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/* |
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* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved. |
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* |
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* The University of Notre Dame grants you ("Licensee") a |
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* non-exclusive, royalty free, license to use, modify and |
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* redistribute this software in source and binary code form, provided |
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* that the following conditions are met: |
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* |
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* 1. Redistributions of source code must retain the above copyright |
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* notice, this list of conditions and the following disclaimer. |
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* |
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* 2. Redistributions in binary form must reproduce the above copyright |
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* notice, this list of conditions and the following disclaimer in the |
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* documentation and/or other materials provided with the |
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* distribution. |
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* |
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* This software is provided "AS IS," without a warranty of any |
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* kind. All express or implied conditions, representations and |
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* warranties, including any implied warranty of merchantability, |
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* fitness for a particular purpose or non-infringement, are hereby |
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* excluded. The University of Notre Dame and its licensors shall not |
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* be liable for any damages suffered by licensee as a result of |
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* using, modifying or distributing the software or its |
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* derivatives. In no event will the University of Notre Dame or its |
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* licensors be liable for any lost revenue, profit or data, or for |
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* direct, indirect, special, consequential, incidental or punitive |
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* damages, however caused and regardless of the theory of liability, |
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* arising out of the use of or inability to use software, even if the |
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* University of Notre Dame has been advised of the possibility of |
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* such damages. |
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* |
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* SUPPORT OPEN SCIENCE! If you use OpenMD or its source code in your |
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* research, please cite the appropriate papers when you publish your |
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* work. Good starting points are: |
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* |
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* [1] Meineke, et al., J. Comp. Chem. 26, 252-271 (2005). |
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* [2] Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006). |
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* [3] Sun, Lin & Gezelter, J. Chem. Phys. 128, 24107 (2008). |
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* [4] Vardeman & Gezelter, in progress (2009). |
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*/ |
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|
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#include <stdio.h> |
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#include <math.h> |
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#include <stdlib.h> |
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#include "math/MatVec3.h" |
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|
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/* |
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* Contains various utilities for dealing with 3x3 matrices and |
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* length 3 vectors |
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*/ |
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|
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void identityMat3(RealType A[3][3]) { |
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int i; |
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for (i = 0; i < 3; i++) { |
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A[i][0] = A[i][1] = A[i][2] = 0.0; |
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A[i][i] = 1.0; |
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} |
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} |
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|
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void swapVectors3(RealType v1[3], RealType v2[3]) { |
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int i; |
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for (i = 0; i < 3; i++) { |
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RealType tmp = v1[i]; |
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v1[i] = v2[i]; |
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v2[i] = tmp; |
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} |
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} |
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|
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RealType normalize3(RealType x[3]) { |
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RealType den; |
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int i; |
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if ( (den = norm3(x)) != 0.0 ) { |
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for (i=0; i < 3; i++) |
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{ |
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x[i] /= den; |
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} |
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} |
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return den; |
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} |
