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LSM9DS1-Adafruit-AHRS/calibration/magnetoCalibrator.c

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// This version of Magneto is authored by S. James Remington
// magneto 1.4 magnetometer/accelerometer calibration code
// from http://sailboatinstruments.blogspot.com/2011/08/improved-magnetometer-calibration.html
// tested and works with Code::Blocks 10.05 through 20.03
// Command line version slightly modified from original, sjames_remington@gmail.com
// Now includes option to reject outliers
// comma separated ASCII input data file expected, three columns x, y, z
#include <stdio.h>
#include <math.h>
#include <malloc.h>
#include <string.h>
#include <float.h>
// Forward declarations of mymathlib.com routines
void Multiply_Self_Transpose(double*, double*, int, int);
void Get_Submatrix(double*, int, int, double*, int, int, int);
int Choleski_LU_Decomposition(double*, int);
int Choleski_LU_Inverse(double*, int);
void Multiply_Matrices(double*, double*, int, int, double*, int);
void Identity_Matrix(double*, int);
int Hessenberg_Form_Elementary(double*, double*, int);
static void Hessenberg_Elementary_Transform(double*, double*, int[], int);
void Copy_Vector(double*, double*, int);
int QR_Hessenberg_Matrix(double*, double*, double[], double[], int, int);
static void One_Real_Eigenvalue(double[], double[], double[], int, double);
static void Two_Eigenvalues(double*, double*, double[], double[], int, int, double);
static void Update_Row(double*, double, double, int, int);
static void Update_Column(double*, double, double, int, int);
static void Update_Transformation(double*, double, double, int, int);
static void Double_QR_Iteration(double*, double*, int, int, int, double*, int);
static void Product_and_Sum_of_Shifts(double*, int, int, double*, double*, double*, int);
static int Two_Consecutive_Small_Subdiagonal(double*, int, int, int, double, double);
static void Double_QR_Step(double*, int, int, int, double, double, double*, int);
static void BackSubstitution(double*, double[], double[], int);
static void BackSubstitute_Real_Vector(double*, double[], double[], int, double, int);
static void BackSubstitute_Complex_Vector(double*, double[], double[], int, double, int);
static void Calculate_Eigenvectors(double*, double*, double[], double[], int);
static void Complex_Division(double, double, double, double, double*, double*);
void Transpose_Square_Matrix(double*, int);
//
// *** CODE STARTS HERE ***
//
int main()
{
char buf[120];
double *D, *S, *C, *S11, *S12, *S12t, *S22, *S22_1, *S22a, *S22b, *SS, *E, *d, *U, *SSS;
double *eigen_real, *eigen_imag, *v1, *v2, *v, *Q, *Q_1, *B, *QB, J, hmb, *SSSS;
int *p;
int i, j, index;
double maxval, norm, btqb, *eigen_real3, *eigen_imag3, *Dz, *vdz, *SQ, *A_1, hm, norm1, norm2, norm3;
double x, y, z, x2, nxsrej, xs, xave;
double *raw; //raw obs
int nlines = 0, ngood=0; //count measurements
char filename[64];
FILE *fp, *fp2;
// this version reads comma separated data
printf("\r\nMagneto 1.4 2/10/2021\r\nInput .csv file? ");
scanf("%s",&filename);
fp = fopen(filename, "r");
if (fp == NULL) {printf("file not found"); return 0;}
// get the number of lines in the file
while(fgets(buf, 100, fp) != NULL)
nlines++;
rewind(fp);
//
// calculate mean (norm) and standard deviation for possible outlier rejection
//
xs=0;
xave=0;
for( i = 0; i < nlines; i++)
{
fgets(buf, 100, fp);
index=sscanf(buf, "%lf,%lf,%lf", &x, &y, &z);
printf("%3d %6.0f %6.0f %6.0f\n",i,x,y,z);
x2 = x*x + y*y + z*z;
xs += x2;
xave += sqrt(x2);
}
xave = xave/nlines; //mean vector length
xs = sqrt(xs/nlines -(xave*xave)); //std. dev.
rewind(fp);
// summarize statistics, give user opportunity to reject outlying measurements
printf("\r\nAverage magnitude (default Hm) and sigma of %d vectors = %6.1lf, %6.1lf\r\n",nlines,xave,xs);
printf("\r\nReject outliers? (0 or d, reject if > d*sigma from mean) ");
scanf("%lf",&nxsrej);
printf("Rejection level selected: %4.1f\r\n ",nxsrej);
// scan file again
ngood = nlines; //count good
if (nxsrej > 0) {
printf("\r\nRejecting measurements if abs(vector_length-average)/(std. dev.) > %5.1f\r\n",nxsrej);
// outlier rejection, count remaining lines
for( i = 0; i < nlines; i++)
{
fgets(buf, 100, fp);
index=sscanf(buf, "%lf,%lf,%lf", &x, &y, &z);
x2 = sqrt(x*x + y*y + z*z); //vector length
x2 =fabs(x2 - xave)/xs; //standard deviations from mean
if (x2 > nxsrej) {
ngood--;
printf("reject #%d, %8.1f %8.1f %8.1f\n",i, x,y,z);
}
}
printf("\r\nNumber of measurements to be rejected: %d, accepted: %d\r\n",nlines-ngood,ngood);
rewind(fp);
}
// third time through!
// allocate array space for accepted measurements
D = (double*)malloc(10 * ngood * sizeof(double));
raw = (double*)malloc(3 * ngood * sizeof(double));
j = 0; //array index for good measurements
// printf("\r\nAccepted measurements (file index, internal index, ...)\r\n");
for( i = 0; i < nlines; i++)
{
fgets(buf, 100, fp);
index=sscanf(buf, "%lf,%lf,%lf", &x, &y, &z); //comma separated values
x2 = sqrt(x*x + y*y + z*z); //vector length
x2 = fabs(x2 - xave)/xs; //standard deviation from mean
if ((nxsrej == 0) || (x2 <= nxsrej)) {
// accepted measurement
// printf("%d, %d: %6.1f %6.1f %6.1f\r\n",i,j,x,y,z);
raw[3*j] = x;
raw[3*j+1] = y;
raw[3*j+2] = z;
D[j] = x * x;
D[ngood+j] = y * y;
D[ngood*2+j] = z * z;
D[ngood*3+j] = 2.0 * y * z;
D[ngood*4+j] = 2.0 * x * z;
D[ngood*5+j] = 2.0 * x * y;
D[ngood*6+j] = 2.0 * x;
D[ngood*7+j] = 2.0 * y;
D[ngood*8+j] = 2.0 * z;
D[ngood*9+j] = 1.0;
j++; //count good measurements
}
}
fclose(fp);
printf("\r\n%3d measurements processed, expected %d\r\n",j,ngood);
if (j != ngood) {printf("Internal indexing error!\r\n"); return 0; }
nlines = ngood; //number to process
printf("\r\nExpected norm of local field vector Hm? (Enter 0 for default %8.1f) ",xave);
scanf("%lf",&hm);
if(hm == 0.0) hm=xave;
printf("\r\nSet Hm = %8.1f\r\n",hm);
// allocate memory for matrix S
S = (double*)malloc(10 * 10 * sizeof(double));
Multiply_Self_Transpose(S, D, 10, nlines);
// Create pre-inverted constraint matrix C
C = (double*)malloc(6 * 6 * sizeof(double));
C[0] = 0.0; C[1] = 0.5; C[2] = 0.5; C[3] = 0.0; C[4] = 0.0; C[5] = 0.0;
C[6] = 0.5; C[7] = 0.0; C[8] = 0.5; C[9] = 0.0; C[10] = 0.0; C[11] = 0.0;
C[12] = 0.5; C[13] = 0.5; C[14] = 0.0; C[15] = 0.0; C[16] = 0.0; C[17] = 0.0;
C[18] = 0.0; C[19] = 0.0; C[20] = 0.0; C[21] = -0.25; C[22] = 0.0; C[23] = 0.0;
C[24] = 0.0; C[25] = 0.0; C[26] = 0.0; C[27] = 0.0; C[28] = -0.25; C[29] = 0.0;
C[30] = 0.0; C[31] = 0.0; C[32] = 0.0; C[33] = 0.0; C[34] = 0.0; C[35] = -0.25;
S11 = (double*)malloc(6 * 6 * sizeof(double));
Get_Submatrix(S11, 6, 6, S, 10, 0, 0);
S12 = (double*)malloc(6 * 4 * sizeof(double));
Get_Submatrix(S12, 6, 4, S, 10, 0, 6);
S12t = (double*)malloc(4 * 6 * sizeof(double));
Get_Submatrix(S12t, 4, 6, S, 10, 6, 0);
S22 = (double*)malloc(4 * 4 * sizeof(double));
Get_Submatrix(S22, 4, 4, S, 10, 6, 6);
S22_1 = (double*)malloc(4 * 4 * sizeof(double));
for(i = 0; i < 16; i++)
S22_1[i] = S22[i];
Choleski_LU_Decomposition(S22_1, 4);
Choleski_LU_Inverse(S22_1, 4);
// Calculate S22a = S22_1 * S12t 4*6 = 4x4 * 4x6 C = AB
S22a = (double*)malloc(4 * 6 * sizeof(double));
Multiply_Matrices(S22a, S22_1, 4, 4, S12t, 6);
// Then calculate S22b = S12 * S22a ( 6x6 = 6x4 * 4x6)
S22b = (double*)malloc(6 * 6 * sizeof(double));
Multiply_Matrices(S22b, S12, 6, 4, S22a, 6);
// Calculate SS = S11 - S22b
SS = (double*)malloc(6 * 6 * sizeof(double));
for(i = 0; i < 36; i++)
SS[i] = S11[i] - S22b[i];
E = (double*)malloc(6 * 6 * sizeof(double));
Multiply_Matrices(E, C, 6, 6, SS, 6);
SSS = (double*)malloc(6 * 6 * sizeof(double));
Hessenberg_Form_Elementary(E, SSS, 6);
eigen_real = (double*)malloc(6 * sizeof(double));
eigen_imag = (double*)malloc(6 * sizeof(double));
QR_Hessenberg_Matrix(E, SSS, eigen_real, eigen_imag, 6, 100);
index = 0;
maxval = eigen_real[0];
for(i = 1; i < 6; i++)
{
if(eigen_real[i] > maxval)
{
maxval = eigen_real[i];
index = i;
}
}
v1 = (double*)malloc(6 * sizeof(double));
v1[0] = SSS[index];
v1[1] = SSS[index+6];
v1[2] = SSS[index+12];
v1[3] = SSS[index+18];
v1[4] = SSS[index+24];
v1[5] = SSS[index+30];
// normalize v1
norm = sqrt(v1[0] * v1[0] + v1[1] * v1[1] + v1[2] * v1[2] + v1[3] * v1[3] + v1[4] * v1[4] + v1[5] * v1[5]);
v1[0] /= norm;
