package ch.akuhn.edu.mit.tedlab;
import static ch.akuhn.edu.mit.tedlab.Revision39.storeVals.RETRP;
import static ch.akuhn.edu.mit.tedlab.Revision39.storeVals.RETRQ;
import static ch.akuhn.edu.mit.tedlab.Revision39.storeVals.STORP;
import static ch.akuhn.edu.mit.tedlab.Revision39.storeVals.STORQ;
import java.io.File;
import java.io.FileNotFoundException;
import java.util.Random;
import java.util.Scanner;
public class Revision39 {
long[] svd_longArray(int size, boolean empty, String name) {
return new long[size];
}
double[] svd_doubleArray(int size, boolean empty, String name) {
return new double[size];
}
void svd_beep() {
System.err.print((char) 10);
}
void svd_debug(String fmt, Object... args) {
System.err.printf(fmt, args);
}
void svd_error(String fmt, Object... args) {
svd_beep();
System.err.print("ERROR: ");
System.err.printf(fmt, args);
System.err.println();
}
void svd_fatalError(String fmt, Object... args) {
svd_error(fmt, args);
System.exit(1);
}
boolean stringEndsIn(String s, String t) {
return s.endsWith(t);
}
/**************************************************************
* returns |a| if b is positive; else fsign returns -|a| *
**************************************************************/
double svd_fsign(double a, double b) {
if ((a >= 0.0 && b >= 0.0) || (a < 0.0 && b < 0.0))
return a;
else
return -a;
}
/**************************************************************
* returns the larger of two double precision numbers *
**************************************************************/
double svd_dmax(double a, double b) {
return (a > b) ? a : b;
}
/**************************************************************
* returns the smaller of two double precision numbers *
**************************************************************/
double svd_dmin(double a, double b) {
return (a < b) ? a : b;
}
/**************************************************************
* returns the larger of two integers *
**************************************************************/
int svd_imax(int a, int b) {
return (a > b) ? a : b;
}
/**************************************************************
* returns the smaller of two integers *
**************************************************************/
int svd_imin(int a, int b) {
return (a < b) ? a : b;
}
/**************************************************************
* Function scales a vector by a constant. * Based on Fortran-77 routine
* from Linpack by J. Dongarra *
**************************************************************/
void svd_dscal(int n, double da, double[] dx, int incx) {
if (n <= 0 || incx == 0) return;
int ix = (incx < 0) ? n - 1 : 0;
for (int i=0; i < n; i++) {
dx[ix] *= da;
ix += incx;
}
return;
}
/**************************************************************
* function scales a vector by a constant. * Based on Fortran-77 routine
* from Linpack by J. Dongarra *
**************************************************************/
void svd_datx(int n, double da, double[] dx, int incx, double[] dy,
int incy) {
assert incx == 1 || incx == -1 || incx == 0;
assert incy == 1 || incy == -1 || incy == 0;
if (n <= 0 || incx == 0 || incy == 0) return;
int ix = (incx == 1) ? 0 : n - 1;
int iy = (incy == 1) ? 0 : n - 1;
for (int i = 0; i < n; i++) {
dy[iy] = da * dx[ix];
iy += incy;
ix += incx;
}
}
/**************************************************************
* Function copies a vector x to a vector y * Based on Fortran-77 routine
* from Linpack by J. Dongarra *
**************************************************************/
void svd_dcopy(int n, double[] dx, int incx, double[] dy, int incy) {
svd_dcopy(n, dx, 0, incx, dy, 0, incy);
}
void svd_dcopy(int n, double[] dx, int ix0, int incx, double[] dy, int iy0, int incy) {
assert incx == 1 || incx == -1 || incx == 0;
assert incy == 1 || incy == -1 || incy == 0;
if (n <= 0 || incx == 0 || incy == 0) return;
int ix = (incx == 1) ? ix0 : n - 1 + ix0;
int iy = (incy == 1) ? iy0 : n - 1 + iy0;
for (int i = 0; i < n; i++) {
dy[iy] = dx[ix];
iy += incy;
ix += incx;
}
}
/**************************************************************
* Function forms the dot product of two vectors. * Based on Fortran-77
* routine from Linpack by J. Dongarra *
**************************************************************/
double svd_ddot(int n, double[] dx, int incx, double[] dy, int incy) {
double dot_product = 0.0;
int ix0 = 0;
int iy0 = 0;
assert incx == 1 || incx == -1 || incx == 0;
assert incy == 1 || incy == -1 || incy == 0;
if (n <= 0 || incx == 0 || incy == 0) return 0.0;
int ix = (incx == 1) ? ix0 : n - 1 + ix0;
int iy = (incy == 1) ? iy0 : n - 1 + iy0;
for (int i = 0; i < n; i++) {
dot_product += dy[iy] * dx[ix];
iy += incy;
ix += incx;
}
return dot_product;
}
/**************************************************************
* Constant times a vector plus a vector * Based on Fortran-77 routine from
* Linpack by J. Dongarra *
**************************************************************/
void svd_daxpy(int n, double da, double[] dx, int incx, double[] dy,
int incy) {
if (n <= 0 || incx == 0 || incy == 0) return;
int ix = (incx == 1) ? 0 : n - 1;
int iy = (incy == 1) ? 0 : n - 1;
for (int i = 0; i < n; i++) {
dy[iy] += da * dx[ix];
iy += incy;
ix += incx;
}
}
/*********************************************************************
* Function sorts array1 and array2 into increasing order for array1 *
*********************************************************************/
void svd_dsort2(int igap, int n, double[] array1, double[] array2) {
double temp;
int i, j, index;
if (0 == igap) return;
else {
for (i = igap; i < n; i++) {
j = i - igap;
index = i;
while (j >= 0 && array1[j] > array1[index]) {
temp = array1[j];
array1[j] = array1[index];
array1[index] = temp;
temp = array2[j];
array2[j] = array2[index];
array2[index] = temp;
j -= igap;
index = j + igap;
}
}
}
svd_dsort2(igap/2,n,array1,array2);
}
/**************************************************************
* Function interchanges two vectors * Based on Fortran-77 routine from
* Linpack by J. Dongarra *
**************************************************************/
void svd_dswap(int n, double[] dx, int incx, double[] dy, int incy) {
if (n <= 0 || incx == 0 || incy == 0) return;
int ix = (incx == 1) ? 0 : n - 1;
int iy = (incy == 1) ? 0 : n - 1;
for (int i = 0; i < n; i++) {
double swap = dy[iy];
dy[iy] = dx[ix];
dx[ix] = swap;
iy += incy;
ix += incx;
}
}
/*****************************************************************
* Function finds the index of element having max. absolute value* based on
* FORTRAN 77 routine from Linpack by J. Dongarra *
*****************************************************************/
int svd_idamax(int n, double[] dx, int ix0, int incx) {
int ix,imax;
double dmax;
if (n < 1) return -1;
if (n == 1) return 0;
if (incx == 0) return -1;
ix = (incx < 0) ? ix0 + ((-n+1) * incx) : ix0;
imax = ix;
dmax = fabs(dx[ix]);
for (int i=1; i < n; i++) {
ix += incx;
double dtemp = fabs(dx[ix]);
if (dtemp > dmax) {
dmax = dtemp;
imax = ix;
}
}
return imax;
}
/* Row-major dense matrix. Rows are consecutive vectors. */
class DMat {
int rows;
int cols;
double[][] value; /*
* Accessed by [row][col]. Free value[0] and value to
* free.
*/
DMat(int rows, int cols) {
this.rows = rows;
this.cols = cols;
this.value = new double[rows][cols];
}
}
/* Harwell-Boeing sparse matrix. */
class SMat {
int rows;
int cols;
int vals; /* Total non-zero entries. */
int[] pointr; /* For each col (plus 1), index of first non-zero entry. */
int[] rowind; /* For each nz entry, the row index. */
double[] value; /* For each nz entry, the value. */
SMat(int rows, int cols, int vals) {
this.rows = rows;
this.cols = cols;
this.vals = vals;
this.pointr = new int[cols + 1];
this.rowind = new int[vals];
this.value = new double[vals];
}
}
class SVDRec {
int d; /* Dimensionality (rank) */
DMat Ut; /*
* Transpose of left singular vectors. (d by m) The vectors are
* the rows of Ut.
*/
double[] S; /* Array of singular values. (length d) */
DMat Vt; /*
* Transpose of right singular vectors. (d by n) The vectors are
* the rows of Vt.
