| 1 | package negotiator.boaframework.offeringstrategy.anac2013.TheFawkes;
|
|---|
| 2 |
|
|---|
| 3 | /*
|
|---|
| 4 | * Class: SmoothingCubicSpline
|
|---|
| 5 | * Description: smoothing cubic spline algorithm of Schoenberg
|
|---|
| 6 | * Environment: Java
|
|---|
| 7 | * Software: SSJ
|
|---|
| 8 | * Copyright (C) 2001 Pierre L'Ecuyer and Université de Montréal
|
|---|
| 9 | * Organization: DIRO, Université de Montréal
|
|---|
| 10 | * @author
|
|---|
| 11 |
|
|---|
| 12 | * SSJ is free software: you can redistribute it and/or modify it under
|
|---|
| 13 | * the terms of the GNU General Public License (GPL) as published by the
|
|---|
| 14 | * Free Software Foundation, either version 3 of the License, or
|
|---|
| 15 | * any later version.
|
|---|
| 16 |
|
|---|
| 17 | * SSJ is distributed in the hope that it will be useful,
|
|---|
| 18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of
|
|---|
| 19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
|---|
| 20 | * GNU General Public License for more details.
|
|---|
| 21 |
|
|---|
| 22 | * A copy of the GNU General Public License is available at
|
|---|
| 23 | <a href="http://www.gnu.org/licenses">GPL licence site</a>.
|
|---|
| 24 | */
|
|---|
| 25 | /**
|
|---|
| 26 | * Represents a cubic spline with nodes at <SPAN CLASS="MATH">(<I>x</I><SUB>i</SUB>, <I>y</I><SUB>i</SUB>)</SPAN>
|
|---|
| 27 | * computed with the smoothing cubic spline algorithm of Schoenberg. A smoothing cubic spline is made of <SPAN
|
|---|
| 28 | * CLASS="MATH"><I>n</I> + 1</SPAN> cubic polynomials. The <SPAN CLASS="MATH"><I>i</I></SPAN>th polynomial of such a
|
|---|
| 29 | * spline, for <SPAN CLASS="MATH"><I>i</I> = 1,…, <I>n</I> - 1</SPAN>, is defined as <SPAN
|
|---|
| 30 | * CLASS="MATH"><I>S</I><SUB>i</SUB>(<I>x</I>)</SPAN> while the complete spline is defined as
|
|---|
| 31 | *
|
|---|
| 32 | * <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"> <I>S</I>(<I>x</I>) =
|
|---|
| 33 | * <I>S</I><SUB>i</SUB>(<I>x</I>), for
|
|---|
| 34 | * <I>x</I>∈[<I>x</I><SUB>i-1</SUB>, <I>x</I><SUB>i</SUB>]. </DIV><P></P> For <SPAN CLASS="MATH"><I>x</I> <
|
|---|
| 35 | * <I>x</I><SUB>0</SUB></SPAN> and <SPAN CLASS="MATH"><I>x</I> > <I>x</I><SUB>n-1</SUB></SPAN>, the spline is not
|
|---|
| 36 | * precisely defined, but this class performs extrapolation by using <SPAN CLASS="MATH"><I>S</I><SUB>0</SUB></SPAN> and
|
|---|
| 37 | * <SPAN CLASS="MATH"><I>S</I><SUB>n</SUB></SPAN> linear polynomials. The algorithm which calculates the smoothing
|
|---|
| 38 | * spline is a generalization of the algorithm for an interpolating spline. <SPAN
|
|---|
| 39 | * CLASS="MATH"><I>S</I><SUB>i</SUB></SPAN> is linked to <SPAN CLASS="MATH"><I>S</I><SUB>i+1</SUB></SPAN> at <SPAN
|
|---|
| 40 | * CLASS="MATH"><I>x</I><SUB>i+1</SUB></SPAN> and keeps continuity properties for first and second derivatives at this
|
|---|
| 41 | * point, therefore
|
|---|
| 42 | *
|
|---|
| 43 | * <SPAN CLASS="MATH"><I>S</I><SUB>i</SUB>(<I>x</I><SUB>i+1</SUB>) =
|
|---|
| 44 | * <I>S</I><SUB>i+1</SUB>(<I>x</I><SUB>i+1</SUB>)</SPAN>,
|
|---|
| 45 | *
|
|---|
| 46 | * <SPAN CLASS="MATH"><I>S'</I><SUB>i</SUB>(<I>x</I><SUB>i+1</SUB>) =
|
|---|
| 47 | * <I>S'</I><SUB>i+1</SUB>(<I>x</I><SUB>i+1</SUB>)</SPAN> and <SPAN
|
|---|
| 48 | * CLASS="MATH"><I>S''</I><SUB>i</SUB>(<I>x</I><SUB>i+1</SUB>) =
|
|---|
| 49 | * <I>S''</I><SUB>i+1</SUB>(<I>x</I><SUB>i+1</SUB>)</SPAN>.