v1[1] /= norm;
v1[2] /= norm;
v1[3] /= norm;
v1[4] /= norm;
v1[5] /= norm;
if(v1[0] < 0.0)
{
v1[0] = -v1[0];
v1[1] = -v1[1];
v1[2] = -v1[2];
v1[3] = -v1[3];
v1[4] = -v1[4];
v1[5] = -v1[5];
}
// Calculate v2 = S22a * v1 ( 4x1 = 4x6 * 6x1)
v2 = (double*)malloc(4 * sizeof(double));
Multiply_Matrices(v2, S22a, 4, 6, v1, 1);
v = (double*)malloc(10 * sizeof(double));
v[0] = v1[0];
v[1] = v1[1];
v[2] = v1[2];
v[3] = v1[3];
v[4] = v1[4];
v[5] = v1[5];
v[6] = -v2[0];
v[7] = -v2[1];
v[8] = -v2[2];
v[9] = -v2[3];
Q = (double*)malloc(3 * 3 * sizeof(double));
Q[0] = v[0];
Q[1] = v[5];
Q[2] = v[4];
Q[3] = v[5];
Q[4] = v[1];
Q[5] = v[3];
Q[6] = v[4];
Q[7] = v[3];
Q[8] = v[2];
U = (double*)malloc(3 * sizeof(double));
U[0] = v[6];
U[1] = v[7];
U[2] = v[8];
Q_1 = (double*)malloc(3 * 3 * sizeof(double));
for(i = 0; i < 9; i++)
Q_1[i] = Q[i];
Choleski_LU_Decomposition(Q_1, 3);
Choleski_LU_Inverse(Q_1, 3);
// Calculate B = Q-1 * U ( 3x1 = 3x3 * 3x1)
B = (double*)malloc(3 * sizeof(double));
Multiply_Matrices(B, Q_1, 3, 3, U, 1);
B[0] = -B[0]; // x-axis combined bias
B[1] = -B[1]; // y-axis combined bias
B[2] = -B[2]; // z-axis combined bias
// for(i = 0; i < 3; i++)
printf("\r\nCombined bias vector B:\r\n");
printf("%8.2lf %8.2lf %8.2lf \r\n", B[0],B[1],B[2]);
// First calculate QB = Q * B ( 3x1 = 3x3 * 3x1)
QB = (double*)malloc(3 * sizeof(double));
Multiply_Matrices(QB, Q, 3, 3, B, 1);
// Then calculate btqb = BT * QB ( 1x1 = 1x3 * 3x1)
Multiply_Matrices(&btqb, B, 1, 3, QB, 1);
// Calculate hmb = sqrt(btqb - J).
J = v[9];
hmb = sqrt(btqb - J);
// Calculate SQ, the square root of matrix Q
SSSS = (double*)malloc(3 * 3 * sizeof(double));
Hessenberg_Form_Elementary(Q, SSSS, 3);
eigen_real3 = (double*)malloc(3 * sizeof(double));
eigen_imag3 = (double*)malloc(3 * sizeof(double));
QR_Hessenberg_Matrix(Q, SSSS, eigen_real3, eigen_imag3, 3, 100);
// normalize eigenvectors
norm1 = sqrt(SSSS[0] * SSSS[0] + SSSS[3] * SSSS[3] + SSSS[6] * SSSS[6]);
SSSS[0] /= norm1;
SSSS[3] /= norm1;
SSSS[6] /= norm1;
norm2 = sqrt(SSSS[1] * SSSS[1] + SSSS[4] * SSSS[4] + SSSS[7] * SSSS[7]);
SSSS[1] /= norm2;
SSSS[4] /= norm2;
SSSS[7] /= norm2;
norm3 = sqrt(SSSS[2] * SSSS[2] + SSSS[5] * SSSS[5] + SSSS[8] * SSSS[8]);
SSSS[2] /= norm3;
SSSS[5] /= norm3;
SSSS[8] /= norm3;
Dz = (double*)malloc(3 * 3 * sizeof(double));
for(i = 0; i < 9; i++)
Dz[i] = 0.0;
Dz[0] = sqrt(eigen_real3[0]);
Dz[4] = sqrt(eigen_real3[1]);
Dz[8] = sqrt(eigen_real3[2]);
vdz = (double*)malloc(3 * 3 * sizeof(double));
Multiply_Matrices(vdz, SSSS, 3, 3, Dz, 3);
Transpose_Square_Matrix(SSSS, 3);
SQ = (double*)malloc(3 * 3 * sizeof(double));
Multiply_Matrices(SQ, vdz, 3, 3, SSSS, 3);
// hm = 0.569;
A_1 = (double*)malloc(3 * 3 * sizeof(double));
for(i = 0; i < 9; i++)
A_1[i] = SQ[i] * hm / hmb;
printf("\r\nCorrection matrix Ainv, using Hm=%8.1f:\r\n",hm);
for(i = 0; i < 3; i++)
printf("%9.5lf %9.5lf %9.5lf\r\n", A_1[i*3], A_1[i*3+1], A_1[i*3+2]);
printf("\r\nWhere Hcalc = Ainv*(H-B)\r\n");
printf("\r\nCode initialization statements...\r\n");
printf("\r\n float B[3]\r\n {%8.2lf,%8.2lf,%8.2lf};\r\n",B[0],B[1],B[2]);
printf("\r\n float Ainv[3][3]\r\n {{%9.5lf,%9.5lf,%9.5lf},\r\n",A_1[0],A_1[1],A_1[2]);
printf(" {%9.5lf,%9.5lf,%9.5lf},\r\n",A_1[3],A_1[4],A_1[5]);
printf(" {%9.5lf,%9.5lf,%9.5lf}};\r\n",A_1[6],A_1[7],A_1[8]);
// dump corrected measurements if desired
printf(" output filename for corrected values (csv) ");
scanf("%s",&filename);
if(strlen(filename) > 1) {
fp2 = fopen(filename, "w");
float xc,yc,zc;
xs=0;
for(i = 0; i<nlines; i++){
x = raw[3*i] -B[0];
y = raw[3*i+1]-B[1];
z = raw[3*i+2]-B[2];
xc = A_1[0]*x + A_1[1]*y + A_1[2]*z;
yc = A_1[3]*x + A_1[4]*y + A_1[5]*z;
zc = A_1[6]*x + A_1[7]*y + A_1[8]*z;
xs += xc*xc + yc*yc + zc*zc;
fprintf(fp2,"%8.2f,%8.2f,%8.2f\n",xc,yc,zc);
}
fclose(fp2);
}
printf("\r\nRMS corrected vector length %9.6f\n",sqrt(xs/nlines));
printf("\r\nRMS residual %9.6f\n",1.0-sqrt(xs/nlines)/hm);
free(D);
free(S);
free(C);
free(S11);
free(S12);
free(S12t);
free(S22);
free(S22_1);
free(S22a);
free(S22b);
free(SS);
free(E);
free(U);
free(SSS);
free(eigen_real);
free(eigen_imag);
free(v1);
free(v2);
free(v);
free(Q);
free(Q_1);
free(B);
free(QB);
free(SSSS);
free(eigen_real3);
free(eigen_imag3);
free(Dz);
free(vdz);
free(SQ);
free(A_1);
return 0;
}
// Place here the source code of all the routines which have been forward-declared, available at
// http://www.mymathlib.com/matrices/
////////////////////////////////////////////////////////////////////////////////
// File: copy_vector.c //
// Routine(s): //
// Copy_Vector //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Copy_Vector(double *d, double *s, int n) //
// //
// Description: //
// Copy the n dimensional vector s(source) to the n dimensional //
// vector d(destination). The memory locations of the source and //
// destination vectors must not overlap, otherwise the results //
// are installation dependent. //
// //
// Arguments: //
// double *d Pointer to the first element of the destination vector d. //
// double *s Pointer to the first element of the source vector s. //
// int n The number of elements of the source / destination vectors.//
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// double v[N], vd[N]; //
// //
// (your code to initialize the vector v) //
// //
// Copy_Vector(vd, v, N); //
// printf(" Vector vd is \n"); //
////////////////////////////////////////////////////////////////////////////////
#include <string.h> // required for memcpy()
void Copy_Vector(double *d, double *s, int n)
{
memcpy(d, s, sizeof(double) * n);
}
////////////////////////////////////////////////////////////////////////////////
// File: multiply_self_transpose.c //
// Routine(s): //
// Multiply_Self_Transpose //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Multiply_Self_Transpose(double *C, double *A, int nrows, int ncols ) //
// //
// Description: //
// Post multiply an nrows x ncols matrix A by its transpose. The result //
// is an nrows x nrows square symmetric matrix C, i.e. C = A A', where ' //
// denotes the transpose. //
// The matrix C should be declared as double C[nrows][nrows] in the //
// calling routine. The memory allocated to C should not include any //
// memory allocated to A. //
// //
// Arguments: //
// double *C Pointer to the first element of the matrix C. //
// double *A Pointer to the first element of the matrix A. //
// int nrows The number of rows of matrix A. //
// int ncols The number of columns of the matrices A. //
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// #define M //
// double A[M][N], C[M][M]; //
// //
// (your code to initialize the matrix A) //
// //
// Multiply_Self_Transpose(&C[0][0], &A[0][0], M, N); //
// printf("The matrix C = AA ' is \n"); ... //
////////////////////////////////////////////////////////////////////////////////
void Multiply_Self_Transpose(double *C, double *A, int nrows, int ncols)
{
int i,j,k;
double *pA;
double *p_A = A;
double *pB;
double *pCdiag = C;
double *pC = C;
double *pCt;
for (i = 0; i < nrows; pCdiag += nrows + 1, p_A = pA, i++) {
pC = pCdiag;
pCt = pCdiag;
pB = p_A;
for (j = i; j < nrows; pC++, pCt += nrows, j++) {
pA = p_A;
*pC = 0.0;
for (k = 0; k < ncols; k++) *pC += *(pA++) * *(pB++);
*pCt = *pC;
}
}
}
////////////////////////////////////////////////////////////////////////////////
// File: get_submatrix.c //
// Routine(s): //
// Get_Submatrix //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Get_Submatrix(double *S, int mrows, int mcols, //
// double *A, int ncols, int row, int col) //
// //
// Description: //
// Copy the mrows and mcols of the nrows x ncols matrix A starting with //
// A[row][col] to the submatrix S. //
// Note that S should be declared double S[mrows][mcols] in the calling //
// routine. //
// //
// Arguments: //
// double *S Destination address of the submatrix. //
// int mrows The number of rows of the matrix S. //
// int mcols The number of columns of the matrix S. //
// double *A Pointer to the first element of the matrix A[nrows][ncols]//
// int ncols The number of columns of the matrix A. //
// int row The row of A corresponding to the first row of S. //
// int col The column of A corresponding to the first column of S. //
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// #define M //
// #define NB //
