#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int clalsd_(char *uplo, integer *smlsiz, integer *n, integer *nrhs, real *d__, real *e, complex *b, integer *ldb, real *rcond, integer *rank, complex *work, real *rwork, integer *iwork, integer * info) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CLALSD uses the singular value decomposition of A to solve the least squares problem of finding X to minimize the Euclidean norm of each column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-by-NRHS. The solution X overwrites B. The singular values of A smaller than RCOND times the largest singular value are treated as zero in solving the least squares problem; in this case a minimum norm solution is returned. The actual singular values are returned in D in ascending order. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. Arguments ========= UPLO (input) CHARACTER*1 = 'U': D and E define an upper bidiagonal matrix. = 'L': D and E define a lower bidiagonal matrix. SMLSIZ (input) INTEGER The maximum size of the subproblems at the bottom of the computation tree. N (input) INTEGER The dimension of the bidiagonal matrix. N >= 0. NRHS (input) INTEGER The number of columns of B. NRHS must be at least 1. D (input/output) REAL array, dimension (N) On entry D contains the main diagonal of the bidiagonal matrix. On exit, if INFO = 0, D contains its singular values. E (input/output) REAL array, dimension (N-1) Contains the super-diagonal entries of the bidiagonal matrix. On exit, E has been destroyed. B (input/output) COMPLEX array, dimension (LDB,NRHS) On input, B contains the right hand sides of the least squares problem. On output, B contains the solution X. LDB (input) INTEGER The leading dimension of B in the calling subprogram. LDB must be at least max(1,N). RCOND (input) REAL The singular values of A less than or equal to RCOND times the largest singular value are treated as zero in solving the least squares problem. If RCOND is negative, machine precision is used instead. For example, if diag(S)*X=B were the least squares problem, where diag(S) is a diagonal matrix of singular values, the solution would be X(i) = B(i) / S(i) if S(i) is greater than RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to RCOND*max(S). RANK (output) INTEGER The number of singular values of A greater than RCOND times the largest singular value. WORK (workspace) COMPLEX array, dimension (N * NRHS). RWORK (workspace) REAL array, dimension at least (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2), where NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) IWORK (workspace) INTEGER array, dimension (3*N*NLVL + 11*N). INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute an singular value while working on the submatrix lying in rows and columns INFO/(N+1) through MOD(INFO,N+1). Further Details =============== Based on contributions by Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA Osni Marques, LBNL/NERSC, USA ===================================================================== Test the input parameters. Parameter adjustments */ /* Table of constant values */ static complex c_b1 = {0.f,0.f}; static integer c__1 = 1; static integer c__0 = 0; static real c_b10 = 1.f; static real c_b35 = 0.f; /* System generated locals */ integer b_dim1, b_offset, i__1, i__2, i__3, i__4, i__5, i__6; real r__1; complex q__1; /* Builtin functions */ double r_imag(complex *), log(doublereal), r_sign(real *, real *); /* Local variables */ static integer c__, i__, j, k; static real r__; static integer s, u, z__; static real cs; static integer bx; static real sn; static integer st, vt, nm1, st1; static real eps; static integer iwk; static real tol; static integer difl, difr; static real rcnd; static integer jcol, irwb, perm, nsub, nlvl, sqre, bxst, jrow, irwu, jimag, jreal; extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); static integer irwib; extern /* Subroutine */ int ccopy_(integer *, complex *, integer *, complex *, integer *); static integer poles, sizei, irwrb, nsize; extern /* Subroutine */ int csrot_(integer *, complex *, integer *, complex *, integer *, real *, real *); static integer irwvt, icmpq1, icmpq2; extern /* Subroutine */ int