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lls.c
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/*
* lls.c
*
* Created on: Apr 14, 2014
* Author: Rafat Hussain
*/
#include "lls.h"
int lls_normal(double *A,double *b,int M,int N,double *x) {
int retcode,sc,i,j,c1,l;
double sum;
double *AT,*d,*C,*y,*CT;
// M - data points
// N - Number of parameters
// A - MXN; b size - M vector; AT - NXM
AT = (double*) malloc(sizeof(double) * M * N);
d = (double*) malloc(sizeof(double) * N);
C = (double*) malloc(sizeof(double) * N * N);
y = (double*) malloc(sizeof(double) * N);
CT = (double*) malloc(sizeof(double) * N * N);
retcode = 0;
mtranspose(A,M,N,AT);
mmult(AT,b,d,N,M,1);
mmult(AT,A,C,N,M,N);
sc = chol(C,N);
if (sc == -1) {
return -1;
}
mtranspose(C,N,N,CT);
//Forward Substitution
y[0] = d[0]/CT[0];
for(i = 1; i < N; ++i) {
sum = 0.;
c1 = i*N;
for(j = 0; j < i; ++j) {
sum += y[j] * CT[c1 + j];
}
y[i] = (d[i] - sum)/CT[c1+i];
}
//Back Substitution
x[N - 1] = y[N - 1]/C[N * N - 1];
for (i = N - 2; i >= 0; i--) {
sum = 0.;
c1 = i*(N+1);
l=0;
for(j = i+1; j < N;j++) {
l++;
sum += C[c1 + l] * x[j];
}
x[i] = (y[i] - sum) / C[c1];
}
free(AT);
free(d);
free(C);
free(y);
return retcode;
}
int lls_qr(double *Ai,double *bi,int M,int N,double *xo) {
int j,i,k,u,t,retcode,c1,l;
double *x,*v,*AT,*w,*bvec,*b,*A,*R;
double beta,sum;
retcode = 0;
if (M < N) {
printf("M should be greater than or equal to N");
exit(-1);
}
x = (double*) malloc(sizeof(double) * M);
b = (double*) malloc(sizeof(double) * M);
bvec = (double*) malloc(sizeof(double) * N);
v = (double*) malloc(sizeof(double) * M);
AT = (double*) malloc(sizeof(double) * M * N);
A = (double*) malloc(sizeof(double) * M * N);
w = (double*) malloc(sizeof(double) * 1);
R = (double*) malloc(sizeof(double) * N * N);
for(j = 0; j < M;++j) {
b[j] = bi[j];
}
for(j = 0; j < M*N;++j) {
A[j] = Ai[j];
}
for(j = 0; j < N;++j) {
for(i=j;i < M;++i) {
x[i-j] = A[i*N+j];
}
beta = house(x,M-j,v);
bvec[j] = beta;
for (i=j; i < M; i++) {
t = i * N;
u = 0;
for (k=j; k < N; k++) {
AT[u+i-j] = A[k+t];
u+=(M-j);
}
}
mmult(AT,v,w,N-j,M-j,1);
scale(w,N-j,1,beta);
mmult(v,w,AT,M-j,1,N-j);
for (i=j; i < M; i++) {
t = i *N;
for (k=j; k < N; k++) {
A[t+k] -= AT[(i-j)*(N-j) + k - j];
}
}
if (j < M) {
for(i=j+1;i < M;++i) {
A[i*N+j] = v[i-j];
}
}
}
for(i = 0; i < N;++i) {
t = i *N;
for(j = 0; j < N;++j) {
if (i > j) {
R[t+j] = 0.;
} else {
R[t+j] = A[t+j];
}
}
}
for(j = 0; j < N;++j) {
v[j] = 1;
for(i = j+1; i < M;++i) {
v[i] = A[i * N + j];//edit
}
mmult(b+j,v+j,w,1,M-j,1);
*w = *w * bvec[j];
for(i = j; i < M;++i) {
v[i] = *w * v[i];
}
for(i = j; i < M;++i) {
b[i] = b[i] - v[i];
}
}
//mdisplay(b,1,M);
//back substitution
xo[N - 1] = b[N - 1]/R[N * N - 1];
for (i = N - 2; i >= 0; i--) {
sum = 0.;
c1 = i*(N+1);
l=0;
for(j = i+1; j < N;j++) {
l++;
sum += R[c1 + l] * xo[j];
}
xo[i] = (b[i] - sum) / R[c1];
}
free(x);
free(v);
free(AT);
free(w);
free(bvec);
free(R);
free(b);
free(A);
return retcode;
}
void bidiag(double *A, int M, int N) {
int j,i,k,u,t;
double *x,*v,*AT,*w;