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|
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void matMul3(RealType a[3][3], RealType b[3][3], RealType c[3][3]) { |
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RealType r00, r01, r02, r10, r11, r12, r20, r21, r22; |
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|
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r00 = a[0][0]*b[0][0] + a[0][1]*b[1][0] + a[0][2]*b[2][0]; |
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r01 = a[0][0]*b[0][1] + a[0][1]*b[1][1] + a[0][2]*b[2][1]; |
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r02 = a[0][0]*b[0][2] + a[0][1]*b[1][2] + a[0][2]*b[2][2]; |
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|
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r10 = a[1][0]*b[0][0] + a[1][1]*b[1][0] + a[1][2]*b[2][0]; |
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r11 = a[1][0]*b[0][1] + a[1][1]*b[1][1] + a[1][2]*b[2][1]; |
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r12 = a[1][0]*b[0][2] + a[1][1]*b[1][2] + a[1][2]*b[2][2]; |
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|
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r20 = a[2][0]*b[0][0] + a[2][1]*b[1][0] + a[2][2]*b[2][0]; |
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r21 = a[2][0]*b[0][1] + a[2][1]*b[1][1] + a[2][2]*b[2][1]; |
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r22 = a[2][0]*b[0][2] + a[2][1]*b[1][2] + a[2][2]*b[2][2]; |
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|
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c[0][0] = r00; c[0][1] = r01; c[0][2] = r02; |
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c[1][0] = r10; c[1][1] = r11; c[1][2] = r12; |
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c[2][0] = r20; c[2][1] = r21; c[2][2] = r22; |
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} |
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|
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void matVecMul3(RealType m[3][3], RealType inVec[3], RealType outVec[3]) { |
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RealType a0, a1, a2; |
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|
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a0 = inVec[0]; a1 = inVec[1]; a2 = inVec[2]; |
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|
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outVec[0] = m[0][0]*a0 + m[0][1]*a1 + m[0][2]*a2; |
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outVec[1] = m[1][0]*a0 + m[1][1]*a1 + m[1][2]*a2; |
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outVec[2] = m[2][0]*a0 + m[2][1]*a1 + m[2][2]*a2; |
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} |
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|
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RealType matDet3(RealType a[3][3]) { |
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int i, j, k; |
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RealType determinant; |
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|
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determinant = 0.0; |
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|
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for(i = 0; i < 3; i++) { |
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j = (i+1)%3; |
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k = (i+2)%3; |
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|
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determinant += a[0][i] * (a[1][j]*a[2][k] - a[1][k]*a[2][j]); |
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} |
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|
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return determinant; |
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} |
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|
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void invertMat3(RealType a[3][3], RealType b[3][3]) { |
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|
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int i, j, k, l, m, n; |
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RealType determinant; |
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|
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determinant = matDet3( a ); |
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|
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if (determinant == 0.0) { |
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sprintf( painCave.errMsg, |
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"Can't invert a matrix with a zero determinant!\n"); |
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painCave.isFatal = 1; |
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simError(); |
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} |
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|
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for (i=0; i < 3; i++) { |
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j = (i+1)%3; |
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k = (i+2)%3; |
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for(l = 0; l < 3; l++) { |
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m = (l+1)%3; |
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n = (l+2)%3; |
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|
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b[l][i] = (a[j][m]*a[k][n] - a[j][n]*a[k][m]) / determinant; |
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} |
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} |
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} |
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|
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void transposeMat3(RealType in[3][3], RealType out[3][3]) { |
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RealType temp[3][3]; |
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int i, j; |
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|
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for (i = 0; i < 3; i++) { |
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for (j = 0; j < 3; j++) { |
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temp[j][i] = in[i][j]; |
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} |
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} |
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for (i = 0; i < 3; i++) { |
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for (j = 0; j < 3; j++) { |
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out[i][j] = temp[i][j]; |
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} |
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} |