|
|---|
| 50 | *
|
|---|
| 51 | * <P> The spline is computed with a smoothing parameter <SPAN CLASS="MATH"><I>ρ</I>∈[0, 1]</SPAN> which
|
|---|
| 52 | * represents its accuracy with respect to the initial <SPAN CLASS="MATH">(<I>x</I><SUB>i</SUB>,
|
|---|
| 53 | * <I>y</I><SUB>i</SUB>)</SPAN> nodes. The smoothing spline minimizes
|
|---|
| 54 | *
|
|---|
| 55 | * <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"> <I>L</I> =
|
|---|
| 56 | * <I>ρ</I>∑<SUB>i=0</SUB><SUP>n-1</SUP><I>w</I><SUB>i</SUB>(<I>y</I><SUB>i</SUB>-<I>S</I><SUB>i</SUB>(<I>x</I><SUB>i</SUB>))<SUP>2</SUP>
|
|---|
| 57 | * + (1 -
|
|---|
| 58 | * <I>ρ</I>)∫<SUB>x<SUB>0</SUB></SUB><SUP>x<SUB>n-1</SUB></SUP>(<I>S''</I>(<I>x</I>))<SUP>2</SUP><I>dx</I>
|
|---|
| 59 | * </DIV><P></P> In fact, by setting <SPAN CLASS="MATH"><I>ρ</I> = 1</SPAN>, we obtain the interpolating spline;
|
|---|
| 60 | * and we obtain a linear function by setting <SPAN CLASS="MATH"><I>ρ</I> = 0</SPAN>. The weights <SPAN
|
|---|
| 61 | * CLASS="MATH"><I>w</I><SUB>i</SUB> > 0</SPAN>, which default to 1, can be used to change the contribution of each
|
|---|
| 62 | * point in the error term. A large value <SPAN CLASS="MATH"><I>w</I><SUB>i</SUB></SPAN> will give a large weight to the
|
|---|
| 63 | * <SPAN CLASS="MATH"><I>i</I></SPAN>th point, so the spline will pass closer to it. Here is a small example that uses
|
|---|
| 64 | * smoothing splines:
|
|---|
| 65 | *
|
|---|
| 66 | * <P>
|
|---|
| 67 | *
|
|---|
| 68 | * <DIV CLASS="vcode" ALIGN="LEFT"> <TT>
|
|---|
| 69 | *
|
|---|
| 70 | * <BR> int n; <BR> double[] X = new double[n]; <BR> double[] Y = new
|
|---|
| 71 | * double[n]; <BR> // here, fill arrays X and Y with n data points (x_i, y_i) <BR> //
|
|---|
| 72 | * The points must be sorted with respect to x_i. <BR> <BR> double rho = 0.1;
|
|---|
| 73 | * <BR> SmoothingCubicSpline fit = new SmoothingCubicSpline(X, Y, rho); <BR> <BR> int
|
|---|
| 74 | * m = 40; <BR> double[] Xp = new double[m+1]; // Xp, Yp are spline
|
|---|
| 75 | * points <BR> double[] Yp = new double[m+1]; <BR> double h = (X[n-1] - X[0]) / m;
|
|---|
| 76 | * // step <BR> <BR> for (int i = 0; i <= m; i++) {
|
|---|
| 77 | * <BR> double z = X[0] + i * h; <BR> Xp[i] = z;
|
|---|
| 78 | * <BR> Yp[i] = fit.evaluate(z);
|
|---|
| 79 | * // evaluate spline at z <BR> } <BR> <BR></TT>
|
|---|
| 80 | * </DIV>
|
|---|
| 81 | *
|
|---|
| 82 | */
|
|---|
| 83 | public final class SmoothingCubicSpline
|
|---|
| 84 | {
|
|---|
| 85 | private final SmoothingPolynomial[] splineVector;
|
|---|
| 86 | private final double[] x, y, weight;
|
|---|
| 87 | private final double rho;