double beta;
if (M < N) {
printf("M should be greater than or equal to N");
exit(1);
}
x = (double*) malloc(sizeof(double) * M);
v = (double*) malloc(sizeof(double) * M);
AT = (double*) malloc(sizeof(double) * M * N);
w = (double*) malloc(sizeof(double) * M * M);
for(j = 0; j < N;++j) {
for(i=j;i < M;++i) {
x[i-j] = A[i*N+j];
}
beta = house(x,M-j,v);
for (i=j; i < M; i++) {
t = i * N;
u = 0;
for (k=j; k < N; k++) {
AT[u+i-j] = A[k+t];
u+=(M-j);
}
}
mmult(AT,v,w,N-j,M-j,1);
scale(w,N-j,1,beta);
mmult(v,w,AT,M-j,1,N-j);
for (i=j; i < M; i++) {
t = i *N;
for (k=j; k < N; k++) {
A[t+k] -= AT[(i-j)*(N-j) + k - j];
}
}
for(i=j+1;i < M;++i) {
A[i*N+j] = v[i-j];
}
if (j < N - 2) {
for(i=j+1;i < N;++i) {
x[i-j-1] = A[j*N+i];
}
beta = house(x,N-j-1,v);
for (i=j; i < M; i++) {
t = i * N;
u = (i-j) *(N-j-1);
for (k=j+1; k < N; k++) {
AT[u+k-j-1] = A[k+t];
}
}
mmult(AT,v,w,M-j,N-j-1,1);
scale(w,M-j,1,beta);
mmult(w,v,AT,M-j,1,N-j-1);
for (i=j; i < M; i++) {
t = i * N;
u = (i-j) *(N-j-1);
for (k=j+1; k < N; k++) {
A[k+t] -= AT[u+k-j-1];
}
}
u = 1;
t = j*N;
for(i = j + 2; i < N;++i) {
A[t+i] = v[u];
u++;
}
}
}
free(x);
free(v);
free(AT);
}
void bidiag_orth(double *A, int M, int N,double *U,double *V) {
int j,i,k,u,t;
double *x,*v,*AT,*w;
double beta;
if (M < N) {
printf("M should be greater than or equal to N");
exit(1);
}
x = (double*) malloc(sizeof(double) * M);
v = (double*) malloc(sizeof(double) * M);
AT = (double*) malloc(sizeof(double) * M * M);
w = (double*) malloc(sizeof(double) * M * M);
eye(U,M);
eye(V,N);
for(j = 0; j < N;++j) {
for(i=j;i < M;++i) {
x[i-j] = A[i*N+j];
}
beta = house(x,M-j,v);
//mdisplay(v,M-j,1);
for (i=j; i < M; i++) {
t = i * N;
u = 0;
for (k=j; k < N; k++) {
AT[u+i-j] = A[k+t];
u+=(M-j);
}
}
mmult(AT,v,w,N-j,M-j,1);
scale(w,N-j,1,beta);
mmult(v,w,AT,M-j,1,N-j);
for (i=j; i < M; i++) {
t = i *N;
for (k=j; k < N; k++) {
A[t+k] -= AT[(i-j)*(N-j) + k - j];
}
}
for(i=j+1;i < M;++i) {
A[i*N+j] = v[i-j];
}
for (i=j; i < M; i++) {
t = i * M;
u = (i-j) *(M-j);
for (k=j; k < M; k++) {
AT[u+k-j] = U[k+t];
}
}
mmult(AT,v,w,M-j,M-j,1);
scale(w,M-j,1,beta);
mmult(w,v,AT,M-j,1,M-j);
for (i=j; i < M; i++) {
t = i * M;
u = (i-j) *(M-j);
for (k=j; k < M; k++) {
U[k+t] -= AT[u+k-j];
}
}
//mdisplay(U,M,M);
if (j < N - 2) {
for(i=j+1;i < N;++i) {
x[i-j-1] = A[j*N+i];
}
beta = house(x,N-j-1,v);
for (i=j; i < M; i++) {
t = i * N;
u = (i-j) *(N-j-1);
for (k=j+1; k < N; k++) {
AT[u+k-j-1] = A[k+t];
}
}
mmult(AT,v,w,M-j,N-j-1,1);
scale(w,M-j,1,beta);
mmult(w,v,AT,M-j,1,N-j-1);
for (i=j; i < M; i++) {
t = i * N;
u = (i-j) *(N-j-1);
for (k=j+1; k < N; k++) {
A[k+t] -= AT[u+k-j-1];
}
}
u = 1;
t = j*N;
for(i = j + 2; i < N;++i) {
A[t+i] = v[u];
u++;
}
}
}
free(x);
free(v);
free(AT);
}
int svd_gr(double *A,int M,int N,double *U,double *V,double *q) {
int i,j,k,l,t,t2,ierr,cancel,iter,l1;
double eps,g,x,s,temp,f,h,scale,c,y,z;
double *e;
/*
THIS SUBROUTINE IS THE MODIFIED C TRANSLATION OF THE
EISPACK FORTRAN TRANSLATION OF THE ALGOL PROCEDURE SVD,
NUM. MATH. 14, 403-420(1970) BY GOLUB AND REINSCH.
HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 134-151(1971).
*/
/*
* U = MXN
* V - NXN
* Q - NX1
*/
e = (double*) malloc(sizeof(double) * N);
ierr = 0;
eps = macheps();
g = scale = x = 0.0;
for(i = 0; i < M*N;++i) {
U[i] = A[i];
}
for(i = 0; i < N;++i) {
l = i+1;
e[i] = scale * g;
g = 0.0;
s = 0.0;
scale = 0.0;
if (i < M) {
for(k = i; k < M;++k) {
scale += fabs(U[k*N+i]);
}
if (scale != 0.0) {
for(k = i; k < M;++k) {
t = k * N;
U[t+i] /= scale;
temp = U[t+i];
s += temp*temp;
}
f = U[i*N+i];
g = (f < 0) ? sqrt(s) : -sqrt(s);
h = f * g - s;
U[i*N+i] = f - g;
if (i < N - 1) {
for(j = l; j < N;++j) {
s = 0.0;
for(k = i; k < M;++k) {
t = k * N;
s += U[t+i]*U[t+j];
}
f = s / h;
for(k = i; k < M;++k) {
t = k * N;
U[t+j] += f * U[t+i];
}
}
}
for(k = i; k < M;++k) {
t = k * N;
U[t+i] *= scale;
}
}
}
q[i] = scale * g;
g = 0.0;
s = 0.0;
scale = 0.0;
if (i < M && i != N - 1) {
t = i *N;
for(k = l; k < M;++k) {
scale += fabs(U[t+k]);
}
if (scale != 0.0) {
for(k = l; k < N;++k) {
U[t+k] /= scale;
temp = U[t+k];
s = s + temp*temp;
}
f = U[t+l];
g = (f < 0) ? sqrt(s) : -sqrt(s);
h = f * g - s;
U[t+l] = f - g;
for(k = l;k < N;++k) {
e[k] = U[t+k] / h;
}
for (j = l; j < M; j++) {
s = 0.0;
t2 = j * N;
for (k = l; k < N; k++) {
s += U[t2+k] * U[t+k];
}
for (k = l; k < N; k++) {
U[t2+k] += s * e[k];
}
}
for (k = l; k < N; k++)
U[t+k] *= scale;
}
}
temp = fabs(q[i]) + fabs(e[i]);
if (x < temp) {
x = temp;
}
}
//Accumulating Right Hand Transformations
for(i = N - 1;i >= 0;--i) {
t = i * N;
if (i < N - 1) {
if (g != 0.0) {
h = U[t+i+1] * g;
for(j = l;j < N;++j) {
V[j*N+i] = U[t+j] / h;
}
for(j = l;j < N;++j) {
s = 0.0;
for(k = l; k < N;++k) {
s += U[t+k] * V[k*N+j];
}
for(k = l; k < N;++k) {
V[k*N+j] += (s * V[k*N+i]);
}
}
}
for(j = l; j < N;++j) {
V[t+j] = V[j*N+i] = 0.0;
}
}
V[t+i] = 1.0;
g = e[i];
l = i;
}
//Accumulating Left Hand Transformations
for(i = N - 1;i >= 0;--i) {
t = i * N;
l = i+1;
g = q[i];
if (i < N - 1) {
for(j = l;j < N;++j) {
U[t+j] = 0.0;
}
}
if (g != 0.0) {
if (i != N - 1) {
//h = U[t+i] * g;
for(j = l;j < N;++j) {
s = 0.0;
for(k = l; k < M;++k) {
s += (U[k*N+i] * U[k*N+j]);
}
f = (s / U[t+i]) / g;
for(k = i; k < M;++k) {
U[k*N+j] += (f * U[k*N+i]);
}
}
}
for(j = i; j < M;++j) {
U[j*N+i] = U[j*N+i] / g;
}
} else {
for(j = i; j < M;++j) {
U[j*N+i] = 0.0;
}
}
U[t+i] += 1.0;
}
// mdisplay(U,M,N);
eps = eps * x;
for(k = N - 1; k >= 0; --k) {
iter = 0;
while(1) {
iter++;
if (iter > SVDMAXITER) {
printf("Convergence Not Achieved \n");
return 15;
}
cancel = 1;
for(l = k; l >= 0; --l) {
if (fabs(e[l]) <= eps) {
cancel = 0; //test f convergence
break;
}
if (fabs(q[l-1]) <= eps) {
//Cancel
break;
