //        first knot point.
00039 //      if1 = 2 specifies cond1 is the second derivative at the
00040 //        first knot point.
00041 //      ifn = 1 specifies condn is the first derivative at the
00042 //        nth knot point.
00043 //      ifn = 2 specifies condn is the second derivative at the
00044 //        nth knot point.
00045 //      output values:
00046 //      y2(n) is the second derivative of the spline evaluated at
00047 //        the knot points.
00048     // Parameter adjustments
00049     --y2;
00050     --y;
00051     --x;
00052 
00053     // Function Body
00054     if ( n > 2000 )
00055     {
00056         return 1;
00057     }
00058 //      y2(i) = u(i) + d(i)*y2(i+1), where
00059 //      d(i) is temporarily stored in y2(i) (see below).
00060     if ( if1 == 2 )
00061     {
00062 //        cond1 is second derivative at first point.
00063 //        these two values assure that for above equation with d(i) temporarily
00064 //        stored in y2(i)
00065         y2[1] = 0.;
00066         u[0]  = cond1;
00067     }
00068     else if ( if1 == 1 )
00069     {
00070 //        cond1 is first derivative at first point.
00071 //        special case (Press et al 3.3.5 with A = 1, and B=0)
00072 //        of equations below where
00073 //        a_j = 0
00074 //        b_j = -(x_j+1 - x_j)/3
00075 //        c_j = -(x_j+1 - x_j)/6
00076 //        r_j = cond1 - (y_j+1 - y_j)/(x_j+1 - x_j)
00077 //        u(i) = r(i)/b(i)
00078 //        d(i) = -c(i)/b(i)
00079 //        N.B. d(i) is temporarily stored in y2.
00080         y2[1] = -.5;
00081         u[0]  = 3. / ( x[2] - x[1] ) * ( ( y[2] - y[1] ) / ( x[2] - x[1] ) - cond1 );
00082     }
00083     else
00084     {
00085         return 2;
00086     }
00087 //      if original tri-diagonal system is characterized as
00088 //      a_j y2_j-1 + b_j y2_j + c_j y2_j+1 = r_j
00089 //      Then from Press et al. 3.3.7, we have the unscaled result:
00090 //      a_j = (x_j - x_j-1)/6
00091 //      b_j = (x_j+1 - x_j-1)/3
00092 //      c_j = (x_j+1 - x_j)/6
00093 //      r_j = (y_j+1 - y_j)/(x_j+1 - x_j) - (y_j - y_j-1)/(x_j - x_j-1)
00094 //      In practice, all these values are divided through by b_j/2 to scale
00095 //      them, and from now on we will use these scaled values.
00096 
00097 //      forward elimination step: assume y2(i-1) = u(i-1) + d(i-1)*y2(i).
00098 //      When this is substituted into above tridiagonal equation ==>
00099 //      y2(i) = u(i) + d(i)*y2(i+1), where
00100 //      u(i) = [r(i) - a(i) u(i-1)]/[b(i) + a(i) d(i-1)]
00101 //      d(i) = -c(i)/[b(i) + a(i) d(i-1)]
00102 //      N.B. d(i) is temporarily stored in y2.
00103     i__1 = n - 1;
00104     for ( i__ = 2; i__ <= i__1; ++i__ )
00105     {
00106 //        sig is scaled a(i)
00107         sig = ( x[i__] - x[i__ - 1] ) / ( x[i__ + 1] - x[i__ - 1] );
00108 //        p is denominator = scaled a(i) d(i-1) + scaled  b(i), where scaled
00109 //        b(i) is 2.
00110         p = sig * y2[i__ - 1] + 2.;
00111 //        propagate d(i) equation above.  Note sig-1 = -c(i)
00112         y2[i__] = ( sig - 1. ) / p;
00113 //        propagate scaled u(i) equation above
00114         u[i__ - 1] = ( ( ( y[i__ + 1] - y[i__] ) / ( x[i__ + 1] - x[i__] ) - ( y[i__]
00115                                                                                - y[i__ - 1] ) / ( x[i__] - x[i__ - 1] ) ) * 6. / ( x[i__ + 1] -
00116                                                                                                                                    x[i__ - 1] ) - sig * u[i__ - 2] ) / p;
00117     }
00118     if ( ifn == 2 )
00119     {
00120 //        condn is second derivative at nth point.
00121 //        These two values assure that in the equation below.
00122         qn = 0.;
00123         un = condn;
00124     }
00125     else if ( ifn == 1 )
00126     {
00127 //        specify condn is first derivative at nth point.
00128 //        special case (Press et al 3.3.5 with A = 0, and B=1)
00129 //        implies a_n y2(n-1) + b_n y2(n) = r_n, where
00130 //        a_n = (x_n - x_n-1)/6
00131 //        b_n = (x_n - x_n-1)/3
00132 //        r_n = cond1 - (y_n - y_n-1)/(x_n - x_n-1)
00133 //        use same propagation equation as above, only with c_n = 0
00134 //        ==> d_n = 0 ==> y2(n) = u(n) =>
00135 //        y(n) = [r(n) - a(n) u(n-1)]/[b(n) + a(n) d(n-1)]
00136 //        qn is scaled a_n
00137         qn = .5;
00138 //        un is scaled r_n (N.B. un is not u(n))!  Sorry for the mixed notation.
00139         un = 3. / ( x[n] - x[n - 1] ) * ( condn - ( y[n] - y[n - 1] ) / ( x[n]
00140                                                                           - x[n - 1] ) );
00141     }
00142     else
00143     {
00144         return 3;
00145     }
00146 //      N.B. d(i) is temporarily stored in y2, and everything is
00147 //      scaled by b_n.
00148 //     qn is scaled a_n, 1.d0 is scaled b_n, and un is scaled r_n.
00149     y2[n] = ( un - qn * u[n - 2] ) / ( qn * y2[n - 1] + 1. );
00150 //      back substitution.
00151     for ( k = n - 1; k >= 1; --k )
00152     {
00153         y2[k] = y2[k] * y2[k + 1] + u[k - 1];
00154     }
00155     return 0;
00156 }
00157 
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