|
|---|
| 88 |
|
|---|
| 89 | /**
|
|---|
| 90 | * Constructs a spline with nodes at <SPAN CLASS="MATH">(<I>x</I><SUB>i</SUB>, <I>y</I><SUB>i</SUB>)</SPAN>, with
|
|---|
| 91 | * weights <SPAN CLASS="MATH"><I>w</I><SUB>i</SUB></SPAN> and smoothing factor <SPAN
|
|---|
| 92 | * CLASS="MATH"><I>ρ</I></SPAN> = <TT>rho</TT>. The <SPAN CLASS="MATH"><I>x</I><SUB>i</SUB></SPAN> <SPAN
|
|---|
| 93 | * CLASS="textit">must</SPAN> be sorted in increasing order.
|
|---|
| 94 | *
|
|---|
| 95 | * @param x the <SPAN CLASS="MATH"><I>x</I><SUB>i</SUB></SPAN> coordinates.
|
|---|
| 96 | *
|
|---|
| 97 | * @param y the <SPAN CLASS="MATH"><I>y</I><SUB>i</SUB></SPAN> coordinates.
|
|---|
| 98 | *
|
|---|
| 99 | * @param w the weight for each point, must be <SPAN CLASS="MATH">> 0</SPAN>.
|
|---|
| 100 | *
|
|---|
| 101 | * @param rho the smoothing parameter
|
|---|
| 102 | *
|
|---|
| 103 | * @exception IllegalArgumentException if <TT>x</TT>, <TT>y</TT> and <TT>z</TT> do not have the same length, if rho
|
|---|
| 104 | * has wrong value, or if the spline cannot be calculated.
|
|---|
| 105 | *
|
|---|
| 106 | */
|
|---|
| 107 | public SmoothingCubicSpline( double[] x, double[] y, double[] w, double rho )
|
|---|
| 108 | {
|
|---|
| 109 | if( x.length != y.length )
|
|---|
| 110 | {
|
|---|
| 111 | throw new IllegalArgumentException( "x.length != y.length" );
|
|---|
| 112 | }
|
|---|
| 113 | else if( w != null && x.length != w.length )
|
|---|
| 114 | {
|
|---|
| 115 | throw new IllegalArgumentException( "x.length != w.length" );
|
|---|
| 116 | }
|
|---|
| 117 | else if( rho < 0 || rho > 1 )
|
|---|
| 118 | {
|
|---|
| 119 | throw new IllegalArgumentException( "rho not in [0, 1]" );
|
|---|
| 120 | }
|
|---|
| 121 | else
|
|---|
| 122 | {
|
|---|
| 123 | this.splineVector = new SmoothingPolynomial[x.length + 1];
|
|---|
| 124 | this.rho = rho;
|
|---|
| 125 | this.x = x.clone();
|
|---|
| 126 | this.y = y.clone();
|
|---|
| 127 | this.weight = new double[x.length];
|
|---|
| 128 | if( w == null )
|
|---|
| 129 | {
|
|---|
| 130 | for( int i = 0; i < this.weight.length; i++ )
|
|---|
| 131 | {
|
|---|
| 132 | this.weight[i] = 1;
|
|---|
| 133 | }
|
|---|
| 134 | }
|
|---|
| 135 | else
|
|---|
| 136 | {
|
|---|
| 137 | System.arraycopy( w, 0, this.weight, 0, this.weight.length );
|
|---|
| 138 | }
|
|---|
| 139 | this.resolve();
|
|---|
| 140 | }
|
|---|
| 141 | }
|
|---|
| 142 |
|
|---|
| 143 | /**
|
|---|
| 144 | * Constructs a spline with nodes at <SPAN CLASS="MATH">(<I>x</I><SUB>i</SUB>, <I>y</I><SUB>i</SUB>)</SPAN>, with
|
|---|