}
}
if (cancel) {
c = 0.0;
s = 1.0;
l1 = l - 1;
for(i = l; i <= k;++i) {
f = s*e[i];
e[i] *= c;
if (fabs(f) <= eps) {
break;
}
g = q[i];
h = q[i] = hypot(f,g);
c = g/h;
s = -f/h;
for(j = 0; j < M;++j) {
t = j * N;
y = U[t+l1];
z = U[t+i];
U[t+l1] = y * c + z * s;
U[t+i] = z * c - y * s;
}
}
}
z = q[k];
if (l != k) {
x = q[l];
y = q[k-1];
g = e[k-1];
h = e[k];
f = 0.5 * (((g + z) / h) * ((g - z) / y) + y / h - h / y);
g = hypot(f,1.0);
if (f < 0.0) {
temp = f - g;
} else {
temp = f+g;
}
f = x - (z / x) * z + (h / x) * (y / temp - h);
//Next QR Transformation
c = s = 1.0;
for(i = l+1; i <= k;++i) {
g = e[i];
y = q[i];
h = s * g;
g = c * g;
e[i-1] = z = hypot(f,h);
c = f / z;
s = h / z;
f = x * c + g * s;
g = g * c - x * s;
h = y * s;
y *= c;
for(j = 0; j < N;++j) {
t = j * N;
x = V[t+i-1];
z = V[t+i];
V[t+i-1] = x * c + z * s;
V[t+i] = z * c - x * s;
}
q[i-1] = z = hypot(f,h);
if (z != 0.0) {
c = f / z;
s = h / z;
}
f = c * g + s * y;
x = c * y - s * g;
for(j = 0; j < M;++j) {
t = j * N;
y = U[t+i-1];
z = U[t+i];
U[t+i-1] = y * c + z * s;
U[t+i] = z * c - y * s;
}
}
e[l] = 0.0;
e[k] = f;
q[k] = x;
} else {
//convergence
if (z < 0.0) {
q[k] = -z;
for (j = 0; j < N; j++) {
t = j *N;
V[t+k] = -V[t+k];
}
}
break;
}
}
}
svd_sort(U,M,N,V,q);
free(e);
return ierr;
}
int svd_gr2(double *A,int M,int N,double *U,double *V,double *q) {
int i,j,k,l,t,t2,ierr,cancel,iter,l1;
double eps,g,x,s,temp,f,h,tol,c,y,z;
double *e;
/*
THIS SUBROUTINE IS THE MODIFIED C TRANSLATION OF THE
EISPACK FORTRAN TRANSLATION OF THE ALGOL PROCEDURE SVD,
NUM. MATH. 14, 403-420(1970) BY GOLUB AND REINSCH.
HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 134-151(1971).
*/
/*
* U = MXN
* V - NXN
* Q - NX1
*/
if (M < N) {
printf("Rows (M) should be greater than Columns (B) \n");
printf("Retry By Transposing the Input Matrix");
return -1;
}
e = (double*) malloc(sizeof(double) * N);
ierr = 0;
eps = macheps();
tol = eps * eps;
g = x = 0.0;
for(i = 0; i < M*N;++i) {
U[i] = A[i];
}
for(i = 0; i < N;++i) {
l = i+1;
e[i] = g;
s = 0.0;
for(k = i; k < M;++k) {
t = k * N;
temp = U[t+i];
s += temp*temp;
}
if (s < tol) {
g = 0.0;
} else {
f = U[i*N+i];
g = (f < 0) ? sqrt(s) : -sqrt(s);
h = f * g - s;
U[i*N+i] = f - g;
for(j = l; j < N;++j) {
s = 0.0;
for(k = i; k < M;++k) {
t = k * N;
s += (U[t+i]*U[t+j]);
}
f = s / h;
for(k = i; k < M;++k) {
t = k * N;
U[t+j] += (f * U[t+i]);
}
}
}
q[i] = g;
s = 0.0;
t = i * N;
for(k = l; k < N;++k) {
temp = U[t+k];
s = s + temp*temp;
}
if (s < tol) {
g = 0.0;
} else {
f = U[t+l];
g = (f < 0) ? sqrt(s) : -sqrt(s);
h = f * g - s;
U[t+l] = f - g;
for(k = l;k < N;++k) {
e[k] = U[t+k] / h;
}