// #define MB //
// double A[M][N], B[MB][NB]; //
// int row, col; //
// //
// (your code to set the matrix A, the row number row and column number //
// col) //
// //
// if ( (row >= 0) && (col >= 0) && ((row + MB) < M) && ((col + NB) < N) )//
// Get_Submatrix(&B[0][0], MB, NB, &A[0][0], N, row, col); //
// printf("The submatrix B is \n"); ... } //
////////////////////////////////////////////////////////////////////////////////
#include <string.h> // required for memcpy()
void Get_Submatrix(double *S, int mrows, int mcols,
double *A, int ncols, int row, int col)
{
int number_of_bytes = sizeof(double) * mcols;
for (A += row * ncols + col; mrows > 0; A += ncols, S+= mcols, mrows--)
memcpy(S, A, number_of_bytes);
}
////////////////////////////////////////////////////////////////////////////////
// File: choleski.c //
// Contents: //
// Choleski_LU_Decomposition //
// Choleski_LU_Solve //
// Choleski_LU_Inverse //
// //
// Required Externally Defined Routines: //
// Lower_Triangular_Solve //
// Lower_Triangular_Inverse //
// Upper_Triangular_Solve //
////////////////////////////////////////////////////////////////////////////////
#include <math.h> // required for sqrt()
// Required Externally Defined Routines
int Lower_Triangular_Solve(double *L, double B[], double x[], int n);
int Lower_Triangular_Inverse(double *L, int n);
int Upper_Triangular_Solve(double *U, double B[], double x[], int n);
////////////////////////////////////////////////////////////////////////////////
// int Choleski_LU_Decomposition(double *A, int n) //
// //
// Description: //
// This routine uses Choleski's method to decompose the n x n positive //
// definite symmetric matrix A into the product of a lower triangular //
// matrix L and an upper triangular matrix U equal to the transpose of L. //
// The original matrix A is replaced by L and U with L stored in the //
// lower triangular part of A and the transpose U in the upper triangular //
// part of A. The original matrix A is therefore destroyed. //
// //
// Choleski's decomposition is performed by evaluating, in order, the //
// following pair of expressions for k = 0, ... ,n-1 : //
// L[k][k] = sqrt( A[k][k] - ( L[k][0] ^ 2 + ... + L[k][k-1] ^ 2 ) ) //
// L[i][k] = (A[i][k] - (L[i][0]*L[k][0] + ... + L[i][k-1]*L[k][k-1])) //
// / L[k][k] //
// and subsequently setting //
// U[k][i] = L[i][k], for i = k+1, ... , n-1. //
// //
// After performing the LU decomposition for A, call Choleski_LU_Solve //
// to solve the equation Ax = B or call Choleski_LU_Inverse to calculate //
// the inverse of A. //
// //
// Arguments: //
// double *A On input, the pointer to the first element of the matrix //
// A[n][n]. On output, the matrix A is replaced by the lower //
// and upper triangular Choleski factorizations of A. //
// int n The number of rows and/or columns of the matrix A. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - The matrix A is not positive definite symmetric (within //
// working accuracy). //
// //
// Example: //
// #define N //
// double A[N][N]; //
// //
// (your code to initialize the matrix A) //
// err = Choleski_LU_Decomposition((double *) A, N); //
// if (err < 0) printf(" Matrix A is singular\n"); //
// else { printf(" The LLt decomposition of A is \n"); //
// ... //
////////////////////////////////////////////////////////////////////////////////
// //
int Choleski_LU_Decomposition(double *A, int n)
{
int i, k, p;
double *p_Lk0; // pointer to L[k][0]
double *p_Lkp; // pointer to L[k][p]
double *p_Lkk; // pointer to diagonal element on row k.
double *p_Li0; // pointer to L[i][0]
double reciprocal;
for (k = 0, p_Lk0 = A; k < n; p_Lk0 += n, k++) {
// Update pointer to row k diagonal element.
p_Lkk = p_Lk0 + k;
// Calculate the difference of the diagonal element in row k
// from the sum of squares of elements row k from column 0 to
// column k-1.
for (p = 0, p_Lkp = p_Lk0; p < k; p_Lkp += 1, p++)
*p_Lkk -= *p_Lkp * *p_Lkp;
// If diagonal element is not positive, return the error code,
// the matrix is not positive definite symmetric.
if ( *p_Lkk <= 0.0 ) return -1;
// Otherwise take the square root of the diagonal element.
*p_Lkk = sqrt( *p_Lkk );
reciprocal = 1.0 / *p_Lkk;
// For rows i = k+1 to n-1, column k, calculate the difference
// between the i,k th element and the inner product of the first
// k-1 columns of row i and row k, then divide the difference by
// the diagonal element in row k.
// Store the transposed element in the upper triangular matrix.
p_Li0 = p_Lk0 + n;
for (i = k + 1; i < n; p_Li0 += n, i++) {
for (p = 0; p < k; p++)
*(p_Li0 + k) -= *(p_Li0 + p) * *(p_Lk0 + p);
*(p_Li0 + k) *= reciprocal;
*(p_Lk0 + i) = *(p_Li0 + k);
}
}
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// int Choleski_LU_Solve(double *LU, double *B, double *x, int n) //
// //
// Description: //
// This routine uses Choleski's method to solve the linear equation //
// Ax = B. This routine is called after the matrix A has been decomposed //
// into a product of a lower triangular matrix L and an upper triangular //
// matrix U which is the transpose of L. The matrix A is the product LU. //
// The solution proceeds by solving the linear equation Ly = B for y and //
// subsequently solving the linear equation Ux = y for x. //
// //
// Arguments: //
// double *LU Pointer to the first element of the matrix whose elements //
// form the lower and upper triangular matrix factors of A. //
// double *B Pointer to the column vector, (n x 1) matrix, B //
// double *x Solution to the equation Ax = B. //
// int n The number of rows and/or columns of the matrix LU. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - The matrix L is singular. //
// //
// Example: //
// #define N //
// double A[N][N], B[N], x[N]; //
// //
// (your code to create matrix A and column vector B) //
// err = Choleski_LU_Decomposition(&A[0][0], N); //
// if (err < 0) printf(" Matrix A is singular\n"); //
// else { //
// err = Choleski_LU_Solve(&A[0][0], B, x, n); //
// if (err < 0) printf(" Matrix A is singular\n"); //
// else printf(" The solution is \n"); //
// ... //
// } //
////////////////////////////////////////////////////////////////////////////////
// //
int Choleski_LU_Solve(double *LU, double B[], double x[], int n)
{
// Solve the linear equation Ly = B for y, where L is a lower
// triangular matrix.
if ( Lower_Triangular_Solve(LU, B, x, n) < 0 ) return -1;
// Solve the linear equation Ux = y, where y is the solution
// obtained above of Ly = B and U is an upper triangular matrix.
return Upper_Triangular_Solve(LU, x, x, n);
}
////////////////////////////////////////////////////////////////////////////////
// int Choleski_LU_Inverse(double *LU, int n) //
// //
// Description: //
// This routine uses Choleski's method to find the inverse of the matrix //
// A. This routine is called after the matrix A has been decomposed //
// into a product of a lower triangular matrix L and an upper triangular //
// matrix U which is the transpose of L. The matrix A is the product of //
// the L and U. Upon completion, the inverse of A is stored in LU so //
// that the matrix LU is destroyed. //
// //
// Arguments: //
// double *LU On input, the pointer to the first element of the matrix //
// whose elements form the lower and upper triangular matrix //
// factors of A. On output, the matrix LU is replaced by the //
// inverse of the matrix A equal to the product of L and U. //
// int n The number of rows and/or columns of the matrix LU. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - The matrix L is singular. //
// //
// Example: //
// #define N //
// double A[N][N], B[N], x[N]; //
// //
// (your code to create matrix A and column vector B) //
// err = Choleski_LU_Decomposition(&A[0][0], N); //
// if (err < 0) printf(" Matrix A is singular\n"); //
// else { //
// err = Choleski_LU_Inverse(&A[0][0], n); //
// if (err < 0) printf(" Matrix A is singular\n"); //
// else printf(" The inverse is \n"); //
// ... //
// } //
////////////////////////////////////////////////////////////////////////////////
// //
int Choleski_LU_Inverse(double *LU, int n)
{
int i, j, k;
double *p_i, *p_j, *p_k;
double sum;
if ( Lower_Triangular_Inverse(LU, n) < 0 ) return -1;
// Premultiply L inverse by the transpose of L inverse.