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} |
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|
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void printMat3(RealType A[3][3] ){ |
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|
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fprintf(stderr, "[ %g, %g, %g ]\n[ %g, %g, %g ]\n[ %g, %g, %g ]\n", |
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A[0][0] , A[0][1] , A[0][2], |
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A[1][0] , A[1][1] , A[1][2], |
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A[2][0] , A[2][1] , A[2][2]) ; |
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} |
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|
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void printMat9(RealType A[9] ){ |
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|
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fprintf(stderr, "[ %g, %g, %g ]\n[ %g, %g, %g ]\n[ %g, %g, %g ]\n", |
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A[0], A[1], A[2], |
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A[3], A[4], A[5], |
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A[6], A[7], A[8]); |
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} |
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|
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RealType matTrace3(RealType m[3][3]){ |
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RealType trace; |
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trace = m[0][0] + m[1][1] + m[2][2]; |
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|
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return trace; |
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} |
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|
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void crossProduct3(RealType a[3],RealType b[3], RealType out[3]){ |
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|
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out[0] = a[1] * b[2] - a[2] * b[1]; |
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out[1] = a[2] * b[0] - a[0] * b[2] ; |
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out[2] = a[0] * b[1] - a[1] * b[0]; |
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|
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} |
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|
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RealType dotProduct3(RealType a[3], RealType b[3]){ |
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return a[0]*b[0] + a[1]*b[1]+ a[2]*b[2]; |
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} |
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|
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/*----------------------------------------------------------------------------*/ |
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/* Extract the eigenvalues and eigenvectors from a 3x3 matrix.*/ |
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/* The eigenvectors (the columns of V) will be normalized. */ |
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/* The eigenvectors are aligned optimally with the x, y, and z*/ |
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/* axes respectively.*/ |
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|
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void diagonalize3x3(const RealType A[3][3], RealType w[3], RealType V[3][3]) { |
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int i,j,k,maxI; |
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RealType tmp, maxVal; |
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|
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/* do the matrix[3][3] to **matrix conversion for Jacobi*/ |
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RealType C[3][3]; |
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RealType *ATemp[3],*VTemp[3]; |
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for (i = 0; i < 3; i++) |
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{ |
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C[i][0] = A[i][0]; |
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C[i][1] = A[i][1]; |
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C[i][2] = A[i][2]; |
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ATemp[i] = C[i]; |
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VTemp[i] = V[i]; |
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} |
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|
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/* diagonalize using Jacobi*/ |
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JacobiN(ATemp,3,w,VTemp); |
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|
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/* if all the eigenvalues are the same, return identity matrix*/ |
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if (w[0] == w[1] && w[0] == w[2]) |
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{ |
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identityMat3(V); |
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return; |
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} |
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|
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/* transpose temporarily, it makes it easier to sort the eigenvectors*/ |
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transposeMat3(V,V); |
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|
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/* if two eigenvalues are the same, re-orthogonalize to optimally line*/ |
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/* up the eigenvectors with the x, y, and z axes*/ |
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for (i = 0; i < 3; i++) |
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{ |
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if (w[(i+1)%3] == w[(i+2)%3]) /* two eigenvalues are the same*/ |
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{ |
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/* find maximum element of the independant eigenvector*/ |
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maxVal = fabs(V[i][0]); |
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maxI = 0; |
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for (j = 1; j < 3; j++) |
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{ |
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if (maxVal < (tmp = fabs(V[i][j]))) |
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{ |
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maxVal = tmp; |
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maxI = j; |