for (j = l; j < M; j++) {
s = 0.0;
t2 = j * N;
for (k = l; k < N; k++) {
s += U[t2+k] * U[t+k];
}
for (k = l; k < N; k++) {
U[t2+k] += s * e[k];
}
}
}
temp = fabs(q[i]) + fabs(e[i]);
if (x < temp) {
x = temp;
}
}
//Accumulating Right Hand Transformations
for(i = N - 1;i >= 0;--i) {
t = i * N;
if (i < N - 1) {
if (g != 0.0) {
h = U[t+i+1] * g;
for(j = l;j < N;++j) {
V[j*N+i] = U[t+j] / h;
}
for(j = l;j < N;++j) {
s = 0.0;
for(k = l; k < N;++k) {
s += U[t+k] * V[k*N+j];
}
for(k = l; k < N;++k) {
V[k*N+j] += (s * V[k*N+i]);
}
}
}
for(j = l; j < N;++j) {
V[t+j] = V[j*N+i] = 0.0;
}
}
V[t+i] = 1.0;
g = e[i];
l = i;
}
//Accumulating Left Hand Transformations
for(i = N - 1;i >= 0;--i) {
t = i * N;
l = i+1;
g = q[i];
if (i < N - 1) {
for(j = l;j < N;++j) {
U[t+j] = 0.0;
}
}
if (g != 0.0) {
if (i != N - 1) {
//h = U[t+i] * g;
for(j = l;j < N;++j) {
s = 0.0;
for(k = l; k < M;++k) {
s += (U[k*N+i] * U[k*N+j]);
}
f = (s / U[t+i]) / g;
for(k = i; k < M;++k) {
U[k*N+j] += (f * U[k*N+i]);
}
}
}
for(j = i; j < M;++j) {
U[j*N+i] = U[j*N+i] / g;
}
} else {
for(j = i; j < M;++j) {
U[j*N+i] = 0.0;
}
}
U[t+i] += 1.0;
}
// mdisplay(U,M,N);
eps = eps * x;
for(k = N - 1; k >= 0; --k) {
iter = 0;
while(1) {
iter++;
if (iter > SVDMAXITER) {
printf("Convergence Not Achieved \n");
return 15;
}
cancel = 1;
for(l = k; l >= 0; --l) {
if (fabs(e[l]) <= eps) {
cancel = 0; //test f convergence
break;
}
if (fabs(q[l-1]) <= eps) {
//Cancel
break;
}
}
if (cancel) {
c = 0.0;
s = 1.0;
l1 = l - 1;
for(i = l; i <= k;++i) {
f = s*e[i];
e[i] *= c;
if (fabs(f) <= eps) {
break;
}
g = q[i];
h = q[i] = hypot(f,g);
c = g/h;
s = -f/h;
for(j = 0; j < M;++j) {
t = j * N;
y = U[t+l1];
z = U[t+i];
U[t+l1] = y * c + z * s;
U[t+i] = z * c - y * s;
}
}
}
z = q[k];
if (l != k) {
x = q[l];
y = q[k-1];
g = e[k-1];
h = e[k];
f = 0.5 * (((g + z) / h) * ((g - z) / y) + y / h - h / y);
g = hypot(f,1.0);
if (f < 0.0) {
temp = f - g;
} else {
temp = f+g;
}
f = x - (z / x) * z + (h / x) * (y / temp - h);
//Next QR Transformation
c = s = 1.0;
for(i = l+1; i <= k;++i) {
g = e[i];
y = q[i];
h = s * g;
g = c * g;
e[i-1] = z = hypot(f,h);
c = f / z;
s = h / z;
f = x * c + g * s;
g = g * c - x * s;
h = y * s;
y *= c;
for(j = 0; j < N;++j) {
t = j * N;
x = V[t+i-1];
z = V[t+i];
V[t+i-1] = x * c + z * s;
V[t+i] = z * c - x * s;
}
q[i-1] = z = hypot(f,h);
if (z != 0.0) {
c = f / z;
s = h / z;
}
f = c * g + s * y;
x = c * y - s * g;
for(j = 0; j < M;++j) {
t = j * N;
y = U[t+i-1];
z = U[t+i];
U[t+i-1] = y * c + z * s;
U[t+i] = z * c - y * s;
}
}
e[l] = 0.0;
e[k] = f;
q[k] = x;
} else {
//convergence
if (z < 0.0) {
q[k] = -z;
for (j = 0; j < N; j++) {
t = j *N;
V[t+k] = -V[t+k];
}