| 145 | * weights <SPAN CLASS="MATH">= 1</SPAN> and smoothing factor <SPAN CLASS="MATH"><I>ρ</I></SPAN> =
|
|---|
| 146 | * <TT>rho</TT>. The <SPAN CLASS="MATH"><I>x</I><SUB>i</SUB></SPAN> <SPAN CLASS="textit">must</SPAN> be sorted in
|
|---|
| 147 | * increasing order.
|
|---|
| 148 | *
|
|---|
| 149 | * @param x the <SPAN CLASS="MATH"><I>x</I><SUB>i</SUB></SPAN> coordinates.
|
|---|
| 150 | *
|
|---|
| 151 | * @param y the <SPAN CLASS="MATH"><I>y</I><SUB>i</SUB></SPAN> coordinates.
|
|---|
| 152 | *
|
|---|
| 153 | * @param rho the smoothing parameter
|
|---|
| 154 | *
|
|---|
| 155 | * @exception IllegalArgumentException if <TT>x</TT> and <TT>y</TT> do not have the same length, if rho has wrong
|
|---|
| 156 | * value, or if the spline cannot be calculated.
|
|---|
| 157 | *
|
|---|
| 158 | */
|
|---|
| 159 | public SmoothingCubicSpline( double[] x, double[] y, double rho )
|
|---|
| 160 | {
|
|---|
| 161 | this( x, y, null, rho );
|
|---|
| 162 | }
|
|---|
| 163 |
|
|---|
| 164 | /**
|
|---|
| 165 | * Evaluates and returns the value of the spline at <SPAN CLASS="MATH"><I>z</I></SPAN>.
|
|---|
| 166 | *
|
|---|
| 167 | * @param z argument of the spline.
|
|---|
| 168 | *
|
|---|
| 169 | * @return value of spline.
|
|---|
| 170 | *
|
|---|
| 171 | */
|
|---|
| 172 | public double evaluate( double z )
|
|---|
| 173 | {
|
|---|
| 174 | double returned;
|
|---|
| 175 | int i = this.getFitPolynomialIndex( z );
|
|---|
| 176 | if( i == 0 )
|
|---|
| 177 | {
|
|---|
| 178 | returned = this.splineVector[i].evaluate( z - x[0] );
|
|---|
| 179 | }
|
|---|
| 180 | else
|
|---|
| 181 | {
|
|---|
| 182 | returned = this.splineVector[i].evaluate( z - x[i - 1] );
|
|---|
| 183 | }
|
|---|
| 184 | if( Double.isNaN( returned ) || returned < 0 )
|
|---|
| 185 | {
|
|---|
| 186 | returned = 0;
|
|---|
| 187 | }
|
|---|
| 188 | else if( Double.isInfinite( returned ) || returned > 1 )
|
|---|
| 189 | {
|
|---|
| 190 | returned = 1;
|
|---|
| 191 | }
|
|---|
| 192 | return returned;
|
|---|
| 193 | }
|
|---|
| 194 |
|
|---|
| 195 | /**
|
|---|
| 196 | * Returns the index of <SPAN CLASS="MATH"><I>P</I></SPAN>, the {@link Polynomial} instance used to evaluate <SPAN
|
|---|
| 197 | * CLASS="MATH"><I>x</I></SPAN>, in an <TT>ArrayList</TT> table instance returned by
|
|---|
| 198 | * <TT>getSplinePolynomials()</TT>. This index <SPAN CLASS="MATH"><I>k</I></SPAN> gives also the interval in table
|
|---|
| 199 | * <SPAN CLASS="textbf">X</SPAN> which contains the value <SPAN CLASS="MATH"><I>x</I></SPAN> (i.e. such that <SPAN
|
|---|
| 200 | * CLASS="MATH"><I>x</I><SUB>k</SUB> < <I>x</I> <= <I>x</I><SUB>k+1</SUB></SPAN>).
