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sysmat.cpp
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#include "sysmat.h"
/******************************************
* SOLVING A LINEAR MATRIX SYSTEM AX = B *
* with Gauss-Jordan method using full *
* pivoting at each step. During the pro- *
* cess, original AA and BB matrices are *
* destroyed to spare storage location. *
* --------------------------------------- *
* INPUTS: AA MATRIX N*N *
* BB MATRIX N*M *
* --------------------------------------- *
* OUTPUS: AA INVERSE OF AA N*N *
* DET DETERMINANT OF AA *
* BB SOLUTION MATRIX N*M *
* --------------------------------------- *
* NOTA - If M=0 inversion of AA matrix *
* only. *
******************************************/
void MATINV(int N,int M,MAT AA,MAT1 BB,REAL *DET) {
REAL PC[NMAX],PL[NMAX],CS[NMAX];
REAL PV,PAV,temp,TT;
int I,IK,J,JK,K;
// Initializations :
*DET= 1.0;
for (I=0; I<N; I++) {
PC[I]= 0.0;
PL[I]= 0.0;
CS[I]= 0.0;
}
// Main loop
for (K=0; K<N; K++) {
// Searching greatest pivot :
PV=AA[K][K];
IK=K;
JK=K;
PAV= ABS(PV);
for (I=K; I<N; I++)
for (J=K; J<N; J++) {
temp = ABS(AA[I][J]);
if (temp > PAV) {
PV=AA[I][J];
PAV= ABS(PV);
IK=I;
JK=J;
}
}
// Search terminated, the pivot is in location I=IK, J=JK.
// Memorizing pivot location:
PC[K]=JK;
PL[K]=IK;
// DETERMINANT DET is actualised
// If DET=0, ERROR MESSAGE and STOP
// Machine dependant EPSMACH equals here 1e-20
if (IK!=K) *DET=-*DET;
if (JK!=K) *DET=-*DET;
*DET=*DET*PV;
temp= ABS(*DET);
if (temp < MACH_EPS) {
// fprintf(fp2,"\n The determinant equals ZERO !!!\n");
return;
}
// POSITIONNING PIVOT IN K,K:
if(IK!=K)
for (I=0; I<N; I++) {
// EXCHANGE LINES IK and K of matrix AA:
TT=AA[IK][I];
AA[IK][I]=AA[K][I];
AA[K][I]=TT;
}
if(M!=0)
for (I=0; I<M; I++) {
TT=BB[IK][I];
BB[IK][I]=BB[K][I];
BB[K][I]=TT;
}
// PIVOT is at correct line
if(JK!=K)
for (I=0; I<N; I++) {
// EXCHANGE COLUMNS JK and K of matrix AA:
TT=AA[I][JK];
AA[I][JK]=AA[I][K];
AA[I][K]=TT;
}
// The PIVOT is at correct column.
// and is located in K,K.
// Column K of matrix AA is stored in CS vector
// then column K is set to zero.
for (I=0; I<N; I++) {
CS[I]=AA[I][K];
AA[I][K]= 0.0;
}
CS[K]= 0.0;
AA[K][K]= 1.0;
// Line K of matrix AA is modified:
temp= ABS(PV);
if(temp < MACH_EPS) {
// fprintf(fp2,"\n PIVOT TOO SMALL - STOP\n");
return;
}
for (I=0; I<N; I++)
AA[K][I]=AA[K][I]/PV;
if (M!=0)
for (I=0; I<M; I++)
BB[K][I]=BB[K][I]/PV;
// Other lines of matrix AA are modified:
for (J=0; J<N; J++) {
if (J==K) J++;
for (I=0; I<N; I++)
// Line J of matrix AA is modified:
AA[J][I]=AA[J][I]-CS[J]*AA[K][I];
if (M!=0)
for (I=0; I<M; I++)
BB[J][I]=BB[J][I]-CS[J]*BB[K][I];
}
// Line K is ready
} //End of K loop
// MATRIX AA INVERSION IS DONE - REARRANGEMENT OF MATRIX AA
// EXCHANGE LINES
for (I=N-1; I>=0; I--) {
IK=(int) PC[I];
if (IK==I) goto fin1;
// EXCHANGE LINES I and PC(I) of matrix AA:
for (J=0; J<N; J++) {
TT=AA[I][J];
AA[I][J]=AA[IK][J];
AA[IK][J]=TT;
}
if (M!=0)
for (J=0; J<M; J++) {
TT=BB[I][J];
BB[I][J]=BB[IK][J];
BB[IK][J]=TT;
}
// NO MORE EXCHANGE is NECESSARY
// Go to next line
fin1: ;
} // for i
// EXCHANGE COLUMNS:
for (J=N-1; J>=0; J--) {
JK=(int) PL[J];
if (JK==J) goto fin2;
// EXCHANGE COLUMNS J ET PL(J) of matrix AA:
for (I=0; I<N; I++) {
TT=AA[I][J];
AA[I][J]=AA[I][JK];
AA[I][JK]=TT;
}
fin2: ;
// NO MORE EXCHANGE is NECESSARY
// Go to next column.