for (i = 0, p_i = LU; i < n; i++, p_i += n) {
for (j = 0, p_j = LU; j <= i; j++, p_j += n) {
sum = 0.0;
for (k = i, p_k = p_i; k < n; k++, p_k += n)
sum += *(p_k + i) * *(p_k + j);
*(p_i + j) = sum;
*(p_j + i) = sum;
}
}
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// File: multiply_matrices.c //
// Routine(s): //
// Multiply_Matrices //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Multiply_Matrices(double *C, double *A, int nrows, int ncols, //
// double *B, int mcols) //
// //
// Description: //
// Post multiply the nrows x ncols matrix A by the ncols x mcols matrix //
// B to form the nrows x mcols matrix C, i.e. C = A B. //
// The matrix C should be declared as double C[nrows][mcols] in the //
// calling routine. The memory allocated to C should not include any //
// memory allocated to A or B. //
// //
// Arguments: //
// double *C Pointer to the first element of the matrix C. //
// double *A Pointer to the first element of the matrix A. //
// int nrows The number of rows of the matrices A and C. //
// int ncols The number of columns of the matrices A and the //
// number of rows of the matrix B. //
// double *B Pointer to the first element of the matrix B. //
// int mcols The number of columns of the matrices B and C. //
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// #define M //
// #define NB //
// double A[M][N], B[N][NB], C[M][NB]; //
// //
// (your code to initialize the matrices A and B) //
// //
// Multiply_Matrices(&C[0][0], &A[0][0], M, N, &B[0][0], NB); //
// printf("The matrix C is \n"); ... //
////////////////////////////////////////////////////////////////////////////////
void Multiply_Matrices(double *C, double *A, int nrows, int ncols,
double *B, int mcols)
{
double *pB;
double *p_B;
int i,j,k;
for (i = 0; i < nrows; A += ncols, i++)
for (p_B = B, j = 0; j < mcols; C++, p_B++, j++) {
pB = p_B;
*C = 0.0;
for (k = 0; k < ncols; pB += mcols, k++)
*C += *(A+k) * *pB;
}
}
////////////////////////////////////////////////////////////////////////////////
// File: identity_matrix.c //
// Routine(s): //
// Identity_Matrix //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Identity_Matrix(double *A, int n) //
// //
// Description: //
// Set the square n x n matrix A equal to the identity matrix, i.e. //
// A[i][j] = 0 if i != j and A[i][i] = 1. //
// //
// Arguments: //
// double *A Pointer to the first element of the matrix A. //
// int n The number of rows and columns of the matrix A. //
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// double A[N][N]; //
// //
// Identity_Matrix(&A[0][0], N); //
// printf("The matrix A is \n"); ... //
////////////////////////////////////////////////////////////////////////////////
void Identity_Matrix(double *A, int n)
{
int i,j;
for (i = 0; i < n - 1; i++) {
*A++ = 1.0;
for (j = 0; j < n; j++) *A++ = 0.0;
}
*A = 1.0;
}
////////////////////////////////////////////////////////////////////////////////
// File: hessenberg_elementary.c //
// Routine(s): //
// Hessenberg_Form_Elementary //
// Hessenberg_Elementary_Transform //
////////////////////////////////////////////////////////////////////////////////
#include <stdlib.h> // required for malloc() and free()
#include <math.h> // required for fabs()
// Required Externally Defined Routines
void Interchange_Rows(double *A, int row1, int row2, int ncols);
void Interchange_Columns(double *A, int col1, int col2, int nrows, int ncols);
void Identity_Matrix(double *A, int n);
void Copy_Vector(double *d, double *s, int n);
// Internally Defined Routines
static void Hessenberg_Elementary_Transform(double* H, double *S,
int perm[], int n);
////////////////////////////////////////////////////////////////////////////////
// int Hessenberg_Form_Elementary(double *A, double *S, int n) //
// //
// Description: //
// This program transforms the square matrix A to a similar matrix in //
// Hessenberg form by a multiplying A on the right by a sequence of //
// elementary transformations and on the left by the sequence of inverse //
// transformations. //
// Def: Two matrices A and B are said to be similar if there exists a //
// nonsingular matrix S such that A S = S B. //
// Def A Hessenberg matrix is the sum of an upper triangular matrix and //
// a matrix all of whose components are 0 except possibly on its //
// subdiagonal. A Hessenberg matrix is sometimes said to be almost //
// upper triangular. //
// The algorithm proceeds by successivly selecting columns j = 0,...,n-3 //
// and then assuming that columns 0, ..., j-1 have been reduced to Hessen-//
// berg form, for rows j+1 to n-1, select that row j' for which |a[j'][j]|//
// is a maximum and interchange rows j+1 and j' and columns j+1 and j'. //
// Then for each i = j+2 to n-1, let x = a[i][j] / a[j+1][j] subtract //
// x * row j+1 from row i and add x * column i to column j+1. //
// //
// Arguments: //
// double *A On input a pointer to the first element of the matrix //
// A[n][n]. The matrix A is replaced with the matrix H, //
// a matrix in Hessenberg form similar to A. //
// double *S On output the transform such that A S = S H. //
// The matrix S should be dimensioned at least n x n in the //
// calling routine. //
// int n The number of rows or columns of the matrix A. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - Not enough memory //
// //
// Example: //
// #define N //
// double A[N][N], S[N][N]; //
// //
// (your code to create the matrix A) //
// if (Hessenberg_Form_Elementary(&A[0][0], (double*)S, N ) < 0) { //
// printf("Not enough memory\n"); exit(0); //
// } //
// //
////////////////////////////////////////////////////////////////////////////////
// //
int Hessenberg_Form_Elementary(double *A, double* S, int n)
{
int i, j, k, col, row;
int* perm;
double *p_row, *pS_row;
double max;
double s;
double *pA, *pB, *pC, *pS;
// n x n matrices for which n <= 2 are already in Hessenberg form
if (n <= 1) { *S = 1.0; return 0; }
if (n == 2) { *S++ = 1.0; *S++ = 0.0; *S++ = 1.0; *S = 0.0; return 0; }
// Allocate working memory
perm = (int*) malloc(n * sizeof(int));
if (perm == NULL) return -1; // not enough memory
// For each column use Elementary transformations
// to zero the entries below the subdiagonal.
p_row = A + n;
pS_row = S + n;
for (col = 0; col < (n - 2); p_row += n, pS_row += n, col++) {
// Find the row in column "col" with maximum magnitude where
// row >= col + 1.
row = col + 1;
perm[row] = row;
for (pA = p_row + col, max = 0.0, i = row; i < n; pA += n, i++)
if (fabs(*pA) > max) { perm[row] = i; max = fabs(*pA); }
// If perm[row] != row, then interchange row "row" and row
// perm[row] and interchange column "row" and column perm[row].
if ( perm[row] != row ) {
Interchange_Rows(A, row, perm[row], n);
Interchange_Columns(A, row, perm[row], n, n);
}
// Zero out the components lying below the subdiagonal.
pA = p_row + n;
pS = pS_row + n;
for (i = col + 2; i < n; pA += n, pS += n, i++) {
s = *(pA + col) / *(p_row + col);
for (j = 0; j < n; j++)
*(pA + j) -= *(p_row + j) * s;
*(pS + col) = s;
for (j = 0, pB = A + col + 1, pC = A + i; j < n; pB +=n, pC += n, j++)
*pB += s * *pC;
}
}
pA = A + n + n;
pS = S + n + n;
for (i = 2; i < n; pA += n, pS += n, i++) Copy_Vector(pA, pS, i - 1);
Hessenberg_Elementary_Transform(A, S, perm, n);
free(perm);
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// static void Hessenberg_Elementary_Transform(double* H, double *S, //
// int perm[], int n) //
// //
// Description: //
// Given a n x n matrix A, let H be the matrix in Hessenberg form similar //
// to A, i.e. A S = S H. If v is an eigenvector of H with eigenvalue z, //
// i.e. Hv = zv, then ASv = SHv = z Sv, i.e. Sv is the eigenvector of A //
// with corresponding eigenvalue z. //
// This routine returns S where S is the similarity transformation such //
// that A S = S H. //
// //
// Arguments: //
// double* H On input a matrix in Hessenberg form with transformation //
// elements stored below the subdiagonal part. //
// On output the matrix in Hessenberg form with elements //
// below the subdiagonal zeroed out. //
// double* S On output, the transformations matrix such that //
// A S = S H. //
// int perm[] Array of row/column interchanges. //
// int n The order of the matrices H and S. //
// //
// Return Values: //
// void //
// //
////////////////////////////////////////////////////////////////////////////////
// //
static void Hessenberg_Elementary_Transform(double *H, double* S, int perm[],
int n)
{
int i, j;
double *pS, *pH;
double x;
Identity_Matrix(S, n);
for (i = n - 2; i >= 1; i--) {
pH = H + n * (i + 1);
pS = S + n * (i + 1);
for (j = i + 1; j < n; pH += n, pS += n, j++) {
*(pS + i) = *(pH + i - 1);
*(pH + i - 1) = 0.0;
}
if (perm[i] != i) {
pS = S + n * i;
pH = S + n * perm[i];
for (j = i; j < n; j++) {
*(pS + j) = *(pH + j);
*(pH + j) = 0.0;
}
*(pH + i) = 1.0;
}
}
}
////////////////////////////////////////////////////////////////////////////////
// File: qr_hessenberg_matrix.c //
// Routine(s): //
// QR_Hessenberg_Matrix //
////////////////////////////////////////////////////////////////////////////////
#include <math.h> // required for fabs() and sqrt()
#include <float.h> // required for DBL_EPSILON
// Internally Defined Routines
static void One_Real_Eigenvalue(double Hrow[], double eigen_real[],
double eigen_imag[], int row, double shift);
static void Two_Eigenvalues(double *H, double *S, double eigen_real[],
double eigen_imag[], int n, int k, double t);
static void Update_Row(double *Hrow, double cos, double sin, int n, int k);
static void Update_Column(double* H, double cos, double sin, int n, int k);
static void Update_Transformation(double *S, double cos, double sin,
int n, int k);
static void Double_QR_Iteration(double *H, double *S, int row, int min_row,
int n, double* shift, int iteration);
static void Product_and_Sum_of_Shifts(double *H, int n, int max_row,
double* shift, double *trace, double *det, int iteration);
static int Two_Consecutive_Small_Subdiagonal(double* H, int min_row,
int max_row, int n, double trace, double det);