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} |
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} |
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/* swap the eigenvector into its proper position*/ |
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if (maxI != i) |
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{ |
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tmp = w[maxI]; |
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w[maxI] = w[i]; |
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w[i] = tmp; |
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swapVectors3(V[i],V[maxI]); |
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} |
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/* maximum element of eigenvector should be positive*/ |
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if (V[maxI][maxI] < 0) |
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{ |
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V[maxI][0] = -V[maxI][0]; |
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V[maxI][1] = -V[maxI][1]; |
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V[maxI][2] = -V[maxI][2]; |
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} |
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|
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/* re-orthogonalize the other two eigenvectors*/ |
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j = (maxI+1)%3; |
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k = (maxI+2)%3; |
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|
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V[j][0] = 0.0; |
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V[j][1] = 0.0; |
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V[j][2] = 0.0; |
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V[j][j] = 1.0; |
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crossProduct3(V[maxI],V[j],V[k]); |
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normalize3(V[k]); |
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crossProduct3(V[k],V[maxI],V[j]); |
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|
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/* transpose vectors back to columns*/ |
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transposeMat3(V,V); |
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return; |
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} |
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} |
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|
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/* the three eigenvalues are different, just sort the eigenvectors*/ |
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/* to align them with the x, y, and z axes*/ |
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|
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/* find the vector with the largest x element, make that vector*/ |
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/* the first vector*/ |
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maxVal = fabs(V[0][0]); |
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maxI = 0; |
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for (i = 1; i < 3; i++) |
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{ |
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if (maxVal < (tmp = fabs(V[i][0]))) |
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{ |
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maxVal = tmp; |
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maxI = i; |
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} |
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} |
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/* swap eigenvalue and eigenvector*/ |
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if (maxI != 0) |
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{ |
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tmp = w[maxI]; |
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w[maxI] = w[0]; |
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w[0] = tmp; |
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swapVectors3(V[maxI],V[0]); |
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} |
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/* do the same for the y element*/ |
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if (fabs(V[1][1]) < fabs(V[2][1])) |
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{ |
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tmp = w[2]; |
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w[2] = w[1]; |
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w[1] = tmp; |
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swapVectors3(V[2],V[1]); |
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} |
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|
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/* ensure that the sign of the eigenvectors is correct*/ |
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for (i = 0; i < 2; i++) |
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{ |
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if (V[i][i] < 0) |
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{ |
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V[i][0] = -V[i][0]; |
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V[i][1] = -V[i][1]; |
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V[i][2] = -V[i][2]; |
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} |
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} |
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/* set sign of final eigenvector to ensure that determinant is positive*/ |
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if (matDet3(V) < 0) |
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{ |
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V[2][0] = -V[2][0]; |
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V[2][1] = -V[2][1]; |
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V[2][2] = -V[2][2]; |
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} |
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|
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/* transpose the eigenvectors back again*/ |
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transposeMat3(V,V); |
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} |
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|
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|
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#define MAT_ROTATE(a,i,j,k,l) g=a[i][j];h=a[k][l];a[i][j]=g-s*(h+g*tau); a[k][l]=h+s*(g-h*tau); |
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|
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#define MAX_ROTATIONS 20 |