*/
SVDRec() {
}
}
/**************************************************************
* multiplication of matrix B by vector x, where B = A'A, * and A is nrow by
* ncol (nrow >> ncol). Hence, B is of order * n = ncol (y stores product
* vector). *
**************************************************************/
void svd_opb(SMat A, double[] x, double[] y, double[] temp) {
int[] pointr = A.pointr;
int[] rowind = A.rowind;
double[] value = A.value;
int n = A.cols;
//SVDCount[SVD_MXV] += 2;
//memset(y, 0, n * sizeof(double));
for (int i = 0; i < n; i++) y[i] = 0;
for (int i = 0; i < A.rows; i++) temp[i] = 0.0;
for (int i = 0; i < A.cols; i++) {
int end = pointr[i+1];
for (int j = pointr[i]; j < end; j++)
temp[rowind[j]] += value[j] * (x[i]);
}
for (int i = 0; i < A.cols; i++) {
int end = pointr[i+1];
for (int j = pointr[i]; j < end; j++)
y[i] += value[j] * temp[rowind[j]];
}
return;
}
/***********************************************************
* multiplication of matrix A by vector x, where A is * nrow by ncol (nrow
* >> ncol). y stores product vector. *
***********************************************************/
void svd_opa(SMat A, double[] x, double[] y) {
int[] pointr = A.pointr, rowind = A.rowind;
double[] value = A.value;
//SVDCount[SVD_MXV]++;
for (int i = 0; i < A.rows; i++) y[i] = 0;
for (int i = 0; i < A.cols; i++) {
int end = pointr[i+1];
for (int j = pointr[i]; j < end; j++)
y[rowind[j]] += value[j] * x[i];
}
return;
}
/***********************************************************************
* * random() * (double precision) *
***********************************************************************/
/***********************************************************************
* Description -----------
*
* This is a translation of a Fortran-77 uniform random number generator.
* The code is based on theory and suggestions given in D. E. Knuth (1969),
* vol 2. The argument to the function should be initialized to an arbitrary
* integer prior to the first call to random. The calling program should not
* alter the value of the argument between subsequent calls to random.
* Random returns values within the interval (0,1).
*
*
* Arguments ---------
*
* (input) iy an integer seed whose value must not be altered by the caller
* between subsequent calls
*
* (output) random a double precision random number between (0,1)
***********************************************************************/
double svd_random2(long[] iy) {
throw null;
// static long m2 = 0;
// static long ia, ic, mic;
// static double halfm, s;
//
// /* If first entry, compute (max int) / 2 */
// if (!m2) {
// m2 = 1 << (8 * (int)sizeof(int) - 2);
// halfm = m2;
//
// /* compute multiplier and increment for linear congruential
// * method */
// ia = 8 * (long)(halfm * atan(1.0) / 8.0) + 5;
// ic = 2 * (long)(halfm * (0.5 - sqrt(3.0)/6.0)) + 1;
// mic = (m2-ic) + m2;
//
// /* s is the scale factor for converting to floating point */
// s = 0.5 / halfm;
// }
//
// /* compute next random number */
// *iy = *iy * ia;
//
// /* for computers which do not allow integer overflow on addition */
// if (*iy > mic) *iy = (*iy - m2) - m2;
//
// *iy = *iy + ic;
//
// /* for computers whose word length for addition is greater than
// * for multiplication */
// if (*iy / 2 > m2) *iy = (*iy - m2) - m2;
//
// /* for computers whose integer overflow affects the sign bit */
// if (*iy < 0) *iy = (*iy + m2) + m2;
//
// return((double)(*iy) * s);
}
/**************************************************************
* * Function finds sqrt(a^2 + b^2) without overflow or * destructive
* underflow. * *
**************************************************************/
/**************************************************************
* Funtions used -------------
*
* UTILITY dmax, dmin
**************************************************************/
double svd_pythag(double a, double b) {
double p, r, s, t, u, temp;
p = svd_dmax(Math.abs(a), Math.abs(b));
if (p != 0.0) {
temp = svd_dmin(Math.abs(a), Math.abs(b)) / p;
r = temp * temp;
t = 4.0 + r;
while (t != 4.0) {
s = r / t;
u = 1.0 + 2.0 * s;
p *= u;
temp = s / u;
r *= temp * temp;
t = 4.0 + r;
}
}
return p;
}
String SVDVersion = "1.34";
long SVDVerbosity = 2;
void svdResetCounters() {
throw null;
}
/**************************** Conversion *************************************/
/* Converts a sparse matrix to a dense one (without affecting the former) */
DMat svdConvertStoD(SMat S) {
throw null;
// int i, c;
// DMat D = svdNewDMat(S->rows, S->cols);
// if (!D) {
// svd_error("svdConvertStoD: failed to allocate D");
// return NULL;
// }
// for (i = 0, c = 0; i < S->vals; i++) {
// while (S->pointr[c + 1] <= i) c++;
// D->value[S->rowind[i]][c] = S->value[i];
// }
// return D;
}
/* Converts a dense matrix to a sparse one (without affecting the dense one) */
SMat svdConvertDtoS(DMat D) {
SMat S;
int i, j, n;
for (i = 0, n = 0; i < D.rows; i++)
for (j = 0; j < D.cols; j++)
if (D.value[i][j] != 0) n++;
S = new SMat(D.rows, D.cols, n);
for (j = 0, n = 0; j < D.cols; j++) {
S.pointr[j] = n;
for (i = 0; i < D.rows; i++)
if (D.value[i][j] != 0) {
S.rowind[n] = i;
S.value[n] = D.value[i][j];
n++;
}
}
S.pointr[S.cols] = S.vals;
return S;
}
/* Transposes a dense matrix. */
DMat svdTransposeD(DMat D) {
int r, c;
DMat N = new DMat(D.cols, D.rows);
for (r = 0; r < D.rows; r++)
for (c = 0; c < D.cols; c++)
N.value[c][r] = D.value[r][c];
return N;
}
/* Efficiently transposes a sparse matrix. */
SMat svdTransposeS(SMat S) {
int r, c, i, j;
SMat N = new SMat(S.cols, S.rows, S.vals);
/* Count number nz in each row. */
for (i = 0; i < S.vals; i++)
N.pointr[S.rowind[i]]++;
/* Fill each cell with the starting point of the previous row. */
N.pointr[S.rows] = S.vals - N.pointr[S.rows - 1];
for (r = S.rows - 1; r > 0; r--)
N.pointr[r] = N.pointr[r + 1] - N.pointr[r - 1];
N.pointr[0] = 0;
/* Assign the new columns and values. */
for (c = 0, i = 0; c < S.cols; c++) {
for (; i < S.pointr[c + 1]; i++) {
r = S.rowind[i];
j = N.pointr[r + 1]++;
N.rowind[j] = c;
N.value[j] = S.value[i];
}
}
return N;
}
/*************************************************************************
(c) Copyright 2003
Douglas Rohde
adapted from SVDPACKC, which is
(c) Copyright 1993
University of Tennessee
All Rights Reserved
*************************************************************************/
static int MAXLL = 2;
enum storeVals {STORQ, RETRQ, STORP, RETRP};
static String[] error_msg = { /* error messages used by function *
* check_parameters */
null,
"",
"ENDL MUST BE LESS THAN ENDR",
"REQUESTED DIMENSIONS CANNOT EXCEED NUM ITERATIONS",
"ONE OF YOUR DIMENSIONS IS LESS THAN OR EQUAL TO ZERO",
"NUM ITERATIONS (NUMBER OF LANCZOS STEPS) IS INVALID",
"REQUESTED DIMENSIONS (NUMBER OF EIGENPAIRS DESIRED) IS INVALID",
"6*N+4*ITERATIONS+1 + ITERATIONS*ITERATIONS CANNOT EXCEED NW",
"6*N+4*ITERATIONS+1 CANNOT EXCEED NW", null};
double[][] LanStore;
double[] OPBTemp;
double eps, eps1, reps, eps34;
long ierr;
/*
double rnm, anorm, tol;
FILE *fp_out1, *fp_out2;
*/
/***********************************************************************
* *
* main() *
* Sparse SVD(A) via Eigensystem of A'A symmetric Matrix *
* (double precision) *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
This sample program uses landr to compute singular triplets of A via
the equivalent symmetric eigenvalue problem
B x = lambda x, where x' = (u',v'), lambda = sigma**2,
where sigma is a singular value of A,
B = A'A , and A is m (nrow) by n (ncol) (nrow >> ncol),
so that {u,sqrt(lambda),v} is a singular triplet of A.
(A' = transpose of A)
User supplied routines: svd_opa, opb, store, timer
svd_opa( x,y) takes an n-vector x and returns A*x in y.
svd_opb(ncol,x,y) takes an n-vector x and returns B*x in y.
Based on operation flag isw, store(n,isw,j,s) stores/retrieves
to/from storage a vector of length n in s.
User should edit timer() with an appropriate call to an intrinsic
timing routine that returns elapsed user time.