static void Double_QR_Step(double *H, int min_row, int max_row, int min_col,
double trace, double det, double *S, int n);
static void Complex_Division(double x, double y, double u, double v,
double* a, double* b);
static void BackSubstitution(double *H, double eigen_real[],
double eigen_imag[], int n);
static void BackSubstitute_Real_Vector(double *H, double eigen_real[],
double eigen_imag[], int row, double zero_tolerance, int n);
static void BackSubstitute_Complex_Vector(double *H, double eigen_real[],
double eigen_imag[], int row, double zero_tolerance, int n);
static void Calculate_Eigenvectors(double *H, double *S, double eigen_real[],
double eigen_imag[], int n);
////////////////////////////////////////////////////////////////////////////////
// int QR_Hessenberg_Matrix( double *H, double *S, double eigen_real[], //
// double eigen_imag[], int n, int max_iteration_count) //
// //
// Description: //
// This program calculates the eigenvalues and eigenvectors of a matrix //
// in Hessenberg form. This routine is adapted from the routine 'hql2' //
// appearing in 'Handbook for Automatic Computation, vol 2: Linear //
// Algebra' published by Springer Verlag (1971) edited by Wilkinson and //
// Reinsch, Contribution II/15 Eigenvectors of Real and Complex Matrices //
// by LR and QR Triangularizations by Peters and Wilkinson. //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the real n x n matrix H in upper//
// Hessenberg form. //
// double *S //
// If H is the primary data matrix, the matrix S should be set //
// to the identity n x n identity matrix on input. If H is //
// derived from an n x n matrix A, then S should be the //
// transformation matrix such that AS = SH. On output, the i-th //
// column of S corresponds to the i-th eigenvalue if that eigen- //
// value is real and the i-th and (i+1)-st columns of S correspond //
// to the i-th eigenvector with the real part in the i-th column //
// and positive imaginary part in the (i+1)-st column if that //
// eigenvalue is complex with positive imaginary part. The //
// eigenvector corresponding to the eigenvalue with negative //
// imaginary part is the complex conjugate of the eigenvector //
// corresponding to the complex conjugate of the eigenvalue. //
// If on input, S was the identity matrix, then the columns are //
// the eigenvectors of H as described, if S was a transformation //
// matrix so that AS = SH, then the columns of S are the //
// eigenvectors of A as described. //
// double eigen_real[] //
// Upon return, eigen_real[i] will contain the real part of the //
// i-th eigenvalue. //
// double eigen_imag[] //
// Upon return, eigen_ima[i] will contain the imaginary part of //
// the i-th eigenvalue. //
// int n //
// The number of rows or columns of the upper Hessenberg matrix A. //
// int max_iteration_count //
// The maximum number of iterations to try to find an eigenvalue //
// before quitting. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - Unable to find an eigenvalue within 'max_iteration_count' //
// iterations. //
// //
// Example: //
// #define N //
// #define MAX_ITERATION_COUNT //
// double H[N][N], S[N][N], eigen_real[N], eigen_imag[N]; //
// int k; //
// //
// (code to initialize H[N][N] and S[N][N]) //
// k = QR_Hessenberg_Matrix( (double*)H, (double*)S, eigen_real, //
// eigen_imag, N, MAX_ITERATION_COUNT); //
// if (k < 0) {printf("Failed"); exit(1);} //
////////////////////////////////////////////////////////////////////////////////
// //
int QR_Hessenberg_Matrix( double *H, double *S, double eigen_real[],
double eigen_imag[], int n, int max_iteration_count)
{
int i;
int row;
int iteration;
int found_eigenvalue;
double shift = 0.0;
double* pH;
for ( row = n - 1; row >= 0; row--) {
found_eigenvalue = 0;
for (iteration = 1; iteration <= max_iteration_count; iteration++) {
// Search for small subdiagonal element
for (i = row, pH = H + row * n; i > 0; i--, pH -= n)
if (fabs(*(pH + i - 1 )) <= DBL_EPSILON *
( fabs(*(pH - n + i - 1)) + fabs(*(pH + i)) ) ) break;
// If the subdiagonal element on row "row" is small, then
// that row element is an eigenvalue. If the subdiagonal
// element on row "row-1" is small, then the eigenvalues
// of the 2x2 diagonal block consisting rows "row-1" and
// "row" are eigenvalues. Otherwise perform a double QR
// iteration.
switch(row - i) {
case 0: // One real eigenvalue
One_Real_Eigenvalue(pH, eigen_real, eigen_imag, i, shift);
found_eigenvalue = 1;
break;
case 1: // Either two real eigenvalues or a complex pair
row--;
Two_Eigenvalues(H, S, eigen_real, eigen_imag, n, row, shift);
found_eigenvalue = 1;
break;
default:
Double_QR_Iteration(H, S, i, row, n, &shift, iteration);
}
if (found_eigenvalue) break;
}
if (iteration > max_iteration_count) return -1;
}
BackSubstitution(H, eigen_real, eigen_imag, n);
Calculate_Eigenvectors(H, S, eigen_real, eigen_imag, n);
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// static void One_Real_Eigenvalue( double Hrow[], double eigen_real[], //
// double eigen_imag[], int row, double shift) //
// //
// Arguments: //
// double Hrow[] //
// Pointer to the row "row" of the matrix in Hessenberg form. //
// double eigen_real[] //
// Array of the real parts of the eigenvalues. //
// double eigen_imag[] //
// Array of the imaginary parts of the eigenvalues. //
// int row //
// The row to which the pointer Hrow[] points of the matrix H. //
// double shift //
// The cumulative exceptional shift of the diagonal elements of //
// the matrix H. //
////////////////////////////////////////////////////////////////////////////////
// //
static void One_Real_Eigenvalue(double Hrow[], double eigen_real[],
double eigen_imag[], int row, double shift)
{
Hrow[row] += shift;
eigen_real[row] = Hrow[row];
eigen_imag[row] = 0.0;
}
////////////////////////////////////////////////////////////////////////////////
// static void Two_Eigenvalues( double *H, double *S, double eigen_real[], //
// double eigen_imag[], int n, int row, double shift) //
// //
// Description: //
// Given the 2x2 matrix A = (a[i][j]), the characteristic equation is: //
// x^2 - Tr(A) x + Det(A) = 0, where Tr(A) = a[0][0] + a[1][1] and //
// Det(A) = a[0][0] * a[1][1] - a[0][1] * a[1][0]. //
// The solution for the eigenvalues x are: //
// x1 = (Tr(A) + sqrt( (Tr(A))^2 + 4 * a[0][1] * a[1][0] ) / 2 and //
// x2 = (Tr(A) - sqrt( (Tr(A))^2 + 4 * a[0][1] * a[1][0] ) / 2. //
// Let p = (a[0][0] - a[1][1]) / 2 and q = p^2 - a[0][1] * a[1][0], then //
// x1 = a[1][1] + p [+|-] sqrt(q) and x2 = a[0][0] + a[1][1] - x1. //
// Choose the sign [+|-] to be the sign of p. //
// If q > 0.0, then both roots are real and the transformation //
// | cos sin | | a[0][0] a[0][1] | | cos -sin | //
// |-sin cos | | a[1][0] a[1][1] | | sin cos | //
// where sin = a[1][0] / r, cos = ( p + sqrt(q) ) / r, where r > 0 is //
// determined sin^2 + cos^2 = 1 transforms the matrix A to an upper //
// triangular matrix with x1 the upper diagonal element and x2 the lower.//
// If q < 0.0, then both roots form a complex conjugate pair. //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the matrix in Hessenberg form. //
// double *S //
// Pointer to the first element of the transformation matrix. //
// double eigen_real[] //
// Array of the real parts of the eigenvalues. //
// double eigen_imag[] //
// Array of the imaginary parts of the eigenvalues. //
// int n //
// The dimensions of the matrix H and S. //
// int row //
// The upper most row of the block diagonal 2 x 2 submatrix of H. //
// double shift //
// The cumulative exceptional shift of the diagonal elements of //
// the matrix H. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Two_Eigenvalues(double *H, double* S, double eigen_real[],
double eigen_imag[], int n, int row, double shift)
{
double p, q, x, discriminant, r;
double cos, sin;
double *Hrow = H + n * row;
double *Hnextrow = Hrow + n;
int nextrow = row + 1;
p = 0.5 * (Hrow[row] - Hnextrow[nextrow]);
x = Hrow[nextrow] * Hnextrow[row];
discriminant = p * p + x;
Hrow[row] += shift;
Hnextrow[nextrow] += shift;
if (discriminant > 0.0) { // pair of real roots
q = sqrt(discriminant);
if (p < 0.0) q = p - q; else q += p;
eigen_real[row] = Hnextrow[nextrow] + q;
eigen_real[nextrow] = Hnextrow[nextrow] - x / q;
eigen_imag[row] = 0.0;
eigen_imag[nextrow] = 0.0;
r = sqrt(Hnextrow[row]*Hnextrow[row] + q * q);
sin = Hnextrow[row] / r;
cos = q / r;
Update_Row(Hrow, cos, sin, n, row);
Update_Column(H, cos, sin, n, row);
Update_Transformation(S, cos, sin, n, row);
}
else { // pair of complex roots
eigen_real[nextrow] = eigen_real[row] = Hnextrow[nextrow] + p;
eigen_imag[row] = sqrt(fabs(discriminant));
eigen_imag[nextrow] = -eigen_imag[row];
}
}
////////////////////////////////////////////////////////////////////////////////
// static void Update_Row(double *Hrow, double cos, double sin, int n, //
// int row) //
// //
// Description: //
// Update rows 'row' and 'row + 1' using the rotation matrix: //
// | cos sin | //
// |-sin cos |. //
// I.e. multiply the matrix H on the left by the identity matrix with //
// the 2x2 diagonal block starting at row 'row' replaced by the above //
// 2x2 rotation matrix. //
// //
// Arguments: //
// double Hrow[] //
// Pointer to the row "row" of the matrix in Hessenberg form. //
// double cos //
// Cosine of the rotation angle. //
// double sin //
// Sine of the rotation angle. //
// int n //
// The dimension of the matrix H. //
// int row //
// The row to which the pointer Hrow[] points of the matrix H //
// in Hessenberg form. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Update_Row(double *Hrow, double cos, double sin, int n, int row)
{
double x;
double *Hnextrow = Hrow + n;
int i;
for (i = row; i < n; i++) {
x = Hrow[i];
Hrow[i] = cos * x + sin * Hnextrow[i];
Hnextrow[i] = cos * Hnextrow[i] - sin * x;
}
}
////////////////////////////////////////////////////////////////////////////////
// static void Update_Column(double* H, double cos, double sin, int n, //
// int col) //
// //
// Description: //