|
|---|
| 201 | *
|
|---|
| 202 | * @return Index of the polynomial check with x in the Polynomial list returned by methodgetSplinePolynomials
|
|---|
| 203 | *
|
|---|
| 204 | */
|
|---|
| 205 | public int getFitPolynomialIndex( double x )
|
|---|
| 206 | {
|
|---|
| 207 | // Algorithme de recherche binaire legerement modifie
|
|---|
| 208 | int j = this.x.length - 1;
|
|---|
| 209 | if( x > this.x[j] )
|
|---|
| 210 | {
|
|---|
| 211 | return j + 1;
|
|---|
| 212 | }
|
|---|
| 213 | int tmp = 0;
|
|---|
| 214 | int i = 0;
|
|---|
| 215 | while( ( i + 1 ) != j )
|
|---|
| 216 | {
|
|---|
| 217 | if( x > this.x[tmp] )
|
|---|
| 218 | {
|
|---|
| 219 | i = tmp;
|
|---|
| 220 | tmp = i + ( ( j - i ) / 2 );
|
|---|
| 221 | }
|
|---|
| 222 | else
|
|---|
| 223 | {
|
|---|
| 224 | j = tmp;
|
|---|
| 225 | tmp = i + ( ( j - i ) / 2 );
|
|---|
| 226 | }
|
|---|
| 227 | if( j == 0 ) // le point est < a x_0, on sort
|
|---|
| 228 | {
|
|---|
| 229 | i--;
|
|---|
| 230 | }
|
|---|
| 231 | }
|
|---|
| 232 | return i + 1;
|
|---|
| 233 | }
|
|---|
| 234 |
|
|---|
| 235 | private void resolve()
|
|---|
| 236 | {
|
|---|
| 237 | /*
|
|---|
| 238 | taken from D.S.G Pollock's paper, "Smoothing with Cubic Splines",
|
|---|
| 239 | Queen Mary, University of London (1993)
|
|---|
| 240 | http://www.qmw.ac.uk/~ugte133/PAPERS/SPLINES.PDF
|
|---|
| 241 | */
|
|---|
| 242 | int n = this.x.length;
|
|---|
| 243 | double[] h = new double[n];
|
|---|
| 244 | double[] r = new double[n];
|
|---|
| 245 | double[] u = new double[n];
|
|---|
| 246 | double[] v = new double[n];
|
|---|
| 247 | double[] w = new double[n];
|
|---|
| 248 | double[] q = new double[n + 1];
|
|---|
| 249 | double[] sigma = new double[this.weight.length];
|
|---|
| 250 |
|
|---|
| 251 | for( int i = 0; i < sigma.length; i++ )
|
|---|
| 252 | {
|
|---|
| 253 | if( this.weight[i] <= 0 )
|
|---|
| 254 | {
|
|---|
| 255 | sigma[i] = 1.0e100; // arbitrary large number to avoid 1/0
|
|---|
| 256 | }
|
|---|
| 257 | else
|
|---|
| 258 | {
|
|---|
| 259 | sigma[i] = 1 / Math.sqrt( this.weight[i] );
|
|---|
| 260 | }
|
|---|
| 261 | }
|
|---|
| 262 |
|
|---|
| 263 | double mu;
|
|---|
| 264 | n = this.x.length - 1;
|
|---|
| 265 | if( this.rho <= 0 )
|
|---|
| 266 | {
|
|---|