External parameters
-------------------
Defined and documented in las2.h
Local parameters
----------------
(input)
endl left end of interval containing unwanted eigenvalues of B
endr right end of interval containing unwanted eigenvalues of B
kappa relative accuracy of ritz values acceptable as eigenvalues
of B
vectors is not equal to 1
r work array
n dimension of the eigenproblem for matrix B (ncol)
dimensions upper limit of desired number of singular triplets of A
iterations upper limit of desired number of Lanczos steps
nnzero number of nonzeros in A
vectors 1 indicates both singular values and singular vectors are
wanted and they can be found in output file lav2;
0 indicates only singular values are wanted
(output)
ritz array of ritz values
bnd array of error bounds
d array of singular values
memory total memory allocated in bytes to solve the B-eigenproblem
Functions used
--------------
BLAS svd_daxpy, svd_dscal, svd_ddot
USER svd_opa, svd_opb, timer
MISC write_header, check_parameters
LAS2 landr
Precision
---------
All floating-point calculations are done in double precision;
variables are declared as long and double.
LAS2 development
----------------
LAS2 is a C translation of the Fortran-77 LAS2 from the SVDPACK
library written by Michael W. Berry, University of Tennessee,
Dept. of Computer Science, 107 Ayres Hall, Knoxville, TN, 37996-1301
31 Jan 1992: Date written
Theresa H. Do
University of Tennessee
Dept. of Computer Science
107 Ayres Hall
Knoxville, TN, 37996-1301
internet: tdo@cs.utk.edu
***********************************************************************/
/***********************************************************************
* *
* check_parameters() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Function validates input parameters and returns error code (long)
Parameters
----------
(input)
dimensions upper limit of desired number of eigenpairs of B
iterations upper limit of desired number of lanczos steps
n dimension of the eigenproblem for matrix B
endl left end of interval containing unwanted eigenvalues of B
endr right end of interval containing unwanted eigenvalues of B
vectors 1 indicates both eigenvalues and eigenvectors are wanted
and they can be found in lav2; 0 indicates eigenvalues only
nnzero number of nonzero elements in input matrix (matrix A)
***********************************************************************/
int check_parameters(SMat A, long dimensions, long iterations,
double endl, double endr, boolean b) {
int error_index;
error_index = 0;
if (endl >/*=*/ endr) error_index = 2;
else if (dimensions > iterations) error_index = 3;
else if (A.cols <= 0 || A.rows <= 0) error_index = 4;
/*else if (n > A->cols || n > A->rows) error_index = 1;*/
else if (iterations <= 0 || iterations > A.cols || iterations > A.rows)
error_index = 5;
else if (dimensions <= 0 || dimensions > iterations) error_index = 6;
if (0 != error_index)
svd_error("svdLAS2 parameter error: %s\n", error_msg[error_index]);
return(error_index);
}
/***********************************************************************
* *
* write_header() *
* Function writes out header of output file containing ritz values *
* *
***********************************************************************/
void write_header(long iterations, long dimensions, double endl, double endr,
boolean b, double kappa, long nrow, long ncol,
long vals) {
printf("SOLVING THE [A^TA] EIGENPROBLEM\n");
printf("NO. OF ROWS = %6d\n", nrow);
printf("NO. OF COLUMNS = %6d\n", ncol);
printf("NO. OF NON-ZERO VALUES = %6d\n", vals);
printf("MATRIX DENSITY = %6.2f%%\n",
((float) vals / nrow) * 100 / ncol);
/* printf("ORDER OF MATRIX A = %5ld\n", n); */
printf("MAX. NO. OF LANCZOS STEPS = %6d\n", iterations);
printf("MAX. NO. OF EIGENPAIRS = %6d\n", dimensions);
printf("LEFT END OF THE INTERVAL = %9.2E\n", endl);
printf("RIGHT END OF THE INTERVAL = %9.2E\n", endr);
printf("KAPPA = %9.2E\n", kappa);
/* printf("WANT S-VECTORS? [T/F] = %c\n", (vectors) ? 'T' : 'F'); */
printf("\n");
return;
}
void printf(String fmt, Object ... args) {
System.out.printf(fmt, args);
}
/***********************************************************************
* *
* landr() *
* Lanczos algorithm with selective orthogonalization *
* Using Simon's Recurrence *
* (double precision) *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
landr() is the LAS2 driver routine that, upon entry,
(1) checks for the validity of input parameters of the
B-eigenproblem
(2) determines several machine constants
(3) makes a Lanczos run
(4) calculates B-eigenvectors (singular vectors of A) if requested
by user
arguments
---------
(input)
n dimension of the eigenproblem for A'A
iterations upper limit of desired number of Lanczos steps
dimensions upper limit of desired number of eigenpairs
nnzero number of nonzeros in matrix A
endl left end of interval containing unwanted eigenvalues of B
endr right end of interval containing unwanted eigenvalues of B
vectors 1 indicates both eigenvalues and eigenvectors are wanted
and they can be found in output file lav2;
0 indicates only eigenvalues are wanted
kappa relative accuracy of ritz values acceptable as eigenvalues
of B (singular values of A)
r work array
(output)
j number of Lanczos steps actually taken
neig number of ritz values stabilized
ritz array to hold the ritz values
bnd array to hold the error bounds
External parameters
-------------------
Defined and documented in las2.h
local parameters
-------------------
ibeta radix for the floating-point representation
it number of base ibeta digits in the floating-point significand
irnd floating-point addition rounded or chopped
machep machine relative precision or round-off error
negeps largest negative integer
wptr array of pointers each pointing to a work space
Functions used
--------------
MISC svd_dmax, machar, check_parameters
LAS2 ritvec, lanso
***********************************************************************/
void fake_memset_127(double[] a) {
double d = Double.longBitsToDouble(0x7f7f7f7f7f7f7f7fL);
for (int n = 0; n < a.length; n++) {
a[n] = d;
}
}
SVDRec svdLAS2A(SMat A, int dimensions) {
double[] end = new double[] {-1.0e-30, 1.0e-30};
double kappa = 1e-6;
if (A == null) {
svd_error("svdLAS2A called with NULL array\n");
return null;
}
return svdLAS2(A, dimensions, 0, end, kappa);
}
SVDRec svdLAS2(SMat A, int dimensions, int iterations, double[] end,
double kappa) {
boolean transpose = false;
//long ibeta, it, irnd, machep, negep, nsig;
int n, m, i, steps;
double[][] wptr = new double[10][];
double[] ritz;
double[] bnd;
SVDRec R = null;
//svdResetCounters();
m = svd_imin(A.rows, A.cols);
if (dimensions <= 0 || dimensions > m)
dimensions = m;
if (iterations <= 0 || iterations > m)
iterations = m;
if (iterations < dimensions) iterations = dimensions;
/* Write output header */
if (SVDVerbosity > 0)
write_header(iterations, dimensions, end[0], end[1], true, kappa, A.rows,
A.cols, A.vals);
/* Check parameters */
if (0 != check_parameters(A, dimensions, iterations, end[0], end[1], true))
return null;
/* If A is wide, the SVD is computed on its transpose for speed. */
if (A.cols >= A.rows * 1.2) {
if (SVDVerbosity > 0) printf("TRANSPOSING THE MATRIX FOR SPEED\n");
transpose = true;
A = svdTransposeS(A);
}
n = A.cols;
/* BEGIN Compute machine precision */
/*long[] machar_result =*/ machar(/* &ibeta, &it, &irnd, &machep, &negep */);
//ibeta = machar_result[0];
//it = machar_result[1];
//irnd = machar_result[2];
//machep = machar_result[3];
//negep = machar_result[4];
/* END Compute machine precision */
eps1 = eps * Math.sqrt((double) n);
reps = Math.sqrt(eps);
eps34 = reps * Math.sqrt(reps);
/* Allocate temporary space. */
wptr[0] = new double[n];
wptr[1] = new double[n];
wptr[2] = new double[n];
wptr[3] = new double[n];
wptr[4] = new double[n];
wptr[5] = new double[n];
wptr[6] = new double[iterations];
wptr[7] = new double[iterations];
wptr[8] = new double[iterations];
wptr[9] = new double[iterations + 1];
ritz = new double[iterations + 1];
bnd = new double[iterations + 1];
fake_memset_127(bnd);
LanStore = new double[iterations + MAXLL][];
OPBTemp = svd_doubleArray(A.rows, false, "las2: OPBTemp");
/* Actually run the lanczos thing: */
int[] ref_neig = new int[] { 0 }; // XXX wrap neig
steps = lanso(A, iterations, dimensions, end[0], end[1], ritz, bnd, wptr,
ref_neig, n);
int neig = ref_neig[0]; // XXX unwrap neig
/* Print some stuff. */
if (SVDVerbosity > 0) {
printf("NUMBER OF LANCZOS STEPS = %6d\n" +
"RITZ VALUES STABILIZED = %6d\n", steps + 1, neig);
}
if (SVDVerbosity > 2) {
printf("\nCOMPUTED RITZ VALUES (ERROR BNDS)\n");
for (i = 0; i <= steps; i++)
printf("%3d %22.14E (%11.2E)\n", i + 1, ritz[i], bnd[i]);
}
wptr[0] = null;
wptr[1] = null;
wptr[2] = null;
wptr[3] = null;
wptr[4] = null;
wptr[7] = null;
wptr[8] = null;
/* Compute eigenvectors */
kappa = svd_dmax(fabs(kappa), eps34);
R = new SVDRec();
R.d = /*svd_imin(nsig, dimensions)*/dimensions;
R.Ut = new DMat(R.d, A.rows);
R.S = svd_doubleArray(R.d, true, "las2: R->s");
R.Vt = new DMat(R.d, A.cols);
/*nsig =*/ ritvec(n, A, R, kappa, ritz, bnd, wptr[6], wptr[9], wptr[5], steps,
neig);
if (SVDVerbosity > 1) {
printf("\nSINGULAR VALUES: ");
svdWriteDenseArray(R.S, R.d, "-", false);
if (SVDVerbosity > 2) {
printf("\nLEFT SINGULAR VECTORS (transpose of U): ");
// svdWriteDenseMatrix(R.Ut, "-", SVD_F_DT); TODO outout
printf("\nRIGHT SINGULAR VECTORS (transpose of V): ");
// svdWriteDenseMatrix(R.Vt, "-", SVD_F_DT); TODO output
}
} else if (SVDVerbosity > 0)
printf("SINGULAR VALUES FOUND = %6d\n", R.d);
/* This swaps and transposes the singular matrices if A was transposed. */
if (transpose) {
DMat swap = R.Ut;
R.Ut = R.Vt;
R.Vt = swap;
}
return R;
}
void svdWriteDenseArray(double[] s, int d, String string, boolean b) {
System.out.println("Declare victory!"); // TODO better print
}
/***********************************************************************
* *
* ritvec() *
* Function computes the singular vectors of matrix A *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
This function is invoked by landr() only if eigenvectors of the A'A
eigenproblem are desired. When called, ritvec() computes the
singular vectors of A and writes the result to an unformatted file.