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|
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/* Jacobi iteration for the solution of eigenvectors/eigenvalues of a nxn*/ |
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/* real symmetric matrix. Square nxn matrix a; size of matrix in n;*/ |
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/* output eigenvalues in w; and output eigenvectors in v. Resulting*/ |
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/* eigenvalues/vectors are sorted in decreasing order; eigenvectors are*/ |
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/* normalized.*/ |
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int JacobiN(RealType **a, int n, RealType *w, RealType **v) { |
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|
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int i, j, k, iq, ip, numPos; |
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int ceil_half_n; |
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RealType tresh, theta, tau, t, sm, s, h, g, c, tmp; |
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RealType bspace[4], zspace[4]; |
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RealType *b = bspace; |
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RealType *z = zspace; |
| 362 |
|
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|
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/* only allocate memory if the matrix is large*/ |
| 365 |
if (n > 4) |
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{ |
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b = (RealType *) calloc(n, sizeof(RealType)); |
| 368 |
z = (RealType *) calloc(n, sizeof(RealType)); |
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} |
| 370 |
|
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/* initialize*/ |
| 372 |
for (ip=0; ip<n; ip++) |
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{ |
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for (iq=0; iq<n; iq++) |
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{ |
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v[ip][iq] = 0.0; |
| 377 |
} |
| 378 |
v[ip][ip] = 1.0; |
| 379 |
} |
| 380 |
for (ip=0; ip<n; ip++) |
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{ |
| 382 |
b[ip] = w[ip] = a[ip][ip]; |
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z[ip] = 0.0; |
| 384 |
} |
| 385 |
|
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/* begin rotation sequence*/ |
| 387 |
for (i=0; i<MAX_ROTATIONS; i++) |
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{ |
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sm = 0.0; |
| 390 |
for (ip=0; ip<n-1; ip++) |
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{ |
| 392 |
for (iq=ip+1; iq<n; iq++) |
| 393 |
{ |
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sm += fabs(a[ip][iq]); |
| 395 |
} |
| 396 |
} |
| 397 |
if (sm == 0.0) |
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{ |
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break; |
| 400 |
} |
| 401 |
|
| 402 |
if (i < 3) /* first 3 sweeps*/ |
| 403 |
{ |
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tresh = 0.2*sm/(n*n); |
| 405 |
} |
| 406 |
else |
| 407 |
{ |
| 408 |
tresh = 0.0; |
| 409 |
} |
| 410 |
|
| 411 |
for (ip=0; ip<n-1; ip++) |
| 412 |
{ |
| 413 |
for (iq=ip+1; iq<n; iq++) |
| 414 |
{ |
| 415 |
g = 100.0*fabs(a[ip][iq]); |
| 416 |
|
| 417 |
/* after 4 sweeps*/ |
| 418 |
if (i > 3 && (fabs(w[ip])+g) == fabs(w[ip]) |
| 419 |
&& (fabs(w[iq])+g) == fabs(w[iq])) |
| 420 |
{ |
| 421 |
a[ip][iq] = 0.0; |
| 422 |
} |
| 423 |
else if (fabs(a[ip][iq]) > tresh) |
| 424 |
{ |
| 425 |
h = w[iq] - w[ip]; |
| 426 |
if ( (fabs(h)+g) == fabs(h)) |
| 427 |
{ |
| 428 |
t = (a[ip][iq]) / h; |
| 429 |
} |
| 430 |
else |
| 431 |
{ |
| 432 |
theta = 0.5*h / (a[ip][iq]); |
| 433 |
t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta)); |
| 434 |
if (theta < 0.0) |
| 435 |
{ |
| 436 |
t = -t; |
| 437 |
} |
| 438 |
} |
| 439 |
c = 1.0 / sqrt(1+t*t); |
| 440 |
s = t*c; |
| 441 |
tau = s/(1.0+c); |
| 442 |
h = t*a[ip][iq]; |
| 443 |
z[ip] -= h; |
| 444 |
z[iq] += h; |
| 445 |
w[ip] -= h; |
| 446 |
w[iq] += h; |
| 447 |
a[ip][iq]=0.0; |
| 448 |
|
| 449 |
/* ip already shifted left by 1 unit*/ |
| 450 |
for (j = 0;j <= ip-1;j++) |
| 451 |
{ |
| 452 |
MAT_ROTATE(a,j,ip,j,iq) |
| 453 |
} |
| 454 |
/* ip and iq already shifted left by 1 unit*/ |
| 455 |
for (j = ip+1;j <= iq-1;j++) |
| 456 |
{ |
| 457 |
MAT_ROTATE(a,ip,j,j,iq) |
| 458 |
} |
| 459 |
/* iq already shifted left by 1 unit*/ |
| 460 |
for (j=iq+1; j<n; j++) |
| 461 |
{ |
| 462 |
MAT_ROTATE(a,ip,j,iq,j) |
| 463 |
} |
| 464 |
for (j=0; j<n; j++) |
| 465 |
{ |
| 466 |
MAT_ROTATE(v,j,ip,j,iq) |
| 467 |
} |
| 468 |
} |
| 469 |
} |
| 470 |
} |
| 471 |
|
| 472 |
for (ip=0; ip<n; ip++) |
| 473 |
{ |
| 474 |
b[ip] += z[ip]; |
| 475 |
w[ip] = b[ip]; |
| 476 |
z[ip] = 0.0; |
| 477 |
} |
| 478 |
} |
| 479 |
|
| 480 |
/*// this is NEVER called*/ |
| 481 |
if ( i >= MAX_ROTATIONS ) |
| 482 |
{ |
| 483 |
sprintf( painCave.errMsg, |
| 484 |
"Jacobi: Error extracting eigenfunctions!\n"); |
| 485 |
painCave.isFatal = 1; |
| 486 |
simError(); |
| 487 |
return 0; |
| 488 |
} |
| 489 |
|
| 490 |
/* sort eigenfunctions these changes do not affect accuracy */ |
| 491 |
for (j=0; j<n-1; j++) /* boundary incorrect*/ |
| 492 |
{ |
| 493 |
k = j; |
| 494 |
tmp = w[k]; |
| 495 |
for (i=j+1; i<n; i++) /* boundary incorrect, shifted already*/ |
| 496 |
{ |
| 497 |
if (w[i] >= tmp) /* why exchage if same?*/ |
| 498 |
{ |
| 499 |
k = i; |
| 500 |
tmp = w[k]; |
| 501 |
} |
| 502 |
} |
| 503 |
if (k != j) |
| 504 |
{ |
| 505 |
w[k] = w[j]; |
| 506 |
w[j] = tmp; |
| 507 |
for (i=0; i<n; i++) |
| 508 |
{ |
| 509 |
tmp = v[i][j]; |
| 510 |
v[i][j] = v[i][k]; |
| 511 |
v[i][k] = tmp; |
| 512 |
} |
| 513 |
} |
| 514 |
} |
| 515 |
/* insure eigenvector consistency (i.e., Jacobi can compute vectors that*/ |
| 516 |
/* are negative of one another (.707,.707,0) and (-.707,-.707,0). This can*/ |
| 517 |
/* reek havoc in hyperstreamline/other stuff. We will select the most*/ |
| 518 |
/* positive eigenvector.*/ |
| 519 |
ceil_half_n = (n >> 1) + (n & 1); |
| 520 |
for (j=0; j<n; j++) |
| 521 |
{ |
| 522 |
for (numPos=0, i=0; i<n; i++) |
| 523 |
{ |
| 524 |
if ( v[i][j] >= 0.0 ) |
| 525 |
{ |
| 526 |
numPos++; |
| 527 |
} |
| 528 |
} |
| 529 |
/* if ( numPos < ceil(RealType(n)/RealType(2.0)) )*/ |
| 530 |
if ( numPos < ceil_half_n) |
| 531 |
{ |
| 532 |
for(i=0; i<n; i++) |
| 533 |
{ |
| 534 |
v[i][j] *= -1.0; |
| 535 |
} |
| 536 |
} |
| 537 |
} |
| 538 |
|
| 539 |
if (n > 4) |
| 540 |
{ |
| 541 |
free(b); |
| 542 |
free(z); |
| 543 |
} |
| 544 |
return 1; |
| 545 |
} |
| 546 |
|
| 547 |
#undef MAT_ROTATE |
| 548 |
#undef MAX_ROTATIONS |