// Update columns 'col' and 'col + 1' using the rotation matrix: //
// | cos -sin | //
// | sin cos |. //
// I.e. multiply the matrix H on the right by the identity matrix with //
// the 2x2 diagonal block starting at row 'col' replaced by the above //
// 2x2 rotation matrix. //
// //
// Arguments: //
// double *H //
// Pointer to the matrix in Hessenberg form. //
// double cos //
// Cosine of the rotation angle. //
// double sin //
// Sine of the rotation angle. //
// int n //
// The dimension of the matrix H. //
// int col //
// The left-most column of the matrix H to update. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Update_Column(double* H, double cos, double sin, int n, int col)
{
double x;
int i;
int next_col = col + 1;
for (i = 0; i <= next_col; i++, H += n) {
x = H[col];
H[col] = cos * x + sin * H[next_col];
H[next_col] = cos * H[next_col] - sin * x;
}
}
////////////////////////////////////////////////////////////////////////////////
// static void Update_Transformation(double *S, double cos, double sin, //
// int n, int k) //
// //
// Description: //
// Update columns 'k' and 'k + 1' using the rotation matrix: //
// | cos -sin | //
// | sin cos |. //
// I.e. multiply the matrix S on the right by the identity matrix with //
// the 2x2 diagonal block starting at row 'k' replaced by the above //
// 2x2 rotation matrix. //
// //
// Arguments: //
// double *S //
// Pointer to the row "row" of the matrix in Hessenberg form. //
// double cos //
// Pointer to the first element of the matrix in Hessenberg form. //
// double sin //
// Pointer to the first element of the transformation matrix. //
// int n //
// The dimensions of the matrix H and S. //
// int k //
// The row to which the pointer Hrow[] points of the matrix H. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Update_Transformation(double *S, double cos, double sin,
int n, int k)
{
double x;
int i;
int k1 = k + 1;
for (i = 0; i < n; i++, S += n) {
x = S[k];
S[k] = cos * x + sin * S[k1];
S[k1] = cos * S[k1] - sin * x;
}
}
////////////////////////////////////////////////////////////////////////////////
// static void Double_QR_Iteration(double *H, double *S, int min_row, //
// int max_row, int n, double* shift, int iteration) //
// //
// Description: //
// Calculate the trace and determinant of the 2x2 matrix: //
// | H[k-1][k-1] H[k-1][k] | //
// | H[k][k-1] H[k][k] | //
// unless iteration = 0 (mod 10) in which case increment the shift and //
// decrement the first k elements of the matrix H, then fudge the trace //
// and determinant by trace = 3/2( |H[k][k-1]| + |H[k-1][k-2]| and //
// det = 4/9 trace^2. //
// //
// Arguments: //
// double *H //
// Pointer to the matrix H in Hessenberg form. //
// double *S //
// Pointer to the transformation matrix S. //
// int min_row //
// The top-most row in which the off-diagonal element of H is //
// negligible. If no such row exists, then min_row = 0. //
// int max_row //
// The maximum row of the block 2 x 2 diagonal matrix used to //
// estimate the two eigenvalues for the two implicit shifts. //
// int n //
// The dimensions of the matrix H and S. //
// double *shift //
// The cumulative exceptional shift of the diagonal elements of //
// the matrix H. //
// int iteration //
// Current iteration count. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Double_QR_Iteration(double *H, double *S, int min_row, int max_row,
int n, double* shift, int iteration)
{
int k;
double trace, det;
Product_and_Sum_of_Shifts(H, n, max_row, shift, &trace, &det, iteration);
k = Two_Consecutive_Small_Subdiagonal(H, min_row, max_row, n, trace, det);
Double_QR_Step(H, min_row, max_row, k, trace, det, S, n);
}
////////////////////////////////////////////////////////////////////////////////
// static void Product_and_Sum_of_Shifts(double *H, int n, int max_row, //
// double* shift, double *trace, double *det, int iteration) //
// //
// Description: //
// Calculate the trace and determinant of the 2x2 matrix: //
// | H[k-1][k-1] H[k-1][k] | //
// | H[k][k-1] H[k][k] | //
// unless iteration = 0 (mod 10) in which case increment the shift and //
// decrement the first k elements of the matrix H, then fudge the trace //
// and determinant by trace = 3/2( |H[k][k-1]| + |H[k-1][k-2]| and //
// det = 4/9 trace^2. //
// //
// Arguments: //
// double *H //
// Pointer to the matrix H in Hessenberg form. //
// int n //
// The dimension of the matrix H. //
// int max_row //
// The maximum row of the block 2 x 2 diagonal matrix used to //
// estimate the two eigenvalues for the two implicit shifts. //
// double *shift //
// The cumulative exceptional shift of the diagonal elements of //
// the matrix H. Modified if an exceptional shift occurs. //
// double *trace //
// Returns the trace of the 2 x 2 block diagonal matrix starting //
// at the row/column max_row-1. For an exceptional shift, the //
// trace is set as described above. //
// double *det //
// Returns the determinant of the 2 x 2 block diagonal matrix //
// starting at the row/column max_row-1. For an exceptional shift,//
// the determinant is set as described above. //
// int iteration //
// Current iteration count. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Product_and_Sum_of_Shifts(double *H, int n, int max_row,
double* shift, double *trace, double *det, int iteration)
{
double *pH = H + max_row * n;
double *p_aux;
int i;
int min_col = max_row - 1;
if ( ( iteration % 10 ) == 0 ) {
*shift += pH[max_row];
for (i = 0, p_aux = H; i <= max_row; p_aux += n, i++)
p_aux[i] -= pH[max_row];
p_aux = pH - n;
*trace = fabs(pH[min_col]) + fabs(p_aux[min_col - 1]);
*det = *trace * *trace;
*trace *= 1.5;
}
else {
p_aux = pH - n;
*trace = p_aux[min_col] + pH[max_row];
*det = p_aux[min_col] * pH[max_row] - p_aux[max_row] * pH[min_col];
}
};
////////////////////////////////////////////////////////////////////////////////
// static int Two_Consecutive_Small_Subdiagonal(double* H, int min_row, //
// int max_row, int n, double trace, double det) //
// //
// Description: //
// To reduce the amount of computation in Francis' double QR step search //
// for two consecutive small subdiagonal elements from row nn to row m, //
// where m < nn. //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the matrix in Hessenberg form. //
// int min_row //
// The row in which to end the search (search is from upwards). //
// int max_row //
// The row in which to begin the search. //
// int n //
// The dimension of H. //
// double trace //
// The trace of the lower 2 x 2 block diagonal matrix. //
// double det //
// The determinant of the lower 2 x 2 block diagonal matrix. //
// //
// Return Value: //
// Row with negligible subdiagonal element or min_row if none found. //
////////////////////////////////////////////////////////////////////////////////
// //
static int Two_Consecutive_Small_Subdiagonal(double* H, int min_row,
int max_row, int n, double trace, double det)
{
double x, y ,z, s;
double* pH;
int i, k;
for (k = max_row - 2, pH = H + k * n; k >= min_row; pH -= n, k--) {
x = (pH[k] * ( pH[k] - trace ) + det) / pH[n+k] + pH[k+1];
y = pH[k] + pH[n+k+1] - trace;
z = pH[n + n + k + 1];
s = fabs(x) + fabs(y) + fabs(z);
x /= s;
y /= s;
z /= s;
if (k == min_row) break;
if ( (fabs(pH[k-1]) * (fabs(y) + fabs(z)) ) <=
DBL_EPSILON * fabs(x) *
(fabs(pH[k-1-n]) + fabs(pH[k]) + fabs(pH[n + k + 1])) ) break;
}
for (i = k+2, pH = H + i * n; i <= max_row; pH += n, i++) pH[i-2] = 0.0;
for (i = k+3, pH = H + i * n; i <= max_row; pH += n, i++) pH[i-3] = 0.0;
return k;
};
////////////////////////////////////////////////////////////////////////////////
// static void Double_QR_Step(double *H, int min_row, int max_row, //
// int min_col, double *S, int n) //
// //
// Description: //
// Perform Francis' double QR step from rows 'min_row' to 'max_row' //
// and columns 'min_col' to 'max_row'. //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the matrix in Hessenberg form. //
// int min_row //
// The row in which to begin. //
// int max_row //
// The row in which to end. //
// int min_col //
// The column in which to begin. //
// double trace //
// The trace of the lower 2 x 2 block diagonal matrix. //
// double det //
// The determinant of the lower 2 x 2 block diagonal matrix. //
// double *S //
// Pointer to the first element of the transformation matrix. //
// int n //
// The dimensions of H and S. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Double_QR_Step(double *H, int min_row, int max_row, int min_col,
double trace, double det, double *S, int n)
{
double s, x, y, z;
double a, b, c;
double *pH;
double *tH;
double *pS;
int i,j,k;
int last_test_row_col = max_row - 1;
k = min_col;
pH = H + min_col * n;
a = (pH[k] * ( pH[k] - trace ) + det) / pH[n+k] + pH[k+1];
b = pH[k] + pH[n+k+1] - trace;
c = pH[n + n + k + 1];
s = fabs(a) + fabs(b) + fabs(c);
a /= s;
b /= s;
c /= s;
for (; k <= last_test_row_col; k++, pH += n) {
if ( k > min_col ) {
c = (k == last_test_row_col) ? 0.0 : pH[n + n + k - 1];
x = fabs(pH[k-1]) + fabs(pH[n + k - 1]) + fabs(c);
if ( x == 0.0 ) continue;
a = pH[k - 1] / x;
b = pH[n + k - 1] / x;
c /= x;
}
s = sqrt( a * a + b * b + c * c );
if (a < 0.0) s = -s;
if ( k > min_col ) pH[k-1] = -s * x;
else if (min_row != min_col) pH[k-1] = -pH[k-1];
a += s;
x = a / s;
y = b / s;
z = c / s;
b /= a;
c /= a;
// Update rows k, k+1, k+2
for (j = k; j < n; j++) {
a = pH[j] + b * pH[n+j];
if ( k != last_test_row_col ) {
a += c * pH[n + n + j];
pH[n + n + j] -= a * z;
}
pH[n + j] -= a * y;
pH[j] -= a * x;
}
// Update column k+1
j = k + 3;
if (j > max_row) j = max_row;
for (i = 0, tH = H; i <= j; i++, tH += n) {
a = x * tH[k] + y * tH[k+1];
if ( k != last_test_row_col ) {
a += z * tH[k+2];
tH[k+2] -= a * c;
}
tH[k+1] -= a * b;
tH[k] -= a;
}
// Update transformation matrix
for (i = 0, pS = S; i < n; pS += n, i++) {
a = x * pS[k] + y * pS[k+1];
if ( k != last_test_row_col ) {
a += z * pS[k+2];
pS[k+2] -= a * c;
}
pS[k+1] -= a * b;
pS[k] -= a;
}
};
}
////////////////////////////////////////////////////////////////////////////////