Parameters
----------
(input)
nrow number of rows of A
steps number of Lanczos iterations performed
fp_out2 pointer to unformatted output file
n dimension of matrix A
kappa relative accuracy of ritz values acceptable as
eigenvalues of A'A
ritz array of ritz values
bnd array of error bounds
alf array of diagonal elements of the tridiagonal matrix T
bet array of off-diagonal elements of T
w1, w2 work space
(output)
xv1 array of eigenvectors of A'A (right singular vectors of A)
ierr error code
0 for normal return from imtql2()
k if convergence did not occur for k-th eigenvalue in
imtql2()
nsig number of accepted ritz values based on kappa
(local)
s work array which is initialized to the identity matrix
of order (j + 1) upon calling imtql2(). After the call,
s contains the orthonormal eigenvectors of the symmetric
tridiagonal matrix T
Functions used
--------------
BLAS svd_dscal, svd_dcopy, svd_daxpy
USER store
imtql2
***********************************************************************/
void rotateArray(double[][] a, int size, int x) {
// TODO fix me, in Java we cannot access a[] as a[][] !!!
int i, j, n, start;
double t1, t2;
if (x == 0) return;
j = start = 0;
t1 = a[0][0];
int len = a.length;
for (i = 0; i < size; i++) {
n = (j >= x) ? j - x : j + size - x;
t2 = a[n % len][n / len];
a[n % len][n / len] = t1;
t1 = t2;
j = n;
if (j == start) {
start = ++j;
t1 = a[j % len][j / len];
}
}
}
long ritvec(int n, SMat A, SVDRec R, double kappa, double[] ritz, double[] bnd,
double[] alf, double[] bet, double[] w2, int steps, long neig) {
int k, x, i, jsq, js, tmp, id2, nsig;
double[] s;
double[] xv2;
double tmp0, tmp1, xnorm;
double[] w1 = R.Vt.value[0];
js = steps + 1;
jsq = js * js;
/*size = sizeof(double) * n;*/
s = svd_doubleArray(jsq, true, "ritvec: s");
xv2 = svd_doubleArray(n, false, "ritvec: xv2");
/* initialize s to an identity matrix */
for (i = 0; i < jsq; i+= (js+1)) {
s[i] = 1.0;
}
svd_dcopy(js, alf, 1, w1, -1);
svd_dcopy(steps, bet, 1, 1, w2, 1, -1); // WAS svd_dcopy(steps, &bet[1], 1, &w2[1], -1);
/* on return from imtql2(), w1 contains eigenvalues in ascending
* order and s contains the corresponding eigenvectors */
imtql2(js, js, w1, w2, s);
if (0 != ierr) return 0; // TODO use exception here?
/*fwrite((char *)&n, sizeof(n), 1, fp_out2);
fwrite((char *)&js, sizeof(js), 1, fp_out2);
fwrite((char *)&kappa, sizeof(kappa), 1, fp_out2);*/
/*id = 0;*/
nsig = 0;
x = 0;
id2 = jsq - js;
for (k = 0; k < js; k++) {
tmp = id2;
if (bnd[k] <= kappa * Math.abs(ritz[k]) && k > js-neig-1) {
if (--x < 0) x = R.d - 1;
w1 = R.Vt.value[x];
for (i = 0; i < n; i++) w1[i] = 0.0;
for (i = 0; i < js; i++) {
store(n, storeVals.RETRQ, i, w2);
svd_daxpy(n, s[tmp], w2, 1, w1, 1);
tmp -= js;
}
/*fwrite((char *)w1, size, 1, fp_out2);*/
/* store the w1 vector row-wise in array xv1;
* size of xv1 is (steps+1) * (nrow+ncol) elements
* and each vector, even though only ncol long,
* will have (nrow+ncol) elements in xv1.
* It is as if xv1 is a 2-d array (steps+1) by
* (nrow+ncol) and each vector occupies a row */
/* j is the index in the R arrays, which are sorted by high to low
singular values. */
/*for (i = 0; i < n; i++) R->Vt->value[x]xv1[id++] = w1[i];*/
/*id += nrow;*/
nsig++;
}
id2++;
}
s = null;
/* Rotate the singular vectors and values. */
/* x is now the location of the highest singular value. */
rotateArray(R.Vt.value, R.Vt.rows * R.Vt.cols,
x * R.Vt.cols);
R.d = svd_imin(R.d, nsig);
for (x = 0; x < R.d; x++) {
/* multiply by matrix B first */
svd_opb(A, R.Vt.value[x], xv2, OPBTemp);
tmp0 = svd_ddot(n, R.Vt.value[x], 1, xv2, 1);
svd_daxpy(n, -tmp0, R.Vt.value[x], 1, xv2, 1);
tmp0 = Math.sqrt(tmp0);
xnorm = Math.sqrt(svd_ddot(n, xv2, 1, xv2, 1));
/* multiply by matrix A to get (scaled) left s-vector */
svd_opa(A, R.Vt.value[x], R.Ut.value[x]);
tmp1 = 1.0 / tmp0;
svd_dscal(A.rows, tmp1, R.Ut.value[x], 1);
xnorm *= tmp1;
bnd[i] = xnorm;
R.S[x] = tmp0;
}
xv2 = null;
return nsig;
}
/***********************************************************************
* *
* lanso() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Function determines when the restart of the Lanczos algorithm should
occur and when it should terminate.