}
// REARRANGEMENT TERMINATED.
return;
} //Matinv()
/******************************************
* MULTIPLICATION OF TWO SQUARE REAL *
* MATRICES *
* --------------------------------------- *
* INPUTS: A MATRIX N*N *
* B MATRIX N*N *
* N INTEGER *
* --------------------------------------- *
* OUTPUTS: C MATRIX N*N PRODUCT A*B *
* *
******************************************/
void MATMUL(MAT A,MAT B, MAT C, int N) {
REAL SUM;
int I,J,K;
for (I=0; I<N; I++)
for (J=0; J<N; J++) {
SUM= 0.0;
for (K=0; K<N; K++)
SUM=SUM+A[I][K]*B[K][J];
C[I][J]=SUM;
}
return;
}
/******************************************
* MULTIPLICATION OF TWO REAL MATRICES *
* --------------------------------------- *
* INPUTS: A MATRIX N*N *
* B MATRIX N*M *
* N INTEGER *
* M INTEGER *
* --------------------------------------- *
* OUTPUTS: C MATRIX N*M PRODUCT A*B *
* *
******************************************/
void MATMUL1(MAT A,MAT1 B, MAT1 C, int N, int M) {
REAL SUM;
int I,J,K;
for (I=0; I<N; I++)
for (J=0; J<M; J++) {
SUM= 0.0;
for (K=0; K<N; K++)
SUM=SUM+A[I][K]*B[K][J];
C[I][J]=SUM;
}
return;
}
#include <math.h>
#ifdef MAX
#undef MAX
#endif
#define MAX(a, b) ((a)>(b)?(a):(b))
#define MATSIZE 3
static double hypot2(double x, double y) {
return sqrt(x*x+y*y);
}
// Symmetric Householder reduction to tridiagonal form.
static void tred2(double V[MATSIZE][MATSIZE], double d[MATSIZE], double e[MATSIZE]) {
// This is derived from the Algol procedures tred2 by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
for (int j = 0; j < MATSIZE; j++) {
d[j] = V[MATSIZE-1][j];
}
// Householder reduction to tridiagonal form.
for (int i = MATSIZE-1; i > 0; i--) {
// Scale to avoid under/overflow.
double scale = 0.0;
double h = 0.0;
for (int k = 0; k < i; k++) {
scale = scale + fabs(d[k]);
}
if (scale == 0.0) {
e[i] = d[i-1];
for (int j = 0; j < i; j++) {
d[j] = V[i-1][j];
V[i][j] = 0.0;
V[j][i] = 0.0;
}
} else {
// Generate Householder vector.
for (int k = 0; k < i; k++) {
d[k] /= scale;
h += d[k] * d[k];
}
double f = d[i-1];
double g = sqrt(h);
if (f > 0) {
g = -g;
}
e[i] = scale * g;
h = h - f * g;
d[i-1] = f - g;
for (int j = 0; j < i; j++) {
e[j] = 0.0;
}
// Apply similarity transformation to remaining columns.
for (int j = 0; j < i; j++) {
f = d[j];
V[j][i] = f;
g = e[j] + V[j][j] * f;
for (int k = j+1; k <= i-1; k++) {
g += V[k][j] * d[k];
e[k] += V[k][j] * f;
}
e[j] = g;
}
f = 0.0;
for (int j = 0; j < i; j++) {
e[j] /= h;
f += e[j] * d[j];
}
double hh = f / (h + h);
for (int j = 0; j < i; j++) {
e[j] -= hh * d[j];
}
for (int j = 0; j < i; j++) {
f = d[j];
g = e[j];
for (int k = j; k <= i-1; k++) {
V[k][j] -= (f * e[k] + g * d[k]);
}
d[j] = V[i-1][j];
V[i][j] = 0.0;
}
}
d[i] = h;
}
// Accumulate transformations.
