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488 lines
16 KiB
C++
488 lines
16 KiB
C++
// This code is in the public domain -- castanyo@yahoo.es
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#include "Matrix.inl"
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#include "Vector.inl"
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#include "nvcore/Array.inl"
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#include <float.h>
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#if !NV_CC_MSVC && !NV_OS_ORBIS
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#include <alloca.h>
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#endif
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using namespace nv;
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// Given a matrix a[1..n][1..n], this routine replaces it by the LU decomposition of a rowwise
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// permutation of itself. a and n are input. a is output, arranged as in equation (2.3.14) above;
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// indx[1..n] is an output vector that records the row permutation effected by the partial
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// pivoting; d is output as -1 depending on whether the number of row interchanges was even
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// or odd, respectively. This routine is used in combination with lubksb to solve linear equations
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// or invert a matrix.
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static bool ludcmp(float **a, int n, int *indx, float *d)
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{
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const float TINY = 1.0e-20f;
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float * vv = (float*)alloca(sizeof(float) * n); // vv stores the implicit scaling of each row.
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*d = 1.0; // No row interchanges yet.
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for (int i = 0; i < n; i++) { // Loop over rows to get the implicit scaling information.
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float big = 0.0;
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for (int j = 0; j < n; j++) {
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big = max(big, fabsf(a[i][j]));
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}
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if (big == 0) {
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return false; // Singular matrix
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}
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// No nonzero largest element.
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vv[i] = 1.0f / big; // Save the scaling.
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}
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for (int j = 0; j < n; j++) { // This is the loop over columns of Crout's method.
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for (int i = 0; i < j; i++) { // This is equation (2.3.12) except for i = j.
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float sum = a[i][j];
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for (int k = 0; k < i; k++) sum -= a[i][k]*a[k][j];
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a[i][j] = sum;
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}
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int imax = -1;
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float big = 0.0; // Initialize for the search for largest pivot element.
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for (int i = j; i < n; i++) { // This is i = j of equation (2.3.12) and i = j+ 1 : : : N
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float sum = a[i][j]; // of equation (2.3.13).
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for (int k = 0; k < j; k++) {
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sum -= a[i][k]*a[k][j];
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}
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a[i][j]=sum;
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float dum = vv[i]*fabs(sum);
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if (dum >= big) {
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// Is the figure of merit for the pivot better than the best so far?
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big = dum;
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imax = i;
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}
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}
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nvDebugCheck(imax != -1);
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if (j != imax) { // Do we need to interchange rows?
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for (int k = 0; k < n; k++) { // Yes, do so...
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swap(a[imax][k], a[j][k]);
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}
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*d = -(*d); // ...and change the parity of d.
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vv[imax]=vv[j]; // Also interchange the scale factor.
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}
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indx[j]=imax;
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if (a[j][j] == 0.0) a[j][j] = TINY;
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// If the pivot element is zero the matrix is singular (at least to the precision of the
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// algorithm). For some applications on singular matrices, it is desirable to substitute
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// TINY for zero.
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if (j != n-1) { // Now, finally, divide by the pivot element.
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float dum = 1.0f / a[j][j];
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for (int i = j+1; i < n; i++) a[i][j] *= dum;
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}
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} // Go back for the next column in the reduction.
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return true;
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}
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// Solves the set of n linear equations Ax = b. Here a[1..n][1..n] is input, not as the matrix
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// A but rather as its LU decomposition, determined by the routine ludcmp. indx[1..n] is input
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// as the permutation vector returned by ludcmp. b[1..n] is input as the right-hand side vector
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// B, and returns with the solution vector X. a, n, and indx are not modified by this routine
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// and can be left in place for successive calls with different right-hand sides b. This routine takes
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// into account the possibility that b will begin with many zero elements, so it is efficient for use
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// in matrix inversion.