Arguments
---------
(input)
n dimension of the eigenproblem for matrix B
iterations upper limit of desired number of lanczos steps
dimensions upper limit of desired number of eigenpairs
endl left end of interval containing unwanted eigenvalues
endr right end of interval containing unwanted eigenvalues
ritz array to hold the ritz values
bnd array to hold the error bounds
wptr array of pointers that point to work space:
wptr[0]-wptr[5] six vectors of length n
wptr[6] array to hold diagonal of the tridiagonal matrix T
wptr[9] array to hold off-diagonal of T
wptr[7] orthogonality estimate of Lanczos vectors at
step j
wptr[8] orthogonality estimate of Lanczos vectors at
step j-1
(output)
j number of Lanczos steps actually taken
neig number of ritz values stabilized
ritz array to hold the ritz values
bnd array to hold the error bounds
ierr (globally declared) error flag
ierr = 8192 if stpone() fails to find a starting vector
ierr = k if convergence did not occur for k-th eigenvalue
in imtqlb()
ierr = 0 otherwise
Functions used
--------------
LAS stpone, error_bound, lanczos_step
MISC svd_dsort2
UTILITY svd_imin, svd_imax
***********************************************************************/
int lanso(SMat A, int iterations, int dimensions, double endl,
double endr, double[] ritz, double[] bnd, double[][] wptr,
int[] neigp, int n) {
double[] alf, eta, oldeta, bet, wrk;
int ll, neig, j = 0, intro = 0, last, i, l, id3, first;
boolean ENOUGH;
alf = wptr[6];
eta = wptr[7];
oldeta = wptr[8];
bet = wptr[9];
wrk = wptr[5];
/* take the first step */
double[] ref_rnm = new double[] { 0d }; // XXX wrap
double[] ref_tol = new double[] { 0d }; // XXX wrap
stpone(A, wptr, ref_rnm, ref_tol, n);
double tol = ref_tol[0]; // XXX unwrap
double rnm = ref_rnm[0]; // XXX unwrap
if (/* !rnm */ 0 == rnm || 0 != ierr) throw null;
eta[0] = eps1;
oldeta[0] = eps1;
ll = 0;
first = 1;
last = svd_imin(dimensions + svd_imax(8, dimensions), iterations);
ENOUGH = false;
/*id1 = 0;*/
while (/*id1 < dimensions && */!ENOUGH) {
if (rnm <= tol) rnm = 0.0;
/* the actual lanczos loop */
int[] ref_ll = new int[] { ll }; // XXX wrap
boolean[] ref_ENOUGH = new boolean[] { ENOUGH }; // XXX wrap
double[] ref2_rnm = new double[] { rnm }; // XXX wrap
double[] ref2_tol = new double[] { tol }; // XXX wrap
j = lanczos_step(A, first, last, wptr, alf, eta, oldeta, bet, ref_ll,
ref_ENOUGH, ref2_rnm, ref2_tol, n);
ll = ref_ll[0]; // XXX unwrap
ENOUGH = ref_ENOUGH[0]; // XXX unwrap
tol = ref2_tol[0]; // XXX unwrap
rnm = ref2_rnm[0]; // XXX unwrap
if (ENOUGH) j = j - 1;
else j = last - 1;
first = j + 1;
bet[j+1] = rnm;
/* analyze T */
l = 0;
for (int id2 = 0; id2 < j; id2++) {
if (l > j) break;
for (i = l; i <= j; i++) if (/* !bet[i+1] */ 0 == bet[i+1]) break;
if (i > j) i = j;
/* now i is at the end of an unreduced submatrix */
svd_dcopy(i-l+1, alf, l, 1, ritz, l, -1); // WAS svd_dcopy(i-l+1, &alf[l], 1, &ritz[l], -1);
svd_dcopy(i-l, bet, l+1, 1, wrk, l+1, -1); // WAS svd_dcopy(i-l, &bet[l+1], 1, &wrk[l+1], -1);
imtqlb(i-l+1, ritz, wrk, bnd, l); // TODO start at l
if (0 != ierr) {
svd_error("svdLAS2: imtqlb failed to converge (ierr = %ld)\n", ierr);
svd_error(" l = %ld i = %ld\n", l, i);
for (id3 = l; id3 <= i; id3++)
svd_error(" %ld %lg %lg %lg\n",
id3, ritz[id3], wrk[id3], bnd[id3]);
}
for (id3 = l; id3 <= i; id3++)
bnd[id3] = rnm * fabs(bnd[id3]);
l = i + 1;
}
/* sort eigenvalues into increasing order */
svd_dsort2((j+1) / 2, j + 1, ritz, bnd);
/* for (i = 0; i < iterations; i++)
printf("%f ", ritz[i]);
printf("\n"); */
/* massage error bounds for very close ritz values */
boolean[] ref2_ENOUGH = new boolean[] { ENOUGH }; // XXX wrap
neig = error_bound(ref2_ENOUGH, endl, endr, ritz, bnd, j, tol);
ENOUGH = ref2_ENOUGH[0]; // XXX unwrap
neigp[0] = neig;
/* should we stop? */
if (neig < dimensions) {
if (/* !neig */ 0 == neig) {
last = first + 9;
intro = first;
} else last = first + svd_imax(3, 1 + ((j - intro) * (dimensions-neig)) /
neig);
last = svd_imin(last, iterations);
} else ENOUGH = true;
ENOUGH = ENOUGH || first >= iterations;
/* id1++; */
/* printf("id1=%d dimen=%d first=%d\n", id1, dimensions, first); */
}
store(n, storeVals.STORQ, j, wptr[1]);
return j;
}
/***********************************************************************
* *
* lanczos_step() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Function embodies a single Lanczos step
Arguments
---------
(input)
n dimension of the eigenproblem for matrix B
first start of index through loop
last end of index through loop
wptr array of pointers pointing to work space
alf array to hold diagonal of the tridiagonal matrix T
eta orthogonality estimate of Lanczos vectors at step j
oldeta orthogonality estimate of Lanczos vectors at step j-1
bet array to hold off-diagonal of T
ll number of intitial Lanczos vectors in local orthog.
(has value of 0, 1 or 2)
enough stop flag
Functions used
--------------
BLAS svd_ddot, svd_dscal, svd_daxpy, svd_datx, svd_dcopy
USER store
LAS purge, ortbnd, startv
UTILITY svd_imin, svd_imax
***********************************************************************/
int lanczos_step(SMat A, int first, int last, double[][] wptr,
double[] alf, double[] eta, double[] oldeta,
double[] bet, int[] ll, boolean[] refEnough, double[] rnmp,
double[] tolp, int n) {
double t;
double[] mid;
double rnm = rnmp[0];
double tol = tolp[0];
double anorm;
int i, j;
for (j=first; j<last; j++) {
mid = wptr[2];
wptr[2] = wptr[1];
wptr[1] = mid;
mid = wptr[3];
wptr[3] = wptr[4];
wptr[4] = mid;
store(n, STORQ, j-1, wptr[2]);
if (j-1 < MAXLL) store(n, STORP, j-1, wptr[4]);
bet[j] = rnm;
/* restart if invariant subspace is found */
if (/* !bet[j] */ 0 == bet[j]) {
rnm = startv(A, wptr, j, n);
if (0 != ierr) return j;
if (/* !rnm */ 0 == rnm) refEnough[0] = true;
}
if (refEnough[0]) {
/* added by Doug... */
/* These lines fix a bug that occurs with low-rank matrices */
mid = wptr[2];
wptr[2] = wptr[1];
wptr[1] = mid;
/* ...added by Doug */
break;
}
/* take a lanczos step */
t = 1.0 / rnm;
svd_datx(n, t, wptr[0], 1, wptr[1], 1);
svd_dscal(n, t, wptr[3], 1);
svd_opb(A, wptr[3], wptr[0], OPBTemp);
svd_daxpy(n, -rnm, wptr[2], 1, wptr[0], 1);
alf[j] = svd_ddot(n, wptr[0], 1, wptr[3], 1);
svd_daxpy(n, -alf[j], wptr[1], 1, wptr[0], 1);
/* orthogonalize against initial lanczos vectors */
if (j <= MAXLL && (Math.abs(alf[j-1]) > 4.0 * Math.abs(alf[j])))
ll[0] = j;
for (i=0; i < svd_imin(ll[0], j-1); i++) {
store(n, RETRP, i, wptr[5]);
t = svd_ddot(n, wptr[5], 1, wptr[0], 1);
store(n, RETRQ, i, wptr[5]);
svd_daxpy(n, -t, wptr[5], 1, wptr[0], 1);
eta[i] = eps1;
oldeta[i] = eps1;
}
/* extended local reorthogonalization */
t = svd_ddot(n, wptr[0], 1, wptr[4], 1);
svd_daxpy(n, -t, wptr[2], 1, wptr[0], 1);
if (bet[j] > 0.0) bet[j] = bet[j] + t;
t = svd_ddot(n, wptr[0], 1, wptr[3], 1);
svd_daxpy(n, -t, wptr[1], 1, wptr[0], 1);
alf[j] = alf[j] + t;
svd_dcopy(n, wptr[0], 1, wptr[4], 1);
rnm = Math.sqrt(svd_ddot(n, wptr[0], 1, wptr[4], 1));
anorm = bet[j] + Math.abs(alf[j]) + rnm;
tol = reps * anorm;
/* update the orthogonality bounds */
ortbnd(alf, eta, oldeta, bet, j, rnm);
/* restore the orthogonality state when needed */
double[] ref_rnm = new double[] { rnm }; // XXX wrap
purge(n, ll[0], wptr[0], wptr[1], wptr[4], wptr[3], wptr[5], eta, oldeta,
j, ref_rnm, tol);
rnm = ref_rnm[0]; // XXX unwrap
if (rnm <= tol) rnm = 0.0;
}
rnmp[0] = rnm;
tolp[0] = tol;
return j;
}
/***********************************************************************
* *
* ortbnd() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Funtion updates the eta recurrence
Arguments
---------
(input)
alf array to hold diagonal of the tridiagonal matrix T
eta orthogonality estimate of Lanczos vectors at step j
oldeta orthogonality estimate of Lanczos vectors at step j-1
bet array to hold off-diagonal of T
n dimension of the eigenproblem for matrix B
j dimension of T
rnm norm of the next residual vector
eps1 roundoff estimate for dot product of two unit vectors
(output)
eta orthogonality estimate of Lanczos vectors at step j+1
oldeta orthogonality estimate of Lanczos vectors at step j
Functions used
--------------
BLAS svd_dswap
***********************************************************************/
void ortbnd(double[] alf, double[] eta, double[] oldeta, double[] bet, int step,
double rnm) {
int i;
if (step < 1) return;
if (/* rnm */ 0 != rnm) {
if (step > 1) {
oldeta[0] = (bet[1] * eta[1] + (alf[0]-alf[step]) * eta[0] -
bet[step] * oldeta[0]) / rnm + eps1;
}
for (i=1; i<=step-2; i++)
oldeta[i] = (bet[i+1] * eta[i+1] + (alf[i]-alf[step]) * eta[i] +
bet[i] * eta[i-1] - bet[step] * oldeta[i])/rnm + eps1;
}
oldeta[step-1] = eps1;
svd_dswap(step, oldeta, 1, eta, 1);
eta[step] = eps1;
return;
}
/***********************************************************************
* *
* purge() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Function examines the state of orthogonality between the new Lanczos
vector and the previous ones to decide whether re-orthogonalization
should be performed
Arguments
---------
(input)
n dimension of the eigenproblem for matrix B
ll number of intitial Lanczos vectors in local orthog.