for (int i = 0; i < MATSIZE-1; i++) {
V[MATSIZE-1][i] = V[i][i];
V[i][i] = 1.0;
double h = d[i+1];
if (h != 0.0) {
for (int k = 0; k <= i; k++) {
d[k] = V[k][i+1] / h;
}
for (int j = 0; j <= i; j++) {
double g = 0.0;
for (int k = 0; k <= i; k++) {
g += V[k][i+1] * V[k][j];
}
for (int k = 0; k <= i; k++) {
V[k][j] -= g * d[k];
}
}
}
for (int k = 0; k <= i; k++) {
V[k][i+1] = 0.0;
}
}
for (int j = 0; j < MATSIZE; j++) {
d[j] = V[MATSIZE-1][j];
V[MATSIZE-1][j] = 0.0;
}
V[MATSIZE-1][MATSIZE-1] = 1.0;
e[0] = 0.0;
}
// Symmetric tridiagonal QL algorithm.
static void tql2(double V[MATSIZE][MATSIZE], double d[MATSIZE], double e[MATSIZE]) {
// This is derived from the Algol procedures tql2, by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
for (int i = 1; i < MATSIZE; i++) {
e[i-1] = e[i];
}
e[MATSIZE-1] = 0.0;
double f = 0.0;
double tst1 = 0.0;
double eps = pow(2.0,-52.0);
for (int l = 0; l < MATSIZE; l++) {
// Find small subdiagonal element
tst1 = MAX(tst1,fabs(d[l]) + fabs(e[l]));
int m = l;
while (m < MATSIZE) {
if (fabs(e[m]) <= eps*tst1) {
break;
}
m++;
}
// If m == l, d[l] is an eigenvalue,
// otherwise, iterate.
if (m > l) {
int iter = 0;
do {
iter = iter + 1; // (Could check iteration count here.)
// Compute implicit shift
double g = d[l];
double p = (d[l+1] - g) / (2.0 * e[l]);
double r = hypot2(p,1.0);
if (p < 0) {
r = -r;
}
d[l] = e[l] / (p + r);
d[l+1] = e[l] * (p + r);
double dl1 = d[l+1];
double h = g - d[l];
for (int i = l+2; i < MATSIZE; i++) {
d[i] -= h;
}
f = f + h;
// Implicit QL transformation.
p = d[m];
double c = 1.0;
double c2 = c;
double c3 = c;
double el1 = e[l+1];
double s = 0.0;
double s2 = 0.0;
for (int i = m-1; i >= l; i--) {
c3 = c2;
c2 = c;
s2 = s;
g = c * e[i];
h = c * p;
r = hypot2(p,e[i]);
e[i+1] = s * r;
s = e[i] / r;
c = p / r;
p = c * d[i] - s * g;
d[i+1] = h + s * (c * g + s * d[i]);
// Accumulate transformation.
for (int k = 0; k < MATSIZE; k++) {
h = V[k][i+1];
V[k][i+1] = s * V[k][i] + c * h;
V[k][i] = c * V[k][i] - s * h;
}
}
p = -s * s2 * c3 * el1 * e[l] / dl1;
e[l] = s * p;
d[l] = c * p;
// Check for convergence.
} while (fabs(e[l]) > eps*tst1);
}
d[l] = d[l] + f;
e[l] = 0.0;
}
// Sort eigenvalues and corresponding vectors.
for (int i = 0; i < MATSIZE-1; i++) {
int k = i;
double p = d[i];
for (int j = i+1; j < MATSIZE; j++) {
if (d[j] < p) {
k = j;
p = d[j];
}
}
if (k != i) {
d[k] = d[i];
d[i] = p;
for (int j = 0; j < MATSIZE; j++) {
p = V[j][i];
V[j][i] = V[j][k];
V[j][k] = p;
}
}
}
}
void eigen_decomposition(double A[MATSIZE][MATSIZE], double V[MATSIZE][MATSIZE], double d[MATSIZE]) {
double e[MATSIZE];
for (int i = 0; i < MATSIZE; i++) {
for (int j = 0; j < MATSIZE; j++) {
V[i][j] = A[i][j];
}
}
tred2(V, d, e);
tql2(V, d, e);
}