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static void lubksb(float **a, int n, int *indx, float b[])
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{
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int ii = 0;
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for (int i=0; i<n; i++) { // When ii is set to a positive value, it will become
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int ip = indx[i]; // the index of the first nonvanishing element of b. We now
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float sum = b[ip]; // do the forward substitution, equation (2.3.6). The
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b[ip] = b[i]; // only new wrinkle is to unscramble the permutation as we go.
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if (ii != 0) {
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for (int j = ii-1; j < i; j++) sum -= a[i][j]*b[j];
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}
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else if (sum != 0.0f) {
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ii = i+1; // A nonzero element was encountered, so from now on we
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}
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b[i] = sum; // will have to do the sums in the loop above.
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}
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for (int i=n-1; i>=0; i--) { // Now we do the backsubstitution, equation (2.3.7).
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float sum = b[i];
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for (int j = i+1; j < n; j++) {
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sum -= a[i][j]*b[j];
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}
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b[i] = sum/a[i][i]; // Store a component of the solution vector X.
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} // All done!
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}
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bool nv::solveLU(const Matrix & A, const Vector4 & b, Vector4 * x)
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{
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nvDebugCheck(x != NULL);
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float m[4][4];
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float *a[4] = {m[0], m[1], m[2], m[3]};
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int idx[4];
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float d;
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for (int y = 0; y < 4; y++) {
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for (int x = 0; x < 4; x++) {
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a[x][y] = A(x, y);
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}
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}
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// Create LU decomposition.
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if (!ludcmp(a, 4, idx, &d)) {
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// Singular matrix.
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return false;
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}
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// Init solution.
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*x = b;
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// Do back substitution.
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lubksb(a, 4, idx, x->component);
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return true;
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}
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// @@ Not tested.
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Matrix nv::inverseLU(const Matrix & A)
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{
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Vector4 Ai[4];
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solveLU(A, Vector4(1, 0, 0, 0), &Ai[0]);
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solveLU(A, Vector4(0, 1, 0, 0), &Ai[1]);
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solveLU(A, Vector4(0, 0, 1, 0), &Ai[2]);
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solveLU(A, Vector4(0, 0, 0, 1), &Ai[3]);
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return Matrix(Ai[0], Ai[1], Ai[2], Ai[3]);
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}
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bool nv::solveLU(const Matrix3 & A, const Vector3 & b, Vector3 * x)
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{
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nvDebugCheck(x != NULL);
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float m[3][3];
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float *a[3] = {m[0], m[1], m[2]};
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int idx[3];
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float d;
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for (int y = 0; y < 3; y++) {
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for (int x = 0; x < 3; x++) {
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a[x][y] = A(x, y);
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}
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}
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// Create LU decomposition.
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if (!ludcmp(a, 3, idx, &d)) {
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// Singular matrix.
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return false;
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}
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// Init solution.
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*x = b;
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// Do back substitution.
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lubksb(a, 3, idx, x->component);
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return true;
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}
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bool nv::solveLU(const Matrix2 & A, const Vector2 & b, Vector2 * x)
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{
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nvDebugCheck(x != NULL);
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float m[2][2];
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float *a[2] = {m[0], m[1]};
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int idx[2];
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float d;
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for (int y = 0; y < 2; y++) {
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for (int x = 0; x < 2; x++) {
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a[x][y] = A(x, y);
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}
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}
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// Create LU decomposition.
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if (!ludcmp(a, 2, idx, &d)) {
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// Singular matrix.
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return false;
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}
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// Init solution.
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*x = b;
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// Do back substitution.
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lubksb(a, 2, idx, x->component);
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return true;
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}
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bool nv::solveCramer(const Matrix & A, const Vector4 & b, Vector4 * x)
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{
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nvDebugCheck(x != NULL);
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*x = transform(inverseCramer(A), b);
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return true; // @@ Return false if determinant(A) == 0 !
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}
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bool nv::solveCramer(const Matrix3 & A, const Vector3 & b, Vector3 * x)
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{
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nvDebugCheck(x != NULL);
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const float det = A.determinant();
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if (equal(det, 0.0f)) { // @@ Use input epsilon.