// static void BackSubstitution(double *H, double eigen_real[], //
// double eigen_imag[], int n) //
// //
// Description: //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the matrix in Hessenberg form. //
// double eigen_real[] //
// The real part of an eigenvalue. //
// double eigen_imag[] //
// The imaginary part of an eigenvalue. //
// int n //
// The dimension of H, eigen_real, and eigen_imag. //
////////////////////////////////////////////////////////////////////////////////
// //
static void BackSubstitution(double *H, double eigen_real[],
double eigen_imag[], int n)
{
double zero_tolerance;
double *pH;
int i, j, row;
// Calculate the zero tolerance
pH = H;
zero_tolerance = fabs(pH[0]);
for (pH += n, i = 1; i < n; pH += n, i++)
for (j = i-1; j < n; j++) zero_tolerance += fabs(pH[j]);
zero_tolerance *= DBL_EPSILON;
// Start Backsubstitution
for (row = n-1; row >= 0; row--) {
if (eigen_imag[row] == 0.0)
BackSubstitute_Real_Vector(H, eigen_real, eigen_imag, row,
zero_tolerance, n);
else if ( eigen_imag[row] < 0.0 )
BackSubstitute_Complex_Vector(H, eigen_real, eigen_imag, row,
zero_tolerance, n);
}
}
////////////////////////////////////////////////////////////////////////////////
// static void BackSubstitute_Real_Vector(double *H, double eigen_real[], //
// double eigen_imag[], int row, double zero_tolerance, int n) //
// //
// Description: //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the matrix in Hessenberg form. //
// double eigen_real[] //
// The real part of an eigenvalue. //
// double eigen_imag[] //
// The imaginary part of an eigenvalue. //
// int row //
// double zero_tolerance //
// Zero substitute. To avoid dividing by zero. //
// int n //
// The dimension of H, eigen_real, and eigen_imag. //
////////////////////////////////////////////////////////////////////////////////
// //
static void BackSubstitute_Real_Vector(double *H, double eigen_real[],
double eigen_imag[], int row, double zero_tolerance, int n)
{
double *pH;
double *pV;
double x,y;
double u[4];
double v[2];
int i,j,k;
k = row;
pH = H + row * n;
pH[row] = 1.0;
for (i = row - 1, pH -= n; i >= 0; i--, pH -= n) {
u[0] = pH[i] - eigen_real[row];
v[0] = pH[row];
pV = H + n * k;
for (j = k; j < row; j++, pV += n) v[0] += pH[j] * pV[row];
if ( eigen_imag[i] < 0.0 ) {
u[3] = u[0];
v[1] = v[0];
} else {
k = i;
if (eigen_imag[i] == 0.0) {
if (u[0] != 0.0) pH[row] = - v[0] / u[0];
else pH[row] = - v[0] / zero_tolerance;
} else {
u[1] = pH[i+1];
u[2] = pH[n+i];
x = (eigen_real[i] - eigen_real[row]);
x *= x;
x += eigen_imag[i] * eigen_imag[i];
pH[row] = (u[1] * v[1] - u[3] * v[0]) / x;
if ( fabs(u[1]) > fabs(u[3]) )
pH[n+row] = -(v[0] + u[0] * pH[row]) / u[1];
else
pH[n+row] = -(v[1] + u[2] * pH[row]) / u[3];
}
}
}
}
////////////////////////////////////////////////////////////////////////////////
// static void BackSubstitute_Complex_Vector(double *H, double eigen_real[], //
// double eigen_imag[], int row, double zero_tolerance, int n) //
// //
// Description: //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the matrix in Hessenberg form. //
// double eigen_real[] //
// The real part of an eigenvalue. //
// double eigen_imag[] //
// The imaginary part of an eigenvalue. //
// int row //
// double zero_tolerance //
// Zero substitute. To avoid dividing by zero. //
// int n //
// The dimension of H, eigen_real, and eigen_imag. //
////////////////////////////////////////////////////////////////////////////////
// //
static void BackSubstitute_Complex_Vector(double *H, double eigen_real[],
double eigen_imag[], int row, double zero_tolerance, int n)
{
double *pH;
double *pV;
double x,y;
double u[4];
double v[2];
double w[2];
int i,j,k;
k = row - 1;
pH = H + n * row;
if ( fabs(pH[k]) > fabs(pH[row-n]) ) {
pH[k-n] = - (pH[row] - eigen_real[row]) / pH[k];
pH[row-n] = -eigen_imag[row] / pH[k];
}
else
Complex_Division(-pH[row-n], 0.0,
pH[k-n]-eigen_real[row], eigen_imag[row], &pH[k-n], &pH[row-n]);
pH[k] = 1.0;
pH[row] = 0.0;
for (i = row - 2, pH = H + n * i; i >= 0; pH -= n, i--) {
u[0] = pH[i] - eigen_real[row];
w[0] = pH[row];
w[1] = 0.0;
pV = H + k * n;
for (j = k; j < row; j++, pV+=n) {
w[0] += pH[j] * pV[row - 1];
w[1] += pH[j] * pV[row];
}
if (eigen_imag[i] < 0.0) {
u[3] = u[0];
v[0] = w[0];
v[1] = w[1];
} else {
k = i;
if (eigen_imag[i] == 0.0) {
Complex_Division(-w[0], -w[1], u[0], eigen_imag[row], &pH[row-1],
&pH[row]);
}
else {
u[1] = pH[i+1];
u[2] = pH[n + i];
x = eigen_real[i] - eigen_real[row];
y = 2.0 * x * eigen_imag[row];
x = x * x + eigen_imag[i] * eigen_imag[i]
- eigen_imag[row] * eigen_imag[row];
if ( x == 0.0 && y == 0.0 )
x = zero_tolerance * ( fabs(u[0]) + fabs(u[1]) + fabs(u[2])
+ fabs(u[3]) + fabs(eigen_imag[row]) );
Complex_Division(u[1]*v[0] - u[3] * w[0] + w[1] * eigen_imag[row],
u[1] * v[1] - u[3] * w[1] - w[0] * eigen_imag[row],
x, y, &pH[row-1], &pH[row]);
if ( fabs(u[1]) > (fabs(u[3]) + fabs(eigen_imag[row])) ) {
pH[n+row-1] = -w[0] - u[0] * pH[row-1]
+ eigen_imag[row] * pH[row] / u[1];
pH[n+row] = -w[1] - u[0] * pH[row]
- eigen_imag[row] * pH[row-1] / u[1];
}
else {
Complex_Division(-v[0] - u[2] * pH[row-1], -v[1] - u[2]*pH[row],
u[3], eigen_imag[row], &pH[n+row-1], &pH[n+row]);
}
}
}
}
}
////////////////////////////////////////////////////////////////////////////////
// static void Calculate_Eigenvectors(double *H, double *S, //
// double eigen_real[], double eigen_imag[], int n) //
// //
// Description: //
// Multiply by transformation matrix. //
// //
// Arguments: //
// double *H //
// Pointer to the first element of the matrix in Hessenberg form. //
// double *S //
// Pointer to the first element of the transformation matrix. //
// double eigen_real[] //
// The real part of an eigenvalue. //
// double eigen_imag[] //
// The imaginary part of an eigenvalue. //
// int n //
// The dimension of H, S, eigen_real, and eigen_imag. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Calculate_Eigenvectors(double *H, double *S, double eigen_real[],
double eigen_imag[], int n)
{
double* pH;
double* pS;
double x,y;
int i,j,k;
for (k = n-1; k >= 0; k--) {
if (eigen_imag[k] < 0.0) {
for (i = 0, pS = S; i < n; pS += n, i++) {
x = 0.0;
y = 0.0;
for (j = 0, pH = H; j <= k; pH += n, j++) {
x += pS[j] * pH[k-1];
y += pS[j] * pH[k];
}
pS[k-1] = x;
pS[k] = y;
}
} else if (eigen_imag[k] == 0.0) {
for (i = 0, pS = S; i < n; i++, pS += n) {
x = 0.0;
for (j = 0, pH = H; j <= k; j++, pH += n)
x += pS[j] * pH[k];
pS[k] = x;
}
}
}
}
////////////////////////////////////////////////////////////////////////////////
// static void Complex_Division(double x, double y, double u, double v, //
// double* a, double* b) //
// //
// Description: //
// a + i b = (x + iy) / (u + iv) //
// = (x * u + y * v) / r^2 + i (y * u - x * v) / r^2, //
// where r^2 = u^2 + v^2. //
// //
// Arguments: //
// double x //
// Real part of the numerator. //
// double y //
// Imaginary part of the numerator. //
// double u //
// Real part of the denominator. //
// double v //
// Imaginary part of the denominator. //
// double *a //
// Real part of the quotient. //
// double *b //
// Imaginary part of the quotient. //
////////////////////////////////////////////////////////////////////////////////
// //
static void Complex_Division(double x, double y, double u, double v,
double* a, double* b)
{
double q = u*u + v*v;
*a = (x * u + y * v) / q;
*b = (y * u - x * v) / q;
}
////////////////////////////////////////////////////////////////////////////////
// File: transpose_square_matrix.c //
// Routine(s): //
// Transpose_Square_Matrix //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Transpose_Square_Matrix( double *A, int n ) //
// //
// Description: //
// Take the transpose of A and store in place. //
// //
// Arguments: //
// double *A Pointer to the first element of the matrix A. //
// int n The number of rows and columns of the matrix A. //
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// double A[N][N]; //
// //
// (your code to initialize the matrix A) //
// //
// Transpose_Square_Matrix( &A[0][0], N); //
// printf("The transpose of A is \n"); ... //
////////////////////////////////////////////////////////////////////////////////
void Transpose_Square_Matrix( double *A, int n )
{
double *pA, *pAt;
double temp;
int i,j;
for (i = 0; i < n; A += n + 1, i++) {
pA = A + 1;
pAt = A + n;
for (j = i+1 ; j < n; pA++, pAt += n, j++) {
temp = *pAt;
*pAt = *pA;
*pA = temp;
}
}
}
///////////////////////////////////////////////////////////////////////////////
// File: lower_triangular.c //
// Routines: //
// Lower_Triangular_Solve //
// Lower_Triangular_Inverse //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// int Lower_Triangular_Solve(double *L, double *B, double x[], int n) //
// //
// Description: //
// This routine solves the linear equation Lx = B, where L is an n x n //
// lower triangular matrix. (The superdiagonal part of the matrix is //
// not addressed.) //
// The algorithm follows: //
// x[0] = B[0]/L[0][0], and //
// x[i] = [B[i] - (L[i][0] * x[0] + ... + L[i][i-1] * x[i-1])] / L[i][i],//
// for i = 1, ..., n-1. //
// //
// Arguments: //
// double *L Pointer to the first element of the lower triangular //
// matrix. //
// double *B Pointer to the column vector, (n x 1) matrix, B. //
// double *x Pointer to the column vector, (n x 1) matrix, x. //
// int n The number of rows or columns of the matrix L. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - The matrix L is singular. //
// //
// Example: //
// #define N //
// double A[N][N], B[N], x[N]; //
// //
// (your code to create matrix A and column vector B) //
// err = Lower_Triangular_Solve(&A[0][0], B, x, n); //
// if (err < 0) printf(" Matrix A is singular\n"); //
// else printf(" The solution is \n"); //
// ... //
////////////////////////////////////////////////////////////////////////////////
// //
int Lower_Triangular_Solve(double *L, double B[], double x[], int n)
{
int i, k;
// Solve the linear equation Lx = B for x, where L is a lower
// triangular matrix.