r residual vector to become next Lanczos vector
q current Lanczos vector
ra previous Lanczos vector
qa previous Lanczos vector
wrk temporary vector to hold the previous Lanczos vector
eta state of orthogonality between r and prev. Lanczos vectors
oldeta state of orthogonality between q and prev. Lanczos vectors
j current Lanczos step
(output)
r residual vector orthogonalized against previous Lanczos
vectors
q current Lanczos vector orthogonalized against previous ones
Functions used
--------------
BLAS svd_daxpy, svd_dcopy, svd_idamax, svd_ddot
USER store
***********************************************************************/
void purge(int n, int ll, double[] r, double[] q, double[] ra,
double[] qa, double[] wrk, double[] eta, double[] oldeta, int step,
double[] rnmp, double tol) {
double t, tq, tr, reps1;
double rnm = rnmp[0];
int k, iteration, i;
boolean flag;
if (step < ll+2) return;
k = svd_idamax(step - (ll+1), eta, ll, 1) + ll; // TODO eta starting at ll
if (Math.abs(eta[k]) > reps) {
reps1 = eps1 / reps;
iteration = 0;
flag = true;
while (iteration < 2 && flag) {
if (rnm > tol) {
/* bring in a lanczos vector t and orthogonalize both
* r and q against it */
tq = 0.0;
tr = 0.0;
for (i = ll; i < step; i++) {
store(n, RETRQ, i, wrk);
t = -svd_ddot(n, qa, 1, wrk, 1);
tq += Math.abs(t);
svd_daxpy(n, t, wrk, 1, q, 1);
t = -svd_ddot(n, ra, 1, wrk, 1);
tr += Math.abs(t);
svd_daxpy(n, t, wrk, 1, r, 1);
}
svd_dcopy(n, q, 1, qa, 1);
t = -svd_ddot(n, r, 1, qa, 1);
tr += Math.abs(t);
svd_daxpy(n, t, q, 1, r, 1);
svd_dcopy(n, r, 1, ra, 1);
rnm = Math.sqrt(svd_ddot(n, ra, 1, r, 1));
if (tq <= reps1 && tr <= reps1 * rnm) flag = false;
}
iteration++;
}
for (i = ll; i <= step; i++) {
eta[i] = eps1;
oldeta[i] = eps1;
}
}
rnmp[0] = rnm;
return;
}
/***********************************************************************
* *
* stpone() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Function performs the first step of the Lanczos algorithm. It also
does a step of extended local re-orthogonalization.
Arguments
---------
(input)
n dimension of the eigenproblem for matrix B
(output)
ierr error flag
wptr array of pointers that point to work space that contains
wptr[0] r[j]
wptr[1] q[j]
wptr[2] q[j-1]
wptr[3] p
wptr[4] p[j-1]
wptr[6] diagonal elements of matrix T
Functions used
--------------
BLAS svd_daxpy, svd_datx, svd_dcopy, svd_ddot, svd_dscal
USER store, opb
LAS startv
***********************************************************************/
double fabs(double a) {
return Math.abs(a);
}
void stpone(SMat A, double[][] wrkptr, double[] rnmp, double[] tolp, int n) {
double t, rnm, anorm;
double[] alf = wrkptr[6];
/* get initial vector; default is random */
rnm = startv(A, wrkptr, 0, n);
if (rnm == 0.0 || ierr != 0) return;
/* normalize starting vector */
t = 1.0 / rnm;
svd_datx(n, t, wrkptr[0], 1, wrkptr[1], 1);
svd_dscal(n, t, wrkptr[3], 1);
/* take the first step */
svd_opb(A, wrkptr[3], wrkptr[0], OPBTemp);
alf[0] = svd_ddot(n, wrkptr[0], 1, wrkptr[3], 1);
svd_daxpy(n, -alf[0], wrkptr[1], 1, wrkptr[0], 1);
t = svd_ddot(n, wrkptr[0], 1, wrkptr[3], 1);
svd_daxpy(n, -t, wrkptr[1], 1, wrkptr[0], 1);
alf[0] += t;
svd_dcopy(n, wrkptr[0], 1, wrkptr[4], 1);
rnm = Math.sqrt(svd_ddot(n, wrkptr[0], 1, wrkptr[4], 1));
anorm = rnm + fabs(alf[0]);
rnmp[0] = rnm;
tolp[0] = reps * anorm;
return;
}
/***********************************************************************
* *
* startv() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Function delivers a starting vector in r and returns |r|; it returns
zero if the range is spanned, and ierr is non-zero if no starting
vector within range of operator can be found.
Parameters
---------
(input)
n dimension of the eigenproblem matrix B
wptr array of pointers that point to work space
j starting index for a Lanczos run
eps machine epsilon (relative precision)
(output)
wptr array of pointers that point to work space that contains
r[j], q[j], q[j-1], p[j], p[j-1]
ierr error flag (nonzero if no starting vector can be found)
Functions used
--------------
BLAS svd_ddot, svd_dcopy, svd_daxpy
USER svd_opb, store
MISC random
***********************************************************************/
double startv(SMat A, double[][] wptr, int step, int n) {
double rnm2, t;
double[] r;
//long irand;
int id, i;
/* get initial vector; default is random */
rnm2 = svd_ddot(n, wptr[0], 1, wptr[0], 1);
Random random = new Random(918273L + step); // irand = 918273 + step;
r = wptr[0];
for (id = 0; id < 3; id++) {
if (id > 0 || step > 0 || rnm2 == 0)
for (i = 0; i < n; i++) r[i] = random.nextDouble(); // svd_random2(&irand);
svd_dcopy(n, wptr[0], 1, wptr[3], 1);
/* apply operator to put r in range (essential if m singular) */
svd_opb(A, wptr[3], wptr[0], OPBTemp);
svd_dcopy(n, wptr[0], 1, wptr[3], 1);
rnm2 = svd_ddot(n, wptr[0], 1, wptr[3], 1);
if (rnm2 > 0.0) break;
}
/* fatal error */
if (rnm2 <= 0.0) {
ierr = 8192;
throw null; // TODO better error handling
}
if (step > 0) {
for (i = 0; i < step; i++) {
store(n, RETRQ, i, wptr[5]);
t = -svd_ddot(n, wptr[3], 1, wptr[5], 1);
svd_daxpy(n, t, wptr[5], 1, wptr[0], 1);
}
/* make sure q[step] is orthogonal to q[step-1] */
t = svd_ddot(n, wptr[4], 1, wptr[0], 1);
svd_daxpy(n, -t, wptr[2], 1, wptr[0], 1);
svd_dcopy(n, wptr[0], 1, wptr[3], 1);
t = svd_ddot(n, wptr[3], 1, wptr[0], 1);
if (t <= eps * rnm2) t = 0.0;
rnm2 = t;
}
return(Math.sqrt(rnm2));
}
/***********************************************************************
* *
* error_bound() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
Function massages error bounds for very close ritz values by placing
a gap between them. The error bounds are then refined to reflect
this.