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return false;
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}
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Matrix3 Ai = inverseCramer(A);
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*x = transform(Ai, b);
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return true;
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}
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bool nv::solveCramer(const Matrix2 & A, const Vector2 & b, Vector2 * x)
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{
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nvDebugCheck(x != NULL);
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const float det = A.determinant();
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if (equal(det, 0.0f)) { // @@ Use input epsilon.
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return false;
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}
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Matrix2 Ai = inverseCramer(A);
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*x = transform(Ai, b);
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return true;
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}
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// Inverse using gaussian elimination. From Jon's code.
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Matrix nv::inverse(const Matrix & m) {
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Matrix A = m;
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Matrix B(identity);
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int i, j, k;
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float max, t, det, pivot;
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det = 1.0;
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for (i=0; i<4; i++) { /* eliminate in column i, below diag */
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max = -1.;
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for (k=i; k<4; k++) /* find pivot for column i */
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if (fabs(A(k, i)) > max) {
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max = fabs(A(k, i));
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j = k;
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}
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if (max<=0.) return B; /* if no nonzero pivot, PUNT */
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if (j!=i) { /* swap rows i and j */
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for (k=i; k<4; k++)
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swap(A(i, k), A(j, k));
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for (k=0; k<4; k++)
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swap(B(i, k), B(j, k));
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det = -det;
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}
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pivot = A(i, i);
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det *= pivot;
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for (k=i+1; k<4; k++) /* only do elems to right of pivot */
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A(i, k) /= pivot;
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for (k=0; k<4; k++)
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B(i, k) /= pivot;
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/* we know that A(i, i) will be set to 1, so don't bother to do it */
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for (j=i+1; j<4; j++) { /* eliminate in rows below i */
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t = A(j, i); /* we're gonna zero this guy */
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for (k=i+1; k<4; k++) /* subtract scaled row i from row j */
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A(j, k) -= A(i, k)*t; /* (ignore k<=i, we know they're 0) */
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for (k=0; k<4; k++)
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B(j, k) -= B(i, k)*t;
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}
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}
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/*---------- backward elimination ----------*/
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for (i=4-1; i>0; i--) { /* eliminate in column i, above diag */
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for (j=0; j<i; j++) { /* eliminate in rows above i */
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t = A(j, i); /* we're gonna zero this guy */
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for (k=0; k<4; k++) /* subtract scaled row i from row j */
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B(j, k) -= B(i, k)*t;
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}
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}
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return B;
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}
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Matrix3 nv::inverse(const Matrix3 & m) {
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Matrix3 A = m;
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Matrix3 B(identity);
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int i, j, k;
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float max, t, det, pivot;
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det = 1.0;
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for (i=0; i<3; i++) { /* eliminate in column i, below diag */
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max = -1.;
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for (k=i; k<3; k++) /* find pivot for column i */
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if (fabs(A(k, i)) > max) {
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max = fabs(A(k, i));
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j = k;
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}
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if (max<=0.) return B; /* if no nonzero pivot, PUNT */
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if (j!=i) { /* swap rows i and j */
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for (k=i; k<3; k++)
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swap(A(i, k), A(j, k));
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for (k=0; k<3; k++)
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swap(B(i, k), B(j, k));
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det = -det;
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}
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pivot = A(i, i);
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det *= pivot;
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for (k=i+1; k<3; k++) /* only do elems to right of pivot */
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A(i, k) /= pivot;
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for (k=0; k<3; k++)
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B(i, k) /= pivot;
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/* we know that A(i, i) will be set to 1, so don't bother to do it */
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for (j=i+1; j<3; j++) { /* eliminate in rows below i */
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t = A(j, i); /* we're gonna zero this guy */
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for (k=i+1; k<3; k++) /* subtract scaled row i from row j */
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A(j, k) -= A(i, k)*t; /* (ignore k<=i, we know they're 0) */
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for (k=0; k<3; k++)
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B(j, k) -= B(i, k)*t;
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}
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}
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/*---------- backward elimination ----------*/
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for (i=3-1; i>0; i--) { /* eliminate in column i, above diag */
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for (j=0; j<i; j++) { /* eliminate in rows above i */
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t = A(j, i); /* we're gonna zero this guy */
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for (k=0; k<3; k++) /* subtract scaled row i from row j */
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B(j, k) -= B(i, k)*t;
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}
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}
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return B;
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}
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#if 0
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// Copyright (C) 1999-2004 Michael Garland.