| 267 | mu = 1.0e100; // arbitrary large number to avoid 1/0
|
|---|
| 268 | }
|
|---|
| 269 | else
|
|---|
| 270 | {
|
|---|
| 271 | mu = ( 2 * ( 1 - this.rho ) ) / ( 3 * this.rho );
|
|---|
| 272 | }
|
|---|
| 273 |
|
|---|
| 274 | h[0] = this.x[1] - this.x[0];
|
|---|
| 275 | r[0] = 3 / h[0];
|
|---|
| 276 | for( int i = 1; i < n; i++ )
|
|---|
| 277 | {
|
|---|
| 278 | h[i] = this.x[i + 1] - this.x[i];
|
|---|
| 279 | r[i] = 3 / h[i];
|
|---|
| 280 | q[i] = ( 3 * ( this.y[i + 1] - y[i] ) / h[i] ) - ( 3 * ( this.y[i] - this.y[i - 1] ) / h[i - 1] );
|
|---|
| 281 | }
|
|---|
| 282 |
|
|---|
| 283 | for( int i = 1; i < n; i++ )
|
|---|
| 284 | {
|
|---|
| 285 | u[i] = ( r[i - 1] * r[i - 1] * sigma[i - 1] ) + ( ( r[i - 1] + r[i] ) * ( r[i - 1] + r[i] ) * sigma[i] )
|
|---|
| 286 | + ( r[i] * r[i] * sigma[i + 1] );
|
|---|
| 287 | u[i] = mu * u[i] + 2 * ( this.x[i + 1] - this.x[i - 1] );
|
|---|
| 288 | v[i] = ( -( r[i - 1] + r[i] ) * r[i] * sigma[i] ) - ( r[i] * ( r[i] + r[i + 1] ) * sigma[i + 1] );
|
|---|
| 289 | v[i] = ( mu * v[i] ) + h[i];
|
|---|
| 290 | w[i] = mu * r[i] * r[i + 1] * sigma[i + 1];
|
|---|
| 291 | }
|
|---|
| 292 | q = Quincunx( u, v, w, q );
|
|---|
| 293 |
|
|---|
| 294 | // extrapolation a gauche
|
|---|
| 295 | double[] params = new double[4];
|
|---|
| 296 | params[0] = this.y[0] - ( mu * r[0] * q[1] * sigma[0] );
|
|---|
| 297 | double dd1 = this.y[1] - ( mu * ( ( -r[0] - r[1] ) * q[1] + r[1] * q[2] ) * sigma[1] );
|
|---|
| 298 | params[1] = ( dd1 - params[0] ) / h[0] - ( q[1] * h[0] / 3 );
|
|---|
| 299 | this.splineVector[0] = new SmoothingPolynomial( params );
|
|---|
| 300 |
|
|---|
| 301 | // premier polynome
|
|---|
| 302 | params[0] = this.y[0] - ( mu * r[0] * q[1] * sigma[0] );
|
|---|
| 303 | double dd2 = this.y[1] - ( mu * ( ( -r[0] - r[1] ) * q[1] + r[1] * q[2] ) * sigma[1] );
|
|---|
| 304 | params[3] = q[1] / ( 3 * h[0] );
|
|---|
| 305 | params[2] = 0;
|
|---|
| 306 | params[1] = ( ( dd2 - params[0] ) / h[0] ) - ( q[1] * h[0] / 3 );
|
|---|
| 307 | this.splineVector[1] = new SmoothingPolynomial( params );
|
|---|
| 308 |
|
|---|
| 309 | // les polynomes suivants
|
|---|
| 310 | int j;
|
|---|
| 311 | for( j = 1; j < n; j++ )
|
|---|
| 312 | {
|
|---|
| 313 | params[3] = ( q[j + 1] - q[j] ) / ( 3 * h[j] );