Arguments
---------
(input)
endl left end of interval containing unwanted eigenvalues
endr right end of interval containing unwanted eigenvalues
ritz array to store the ritz values
bnd array to store the error bounds
enough stop flag
Functions used
--------------
BLAS svd_idamax
UTILITY svd_dmin
***********************************************************************/
int error_bound(boolean[] enough, double endl, double endr,
double[] ritz, double[] bnd, int step, double tol) {
int mid, neig;
int i;
double gapl, gap;
/* massage error bounds for very close ritz values */
mid = svd_idamax(step + 1, bnd, 0, 1);
for (i=((step+1) + (step-1)) / 2; i >= mid + 1; i -= 1)
if (fabs(ritz[i-1] - ritz[i]) < eps34 * fabs(ritz[i]))
if (bnd[i] > tol && bnd[i-1] > tol) {
bnd[i-1] = Math.sqrt(bnd[i] * bnd[i] + bnd[i-1] * bnd[i-1]);
bnd[i] = 0.0;
}
for (i=((step+1) - (step-1)) / 2; i <= mid - 1; i +=1 )
if (fabs(ritz[i+1] - ritz[i]) < eps34 * fabs(ritz[i]))
if (bnd[i] > tol && bnd[i+1] > tol) {
bnd[i+1] = Math.sqrt(bnd[i] * bnd[i] + bnd[i+1] * bnd[i+1]);
bnd[i] = 0.0;
}
/* refine the error bounds */
neig = 0;
gapl = ritz[step] - ritz[0];
for (i = 0; i <= step; i++) {
gap = gapl;
if (i < step) gapl = ritz[i+1] - ritz[i];
gap = svd_dmin(gap, gapl);
if (gap > bnd[i]) bnd[i] = bnd[i] * (bnd[i] / gap);
if (bnd[i] <= 16.0 * eps * fabs(ritz[i])) {
neig++;
if (!enough[0]) enough[0] = endl < ritz[i] && ritz[i] < endr;
}
}
return neig;
}
/***********************************************************************
* *
* imtqlb() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
imtqlb() is a translation of a Fortran version of the Algol
procedure IMTQL1, Num. Math. 12, 377-383(1968) by Martin and
Wilkinson, as modified in Num. Math. 15, 450(1970) by Dubrulle.
Handbook for Auto. Comp., vol.II-Linear Algebra, 241-248(1971).
See also B. T. Smith et al, Eispack Guide, Lecture Notes in
Computer Science, Springer-Verlag, (1976).
The function finds the eigenvalues of a symmetric tridiagonal
matrix by the implicit QL method.
Arguments
---------
(input)
n order of the symmetric tridiagonal matrix
d contains the diagonal elements of the input matrix
e contains the subdiagonal elements of the input matrix in its
last n-1 positions. e[0] is arbitrary
(output)
d contains the eigenvalues in ascending order. if an error
exit is made, the eigenvalues are correct and ordered for
indices 0,1,...ierr, but may not be the smallest eigenvalues.
e has been destroyed.
ierr set to zero for normal return, j if the j-th eigenvalue has
not been determined after 30 iterations.
Functions used
--------------
UTILITY svd_fsign
MISC svd_pythag
***********************************************************************/
void imtqlb(int n, double d[], double e[], double bnd[], int offset) {
double[] dn = new double[n];
System.arraycopy(d, offset, dn, 0, n);
double[] en = new double[n];
System.arraycopy(e, offset, en, 0, n);
double[] bndn = new double[n];
System.arraycopy(bnd, offset, bndn, 0, n);
imtqlb(n, dn, en, bndn);
System.arraycopy(dn, 0, d, offset, n);
System.arraycopy(en, 0, e, offset, n);
System.arraycopy(bndn, 0, bnd, offset, n);
}
void imtqlb(int n, double d[], double e[], double bnd[]) {
long iteration;
int last, i, m, l;
/* various flags */
boolean exchange, convergence, underflow;
double b, test, g, r, s, c, p, f;
if (n == 1) return;
ierr = 0;
bnd[0] = 1.0;
last = n - 1;
for (i = 1; i < n; i++) {
bnd[i] = 0.0;
e[i-1] = e[i];
}
e[last] = 0.0;
for (l = 0; l < n; l++) {
iteration = 0;
while (iteration <= 30) {
for (m = l; m < n; m++) {
convergence = false;
if (m == last) break;
else {
test = fabs(d[m]) + fabs(d[m+1]);
if (test + fabs(e[m]) == test) convergence = true;
}
if (convergence) break;
}
p = d[l];
f = bnd[l];
if (m != l) {
if (iteration == 30) {
ierr = l;
return;
}
iteration += 1;
/*........ form shift ........*/
g = (d[l+1] - p) / (2.0 * e[l]);
r = svd_pythag(g, 1.0);
g = d[m] - p + e[l] / (g + svd_fsign(r, g));
s = 1.0;
c = 1.0;
p = 0.0;
underflow = false;
i = m - 1;
while (underflow == false && i >= l) {
f = s * e[i];
b = c * e[i];
r = svd_pythag(f, g);
e[i+1] = r;
if (r == 0.0) underflow = true;
else {
s = f / r;
c = g / r;
g = d[i+1] - p;
r = (d[i] - g) * s + 2.0 * c * b;
p = s * r;
d[i+1] = g + p;
g = c * r - b;
f = bnd[i+1];
bnd[i+1] = s * bnd[i] + c * f;
bnd[i] = c * bnd[i] - s * f;
i--;
}
} /* end while (underflow != FALSE && i >= l) */
/*........ recover from underflow .........*/
if (underflow) {
d[i+1] -= p;
e[m] = 0.0;
}
else {
d[l] -= p;
e[l] = g;
e[m] = 0.0;
}
} /* end if (m != l) */
else {
/* order the eigenvalues */
exchange = true;
if (l != 0) {
i = l;
while (i >= 1 && exchange == true) {
if (p < d[i-1]) {
d[i] = d[i-1];
bnd[i] = bnd[i-1];
i--;
}
else exchange = false;
}
}
if (exchange) i = 0;
d[i] = p;
bnd[i] = f;
iteration = 31;
}
} /* end while (iteration <= 30) */
} /* end for (l=0; l<n; l++) */
return;
} /* end main */
/***********************************************************************
* *
* imtql2() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
imtql2() is a translation of a Fortran version of the Algol
procedure IMTQL2, Num. Math. 12, 377-383(1968) by Martin and
Wilkinson, as modified in Num. Math. 15, 450(1970) by Dubrulle.
Handbook for Auto. Comp., vol.II-Linear Algebra, 241-248(1971).
See also B. T. Smith et al, Eispack Guide, Lecture Notes in
Computer Science, Springer-Verlag, (1976).
This function finds the eigenvalues and eigenvectors of a symmetric
tridiagonal matrix by the implicit QL method.
Arguments
---------
(input)
nm row dimension of the symmetric tridiagonal matrix
n order of the matrix
d contains the diagonal elements of the input matrix
e contains the subdiagonal elements of the input matrix in its
last n-1 positions. e[0] is arbitrary
z contains the identity matrix
(output)
d contains the eigenvalues in ascending order. if an error
exit is made, the eigenvalues are correct but unordered for
for indices 0,1,...,ierr.
e has been destroyed.
z contains orthonormal eigenvectors of the symmetric
tridiagonal (or full) matrix. if an error exit is made,
z contains the eigenvectors associated with the stored
eigenvalues.
ierr set to zero for normal return, j if the j-th eigenvalue has
not been determined after 30 iterations.