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//
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// Permission is hereby granted, free of charge, to any person obtaining a
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// copy of this software and associated documentation files (the
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// "Software"), to deal in the Software without restriction, including
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// without limitation the rights to use, copy, modify, merge, publish,
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// distribute, and/or sell copies of the Software, and to permit persons
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// to whom the Software is furnished to do so, provided that the above
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// copyright notice(s) and this permission notice appear in all copies of
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// the Software and that both the above copyright notice(s) and this
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// permission notice appear in supporting documentation.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
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// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
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// OF THIRD PARTY RIGHTS. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR
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// HOLDERS INCLUDED IN THIS NOTICE BE LIABLE FOR ANY CLAIM, OR ANY SPECIAL
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// INDIRECT OR CONSEQUENTIAL DAMAGES, OR ANY DAMAGES WHATSOEVER RESULTING
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// FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT,
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// NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION
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// WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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//
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// Except as contained in this notice, the name of a copyright holder
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// shall not be used in advertising or otherwise to promote the sale, use
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// or other dealings in this Software without prior written authorization
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// of the copyright holder.
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// Matrix inversion code for 4x4 matrices using Gaussian elimination
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// with partial pivoting. This is a specialized version of a
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// procedure originally due to Paul Heckbert <ph@cs.cmu.edu>.
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//
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// Returns determinant of A, and B=inverse(A)
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// If matrix A is singular, returns 0 and leaves trash in B.
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//
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#define SWAP(a, b, t) {t = a; a = b; b = t;}
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double invert(Mat4& B, const Mat4& m)
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{
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Mat4 A = m;
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int i, j, k;
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double max, t, det, pivot;
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/*---------- forward elimination ----------*/
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for (i=0; i<4; i++) /* put identity matrix in B */
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for (j=0; j<4; j++)
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B(i, j) = (double)(i==j);
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det = 1.0;
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for (i=0; i<4; i++) { /* eliminate in column i, below diag */
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max = -1.;
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for (k=i; k<4; k++) /* find pivot for column i */
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if (fabs(A(k, i)) > max) {
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max = fabs(A(k, i));
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j = k;
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}
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if (max<=0.) return 0.; /* if no nonzero pivot, PUNT */
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if (j!=i) { /* swap rows i and j */
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for (k=i; k<4; k++)
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SWAP(A(i, k), A(j, k), t);
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for (k=0; k<4; k++)
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SWAP(B(i, k), B(j, k), t);
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det = -det;
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}
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pivot = A(i, i);
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det *= pivot;
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for (k=i+1; k<4; k++) /* only do elems to right of pivot */
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A(i, k) /= pivot;
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for (k=0; k<4; k++)
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B(i, k) /= pivot;
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/* we know that A(i, i) will be set to 1, so don't bother to do it */
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for (j=i+1; j<4; j++) { /* eliminate in rows below i */
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t = A(j, i); /* we're gonna zero this guy */
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for (k=i+1; k<4; k++) /* subtract scaled row i from row j */
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A(j, k) -= A(i, k)*t; /* (ignore k<=i, we know they're 0) */
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for (k=0; k<4; k++)
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B(j, k) -= B(i, k)*t;
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}
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}
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/*---------- backward elimination ----------*/
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for (i=4-1; i>0; i--) { /* eliminate in column i, above diag */
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for (j=0; j<i; j++) { /* eliminate in rows above i */
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t = A(j, i); /* we're gonna zero this guy */
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for (k=0; k<4; k++) /* subtract scaled row i from row j */
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B(j, k) -= B(i, k)*t;
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}
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}
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return det;
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}
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#endif // 0
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