for (k = 0; k < n; L += n, k++) {
if (*(L + k) == 0.0) return -1; // The matrix L is singular
x[k] = B[k];
for (i = 0; i < k; i++) x[k] -= x[i] * *(L + i);
x[k] /= *(L + k);
}
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// int Lower_Triangular_Inverse(double *L, int n) //
// //
// Description: //
// This routine calculates the inverse of the lower triangular matrix L. //
// The superdiagonal part of the matrix is not addressed. //
// The algorithm follows: //
// Let M be the inverse of L, then L M = I, //
// M[i][i] = 1.0 / L[i][i] for i = 0, ..., n-1, and //
// M[i][j] = -[(L[i][j] M[j][j] + ... + L[i][i-1] M[i-1][j])] / L[i][i], //
// for i = 1, ..., n-1, j = 0, ..., i - 1. //
// //
// //
// Arguments: //
// double *L On input, the pointer to the first element of the matrix //
// whose lower triangular elements form the matrix which is //
// to be inverted. On output, the lower triangular part is //
// replaced by the inverse. The superdiagonal elements are //
// not modified. //
// its inverse. //
// int n The number of rows and/or columns of the matrix L. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - The matrix L is singular. //
// //
// Example: //
// #define N //
// double L[N][N]; //
// //
// (your code to create the matrix L) //
// err = Lower_Triangular_Inverse(&L[0][0], N); //
// if (err < 0) printf(" Matrix L is singular\n"); //
// else { //
// printf(" The inverse is \n"); //
// ... //
// } //
////////////////////////////////////////////////////////////////////////////////
// //
int Lower_Triangular_Inverse(double *L, int n)
{
int i, j, k;
double *p_i, *p_j, *p_k;
double sum;
// Invert the diagonal elements of the lower triangular matrix L.
for (k = 0, p_k = L; k < n; p_k += (n + 1), k++) {
if (*p_k == 0.0) return -1;
else *p_k = 1.0 / *p_k;
}
// Invert the remaining lower triangular matrix L row by row.
for (i = 1, p_i = L + n; i < n; i++, p_i += n) {
for (j = 0, p_j = L; j < i; p_j += n, j++) {
sum = 0.0;
for (k = j, p_k = p_j; k < i; k++, p_k += n)
sum += *(p_i + k) * *(p_k + j);
*(p_i + j) = - *(p_i + i) * sum;
}
}
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// File: upper_triangular.c //
// Routines: //
// Upper_Triangular_Solve //
// Upper_Triangular_Inverse //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// int Upper_Triangular_Solve(double *U, double *B, double x[], int n) //
// //
// Description: //
// This routine solves the linear equation Ux = B, where U is an n x n //
// upper triangular matrix. (The subdiagonal part of the matrix is //
// not addressed.) //
// The algorithm follows: //
// x[n-1] = B[n-1]/U[n-1][n-1], and //
// x[i] = [B[i] - (U[i][i+1] * x[i+1] + ... + U[i][n-1] * x[n-1])] //
// / U[i][i], //
// for i = n-2, ..., 0. //
// //
// Arguments: //
// double *U Pointer to the first element of the upper triangular //
// matrix. //
// double *B Pointer to the column vector, (n x 1) matrix, B. //
// double *x Pointer to the column vector, (n x 1) matrix, x. //
// int n The number of rows or columns of the matrix U. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - The matrix U is singular. //
// //
// Example: //
// #define N //
// double A[N][N], B[N], x[N]; //
// //
// (your code to create matrix A and column vector B) //
// err = Upper_Triangular_Solve(&A[0][0], B, x, n); //
// if (err < 0) printf(" Matrix A is singular\n"); //
// else printf(" The solution is \n"); //
// ... //
////////////////////////////////////////////////////////////////////////////////
// //
int Upper_Triangular_Solve(double *U, double B[], double x[], int n)
{
int i, k;
// Solve the linear equation Ux = B for x, where U is an upper
// triangular matrix.
for (k = n-1, U += n * (n - 1); k >= 0; U -= n, k--) {
if (*(U + k) == 0.0) return -1; // The matrix U is singular
x[k] = B[k];
for (i = k + 1; i < n; i++) x[k] -= x[i] * *(U + i);
x[k] /= *(U + k);
}
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// int Upper_Triangular_Inverse(double *U, int n) //
// //
// Description: //
// This routine calculates the inverse of the upper triangular matrix U. //
// The subdiagonal part of the matrix is not addressed. //
// The algorithm follows: //
// Let M be the inverse of U, then U M = I, //
// M[n-1][n-1] = 1.0 / U[n-1][n-1] and //
// M[i][j] = -( U[i][i+1] M[i+1][j] + ... + U[i][j] M[j][j] ) / U[i][i], //
// for i = n-2, ... , 0, j = n-1, ..., i+1. //
// //
// //
// Arguments: //
// double *U On input, the pointer to the first element of the matrix //
// whose upper triangular elements form the matrix which is //
// to be inverted. On output, the upper triangular part is //
// replaced by the inverse. The subdiagonal elements are //
// not modified. //
// int n The number of rows and/or columns of the matrix U. //
// //
// Return Values: //
// 0 Success //
// -1 Failure - The matrix U is singular. //
// //
// Example: //
// #define N //
// double U[N][N]; //
// //
// (your code to create the matrix U) //
// err = Upper_Triangular_Inverse(&U[0][0], N); //
// if (err < 0) printf(" Matrix U is singular\n"); //
// else { //
// printf(" The inverse is \n"); //
// ... //
// } //
////////////////////////////////////////////////////////////////////////////////
// //
int Upper_Triangular_Inverse(double *U, int n)
{
int i, j, k;
double *p_i, *p_j, *p_k;
double sum;
// Invert the diagonal elements of the upper triangular matrix U.
for (k = 0, p_k = U; k < n; p_k += (n + 1), k++) {
if (*p_k == 0.0) return -1;
else *p_k = 1.0 / *p_k;
}
// Invert the remaining upper triangular matrix U.
for (i = n - 2, p_i = U + n * (n - 2); i >=0; p_i -= n, i-- ) {
for (j = n - 1; j > i; j--) {
sum = 0.0;
for (k = i + 1, p_k = p_i + n; k <= j; p_k += n, k++ ) {
sum += *(p_i + k) * *(p_k + j);
}
*(p_i + j) = - *(p_i + i) * sum;
}
}
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// File: interchange_cols.c //
// Routine(s): //
// Interchange_Columns //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Interchange_Columns(double *A, int col1, int col2, int nrows, //
// int ncols) //
// //
// Description: //
// Interchange the columns 'col1' and 'col2' of the nrows x ncols //
// matrix A. //
// //
// Arguments: //
// double *A Pointer to the first element of the matrix A. //
// int col1 The column of A which is to be interchanged with col2. //
// int col2 The column of A which is to be interchanged with col1. //
// int nrows The number of rows matrix A. //
// int ncols The number of columns of the matrix A. //
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// #define M //
// double A[M][N]; //
// int i,j; //
// //
// (your code to initialize the matrix A, the column number i and column //
// number j) //
// //
// if ( (i >= 0) && ( i < N ) && ( j >= 0 ) && (j < N) ) //
// Interchange_Columns(&A[0][0], i, j, M, N); //
// printf("The matrix A is \n"); ... //
////////////////////////////////////////////////////////////////////////////////
void Interchange_Columns(double *A, int col1, int col2, int nrows, int ncols)
{
int i;
double *pA1, *pA2;
double temp;
pA1 = A + col1;
pA2 = A + col2;
for (i = 0; i < nrows; pA1 += ncols, pA2 += ncols, i++) {
temp = *pA1;
*pA1 = *pA2;
*pA2 = temp;
}
}
////////////////////////////////////////////////////////////////////////////////
// File: interchange_rows.c //
// Routine(s): //
// Interchange_Rows //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// void Interchange_Rows(double *A, int row1, int row2, int ncols) //
// //
// Description: //
// Interchange the rows 'row1' and 'row2' of the nrows x ncols matrix A. //
// //
// Arguments: //
// double *A Pointer to the first element of the matrix A. //
// int row1 The row of A which is to be interchanged with row row2. //
// int row2 The row of A which is to be interchanged with row row1. //
// int ncols The number of columns of the matrix A. //
// //
// Return Values: //
// void //
// //
// Example: //
// #define N //
// #define M //
// double A[M][N]; //
// int i, j; //
// //
// (your code to initialize the matrix A, the row number i and row number j) //
// //
// if ( (i >= 0) && ( i < M ) && (j > 0) && ( j < M ) ) //
// Interchange_Rows(&A[0][0], i, j, N); //
// printf("The matrix A is \n"); ... //
////////////////////////////////////////////////////////////////////////////////
void Interchange_Rows(double *A, int row1, int row2, int ncols)
{
int i;
double *pA1, *pA2;
double temp;
pA1 = A + row1 * ncols;
pA2 = A + row2 * ncols;
for (i = 0; i < ncols; i++) {
temp = *pA1;
*pA1++ = *pA2;
*pA2++ = temp;
}
}
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