|
|---|
| 314 | params[2] = q[j];
|
|---|
| 315 | params[1] = ( ( q[j] + q[j - 1] ) * h[j - 1] ) + this.splineVector[j].getCoefficient( 1 );
|
|---|
| 316 | params[0] = ( r[j - 1] * q[j - 1] ) + ( ( -r[j - 1] - r[j] ) * q[j] ) + ( r[j] * q[j + 1] );
|
|---|
| 317 | params[0] = this.y[j] - ( mu * params[0] * sigma[j] );
|
|---|
| 318 | this.splineVector[j + 1] = new SmoothingPolynomial( params );
|
|---|
| 319 | }
|
|---|
| 320 |
|
|---|
| 321 | // extrapolation a droite
|
|---|
| 322 | j = n;
|
|---|
| 323 | params[3] = 0;
|
|---|
| 324 | params[2] = 0;
|
|---|
| 325 | params[1] = this.splineVector[j].derivative( this.x[n] - this.x[n - 1] );
|
|---|
| 326 | params[0] = this.splineVector[j].evaluate( this.x[n] - this.x[n - 1] );
|
|---|
| 327 | this.splineVector[n + 1] = new SmoothingPolynomial( params );
|
|---|
| 328 | }
|
|---|
| 329 |
|
|---|
| 330 | private static double[] Quincunx( double[] u, double[] v, double[] w, double[] q )
|
|---|
| 331 | {
|
|---|
| 332 | u[0] = 0;
|
|---|
| 333 | v[1] /= u[1];
|
|---|
| 334 | w[1] /= u[1];
|
|---|
| 335 | int j;
|
|---|
| 336 | for( j = 2; j < ( u.length - 1 ); j++ )
|
|---|
| 337 | {
|
|---|
| 338 | u[j] = u[j] - ( u[j - 2] * w[j - 2] * w[j - 2] ) - ( u[j - 1] * v[j - 1] * v[j - 1] );
|
|---|
| 339 | v[j] = ( v[j] - ( u[j - 1] * v[j - 1] * w[j - 1] ) ) / u[j];
|
|---|
| 340 | w[j] /= u[j];
|
|---|
| 341 | }
|
|---|
| 342 |
|
|---|
| 343 | // forward substitution
|
|---|
| 344 | q[1] -= v[0] * q[0];
|
|---|
| 345 | for( j = 2; j < ( u.length - 1 ); j++ )
|
|---|
| 346 | {
|
|---|
| 347 | q[j] = q[j] - ( v[j - 1] * q[j - 1] ) - ( w[j - 2] * q[j - 2] );
|
|---|
| 348 | }
|
|---|
| 349 | for( j = 1; j < ( u.length - 1 ); j++ )
|
|---|
| 350 | {
|
|---|
| 351 | q[j] /= u[j];
|
|---|
| 352 | }
|
|---|
| 353 |
|
|---|
| 354 | // backward substitution
|
|---|
| 355 | q[u.length - 1] = 0;
|
|---|
| 356 | for( j = ( u.length - 3 ); j > 0; j-- )
|
|---|
| 357 | {
|
|---|
| 358 | q[j] = q[j] - ( v[j] * q[j + 1] ) - ( w[j] * q[j + 2] );
|
|---|
| 359 | }
|
|---|
| 360 |
|
|---|
| 361 | return q;
|
|---|
| 362 | }
|
|---|
| 363 | }
|
|---|
![(please configure the [header_logo] section in trac.ini)](/pubtrac/Genius/chrome/site/your_project_logo.png){kind=logo}