Functions used
--------------
UTILITY svd_fsign
MISC svd_pythag
***********************************************************************/
void imtql2(int nm, int n, double d[], double e[], double z[]) {
int index, nnm, j, last, l, m, i, k, iteration;
boolean convergence, underflow;
double b, test, g, r, s, c, p, f;
if (n == 1) return;
ierr = 0;
last = n - 1;
for (i = 1; i < n; i++) e[i-1] = e[i];
e[last] = 0.0;
nnm = n * nm;
for (l = 0; l < n; l++) {
iteration = 0;
/* look for small sub-diagonal element */
while (iteration <= 30) {
for (m = l; m < n; m++) {
convergence = false;
if (m == last) break;
else {
test = fabs(d[m]) + fabs(d[m+1]);
if (test + fabs(e[m]) == test) convergence = true;
}
if (convergence) break;
}
if (m != l) {
/* set error -- no convergence to an eigenvalue after
* 30 iterations. */
if (iteration == 30) {
ierr = l;
return;
}
p = d[l];
iteration += 1;
/* form shift */
g = (d[l+1] - p) / (2.0 * e[l]);
r = svd_pythag(g, 1.0);
g = d[m] - p + e[l] / (g + svd_fsign(r, g));
s = 1.0;
c = 1.0;
p = 0.0;
underflow = false;
i = m - 1;
while (underflow == false && i >= l) {
f = s * e[i];
b = c * e[i];
r = svd_pythag(f, g);
e[i+1] = r;
if (r == 0.0) underflow = true;
else {
s = f / r;
c = g / r;
g = d[i+1] - p;
r = (d[i] - g) * s + 2.0 * c * b;
p = s * r;
d[i+1] = g + p;
g = c * r - b;
/* form vector */
for (k = 0; k < nnm; k += n) {
index = k + i;
f = z[index+1];
z[index+1] = s * z[index] + c * f;
z[index] = c * z[index] - s * f;
}
i--;
}
} /* end while (underflow != FALSE && i >= l) */
/*........ recover from underflow .........*/
if (underflow) {
d[i+1] -= p;
e[m] = 0.0;
}
else {
d[l] -= p;
e[l] = g;
e[m] = 0.0;
}
}
else break;
} /*...... end while (iteration <= 30) .........*/
} /*...... end for (l=0; l<n; l++) .............*/
/* order the eigenvalues */
for (l = 1; l < n; l++) {
i = l - 1;
k = i;
p = d[i];
for (j = l; j < n; j++) {
if (d[j] < p) {
k = j;
p = d[j];
}
}
/* ...and corresponding eigenvectors */
if (k != i) {
d[k] = d[i];
d[i] = p;
for (j = 0; j < nnm; j += n) {
p = z[j+i];
z[j+i] = z[j+k];
z[j+k] = p;
}
}
}
return;
} /*...... end main ............................*/
/***********************************************************************
* *
* machar() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
This function is a partial translation of a Fortran-77 subroutine
written by W. J. Cody of Argonne National Laboratory.
It dynamically determines the listed machine parameters of the
floating-point arithmetic. According to the documentation of
the Fortran code, "the determination of the first three uses an
extension of an algorithm due to M. Malcolm, ACM 15 (1972),
pp. 949-951, incorporating some, but not all, of the improvements
suggested by M. Gentleman and S. Marovich, CACM 17 (1974),
pp. 276-277." The complete Fortran version of this translation is
documented in W. J. Cody, "Machar: a Subroutine to Dynamically
Determine Determine Machine Parameters," TOMS 14, December, 1988.
Parameters reported
-------------------
ibeta the radix for the floating-point representation
it the number of base ibeta digits in the floating-point
significand
irnd 0 if floating-point addition chops
1 if floating-point addition rounds, but not in the
ieee style
2 if floating-point addition rounds in the ieee style
3 if floating-point addition chops, and there is
partial underflow
4 if floating-point addition rounds, but not in the
ieee style, and there is partial underflow
5 if floating-point addition rounds in the ieee style,
and there is partial underflow
machep the largest negative integer such that
1.0+float(ibeta)**machep .ne. 1.0, except that
machep is bounded below by -(it+3)
negeps the largest negative integer such that
1.0-float(ibeta)**negeps .ne. 1.0, except that
negeps is bounded below by -(it+3)
***********************************************************************/
// TODO check type of array
long[] machar(/*long[] ibeta, long[] it, long[] irnd, long[] machep, long[] negep*/) {
long ibeta, it, irnd, machep, negep;
double beta, betain, betah, a, b, ZERO, ONE, TWO, temp, tempa,
temp1;
long i, itemp;
ONE = (double) 1;
TWO = ONE + ONE;
ZERO = ONE - ONE;
a = ONE;
temp1 = ONE;
while (temp1 - ONE == ZERO) {
a = a + a;
temp = a + ONE;
temp1 = temp - a;
// b += a; /* to prevent icc compiler error */ XXX Intel rockstar compiler :)
}
b = ONE;
itemp = 0;
while (itemp == 0) {
b = b + b;
temp = a + b;
itemp = (long)(temp - a);
}
ibeta = itemp;
beta = (double) ibeta;
it = 0;
b = ONE;
temp1 = ONE;
while (temp1 - ONE == ZERO) {
it = it + 1;
b = b * beta;
temp = b + ONE;
temp1 = temp - b;
}
irnd = 0;
betah = beta / TWO;
temp = a + betah;
if (temp - a != ZERO) irnd = 1;
tempa = a + beta;
temp = tempa + betah;
if ((irnd == 0) && (temp - tempa != ZERO)) irnd = 2;
negep = it + 3;
betain = ONE / beta;
a = ONE;
for (i = 0; i < negep; i++) a = a * betain;
b = a;
temp = ONE - a;
while (temp-ONE == ZERO) {
a = a * beta;
negep = negep - 1;
temp = ONE - a;
}
negep = -(negep);
machep = -(it) - 3;
a = b;
temp = ONE + a;
while (temp - ONE == ZERO) {
a = a * beta;
machep = machep + 1;
temp = ONE + a;
}
eps = a;
return new long[] { ibeta, it, irnd, machep, negep };
}
/***********************************************************************
* *
* store() *
* *
***********************************************************************/
/***********************************************************************
Description
-----------
store() is a user-supplied function which, based on the input
operation flag, stores to or retrieves from memory a vector.
Arguments
---------
(input)
n length of vector to be stored or retrieved
isw operation flag:
isw = 1 request to store j-th Lanczos vector q(j)
isw = 2 request to retrieve j-th Lanczos vector q(j)
isw = 3 request to store q(j) for j = 0 or 1
isw = 4 request to retrieve q(j) for j = 0 or 1
s contains the vector to be stored for a "store" request
(output)
s contains the vector retrieved for a "retrieve" request
Functions used
--------------
BLAS svd_dcopy
***********************************************************************/
void store(int n, storeVals isw, int j, double[] s) {
/* printf("called store %ld %ld\n", isw, j); */
switch(isw) {
case STORQ:
if (null == LanStore[j + MAXLL]) {
LanStore[j + MAXLL] = svd_doubleArray(n, false, "LanStore[j]");
}
svd_dcopy(n, s, 1, LanStore[j + MAXLL], 1);
break;
case RETRQ:
if (null == LanStore[j + MAXLL]) throw new Error(String.format(
"svdLAS2: store (RETRQ) called on index %d (not allocated)", j + MAXLL));
svd_dcopy(n, LanStore[j + MAXLL], 1, s, 1);
break;
case STORP:
if (j >= MAXLL) {
throw new Error("svdLAS2: store (STORP) called with j >= MAXLL");
}
if (null == LanStore[j]) {
LanStore[j] = svd_doubleArray(n, false, "LanStore[j]");
}
svd_dcopy(n, s, 1, LanStore[j], 1);
break;
case RETRP:
if (j >= MAXLL) {
svd_error("svdLAS2: store (RETRP) called with j >= MAXLL");
break;
}
if (null == LanStore[j]) throw new Error(String.format(
"svdLAS2: store (RETRP) called on index %d (not allocated)", j));
svd_dcopy(n, LanStore[j], 1, s, 1);
break;
}
return;
}
/* File format has a funny header, then first entry index per column, then the
row for each entry, then the value for each entry. Indices count from 1.
Assumes A is initialized. */
SMat svdLoadSparseTextHBFile(File file) throws FileNotFoundException {
int i, x, rows, cols, vals, num_mat;
Scanner scanner = new Scanner(file);
SMat S;
/* Skip the header line: */
scanner.nextLine();
/* Skip the line giving the number of lines in this file: */
scanner.nextLine();
/* Read the line with useful dimensions: */
scanner.next();
rows = scanner.nextInt();
cols = scanner.nextInt();
vals = scanner.nextInt();
num_mat = scanner.nextInt();
scanner.nextLine();
if (num_mat != 0) {
throw new Error("svdLoadSparseTextHBFile: I don't know how to handle a file "
+ "with elemental matrices (last entry on header line 3)");
}
/* Skip the line giving the formats: */
scanner.nextLine();
S = new SMat(rows, cols, vals);
/* Read column pointers. */
for (i = 0; i <= S.cols; i++) {
x = scanner.nextInt();
S.pointr[i] = x - 1;
}
S.pointr[S.cols] = S.vals;
/* Read row indices. */
for (i = 0; i < S.vals; i++) {
x = scanner.nextInt();
S.rowind[i] = x - 1;
}
for (i = 0; i < S.vals; i++) {
S.value[i] = scanner.nextDouble();
}
return S;
}
}