clalsa_(integer *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, real *, integer *, real *, integer *, real *, real *, real *, real *, integer *, integer *, integer *, integer *, real *, real *, real * , real *, integer *, integer *), clascl_(char *, integer *, integer *, real *, real *, integer *, integer *, complex *, integer *, integer *); extern doublereal slamch_(char *); extern /* Subroutine */ int slasda_(integer *, integer *, integer *, integer *, real *, real *, real *, integer *, real *, integer *, real *, real *, real *, real *, integer *, integer *, integer *, integer *, real *, real *, real *, real *, integer *, integer *), clacpy_(char *, integer *, integer *, complex *, integer *, complex *, integer *), claset_(char *, integer *, integer *, complex *, complex *, complex *, integer *), xerbla_( char *, integer *), slascl_(char *, integer *, integer *, real *, real *, integer *, integer *, real *, integer *, integer * ); extern integer isamax_(integer *, real *, integer *); static integer givcol; extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer *, integer *, integer *, real *, real *, real *, integer *, real * , integer *, real *, integer *, real *, integer *), slaset_(char *, integer *, integer *, real *, real *, real *, integer *), slartg_(real *, real *, real *, real *, real * ); static real orgnrm; static integer givnum; extern doublereal slanst_(char *, integer *, real *, real *); extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *); static integer givptr, nrwork, irwwrk, smlszp; --d__; --e; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --work; --rwork; --iwork; /* Function Body */ *info = 0; if (*n < 0) { *info = -3; } else if (*nrhs < 1) { *info = -4; } else if (*ldb < 1 || *ldb < *n) { *info = -8; } if (*info != 0) { i__1 = -(*info); xerbla_("CLALSD", &i__1); return 0; } eps = slamch_("Epsilon"); /* Set up the tolerance. */ if (*rcond <= 0.f || *rcond >= 1.f) { rcnd = eps; } else { rcnd = *rcond; } *rank = 0; /* Quick return if possible. */ if (*n == 0) { return 0; } else if (*n == 1) { if (d__[1] == 0.f) { claset_("A", &c__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb); } else { *rank = 1; clascl_("G", &c__0, &c__0, &d__[1], &c_b10, &c__1, nrhs, &b[ b_offset], ldb, info); d__[1] = dabs(d__[1]); } return 0; } /* Rotate the matrix if it is lower bidiagonal. */ if (*(unsigned char *)uplo == 'L') { i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { slartg_(&d__[i__], &e[i__], &cs, &sn, &r__); d__[i__] = r__; e[i__] = sn * d__[i__ + 1]; d__[i__ + 1] = cs * d__[i__ + 1]; if (*nrhs == 1) { csrot_(&c__1, &b[i__ + b_dim1], &c__1, &b[i__ + 1 + b_dim1], & c__1, &cs, &sn); } else { rwork[(i__ << 1) - 1] = cs; rwork[i__ * 2] = sn; } /* L10: */ } if (*nrhs > 1) { i__1 = *nrhs; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *n - 1; for (j = 1; j <= i__2; ++j) { cs = rwork[(j << 1) - 1]; sn = rwork[j * 2]; csrot_(&c__1, &b[j + i__ * b_dim1], &c__1, &b[j + 1 + i__ * b_dim1], &c__1, &cs, &sn); /* L20: */ } /* L30: */ } } } /* Scale. */ nm1 = *n - 1; orgnrm = slanst_("M", n, &d__[1], &e[1]); if (orgnrm == 0.f) { claset_("A", n, nrhs, &c_b1, &c_b1, &b[b_offset], ldb); return 0; } slascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, &c__1, &d__[1], n, info); slascl_("G", &c__0, &c__0, &orgnrm, &c_b10, &nm1, &c__1, &e[1], &nm1, info); /* If N is smaller than the minimum divide size SMLSIZ, then solve the problem with another solver. */ if (*n <= *smlsiz) { irwu = 1; irwvt = irwu + *n * *n; irwwrk = irwvt + *n * *n; irwrb = irwwrk; irwib = irwrb + *n * *nrhs; irwb = irwib + *n * *nrhs; slaset_("A", n, n, &c_b35, &c_b10, &rwork[irwu], n); slaset_("A", n, n, &c_b35, &c_b10, &rwork[irwvt], n); slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &rwork[irwvt], n, &rwork[irwu], n, &rwork[irwwrk], &c__1, &rwork[irwwrk], info); if (*info != 0) { return 0; } /* In the real version, B is passed to SLASDQ and multiplied internally by Q'. Here B is complex and that product is computed below in two steps (real and imaginary parts). */ j = irwb - 1; i__1 = *nrhs; for (jcol = 1; jcol <= i__1; ++jcol) { i__2 = *n; for (jrow = 1; jrow <= i__2; ++jrow) { ++j; i__3 = jrow + jcol * b_dim1; rwork[j] = b[i__3].r; /* L40: */ } /* L50: */ } sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n, &c_b35, &rwork[irwrb], n); j = irwb - 1; i__1 = *nrhs; for (jcol = 1; jcol <= i__1; ++jcol) { i__2 = *n; for (jrow = 1; jrow <= i__2; ++jrow) { ++j; rwork[j] = r_imag(&b[jrow + jcol * b_dim1]); /* L60: */ } /* L70: */ } sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n, &c_b35, &rwork[irwib], n); jreal = irwrb - 1; jimag = irwib - 1; i__1 = *nrhs; for (jcol = 1; jcol <= i__1; ++jcol) { i__2 = *n; for (jrow = 1; jrow <= i__2; ++jrow) { ++jreal; ++jimag; i__3 = jrow + jcol * b_dim1; i__4 = jreal; i__5 = jimag; q__1.r = rwork[i__4], q__1.i = rwork[i__5]; b[i__3].r = q__1.r, b[i__3].i = q__1.i; /* L80: */ } /* L90: */ } tol = rcnd * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1)); i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if (d__[i__] <= tol) { claset_("A", &c__1, nrhs, &c_b1, &c_b1, &b[i__ + b_dim1], ldb); } else { clascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &b[ i__ + b_dim1], ldb, info); ++(*rank); } /* L100: */ } /* Since B is complex, the following call to SGEMM is performed in two steps (real and imaginary parts). That is for V * B (in the real version of the code V' is stored in WORK). CALL SGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, $ WORK( NWORK ), N ) */ j = irwb - 1; i__1 = *nrhs; for (jcol = 1; jcol <= i__1; ++jcol) { i__2 = *n; for (jrow = 1; jrow <= i__2; ++jrow) { ++j; i__3 = jrow + jcol * b_dim1; rwork[j] = b[i__3].r; /* L110: */ } /* L120: */ } sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb], n, &c_b35, &rwork[irwrb], n); j = irwb - 1; i__1 = *nrhs; for (jcol = 1; jcol <= i__1; ++jcol) { i__2 = *n; for (jrow = 1; jrow <= i__2; ++jrow) { ++j; rwork[j] = r_imag(&b[jrow + jcol * b_dim1]); /* L130: */ } /* L140: */ } sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb], n, &c_b35, &rwork[irwib], n); jreal = irwrb - 1; jimag = irwib - 1; i__1 = *nrhs; for (jcol = 1; jcol <= i__1; ++jcol) { i__2 = *n; for (jrow = 1; jrow <= i__2; ++jrow) { ++jreal; ++jimag; i__3 = jrow + jcol * b_dim1; i__4 = jreal; i__5 = jimag; q__1.r = rwork[i__4], q__1.i = rwork[i__5]; b[i__3].r = q__1.r, b[i__3].i = q__1.i; /* L150: */ } /* L160: */ } /* Unscale. */ slascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n, info); slasrt_("D", n, &d__[1], info); clascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset], ldb, info); return 0; } /* Book-keeping and setting up some constants. */ nlvl = (integer) (log((real) (*n) / (real) (*smlsiz + 1)) / log(2.f)) + 1; smlszp = *smlsiz + 1; u = 1; vt = *smlsiz * *n + 1; difl = vt + smlszp * *n; difr = difl + nlvl * *n; z__ = difr + (nlvl * *n << 1); c__ = z__ + nlvl * *n; s = c__ + *n; poles = s + *n; givnum = poles + (nlvl << 1) * *n; nrwork = givnum + (nlvl << 1) * *n; bx = 1; irwrb = nrwork; irwib = irwrb + *smlsiz * *nrhs; irwb = irwib + *smlsiz * *nrhs; sizei = *n + 1; k = sizei + *n; givptr = k + *n; perm = givptr + *n; givcol = perm + nlvl * *n; iwk = givcol + (nlvl * *n << 1); st = 1; sqre = 0; icmpq1 = 1; icmpq2 = 0; nsub = 0; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if ((r__1 = d__[i__], dabs(r__1)) < eps) { d__[i__] = r_sign(&eps, &d__[i__]); } /* L170: */ } i__1 = nm1; for (i__ = 1; i__ <= i__1; ++i__) { if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) { ++nsub; iwork[nsub] = st; /* Subproblem found. First determine its size and then apply divide and conquer on it. */ if (i__ < nm1) { /* A subproblem with E(I) small for I < NM1. */ nsize = i__ - st + 1; iwork[sizei + nsub - 1] = nsize; } else if ((r__1 = e[i__], dabs(r__1)) >= eps) { /* A subproblem with E(NM1) not too small but I = NM1. */ nsize = *n - st + 1; iwork[sizei + nsub - 1] = nsize; } else { /* A subproblem with E(NM1) small. This implies an 1-by-1 subproblem at D(N), which is not solved explicitly. */ nsize = i__ - st + 1; iwork[sizei + nsub - 1] = nsize; ++nsub; iwork[nsub] = *n; iwork[sizei + nsub - 1] = 1; ccopy_(nrhs, &b[*n + b_dim1], ldb, &work[bx + nm1], n); } st1 = st - 1; if (nsize == 1) { /* This is a 1-by-1 subproblem and is not solved explicitly. */ ccopy_(nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n); } else if (nsize <= *smlsiz) { /* This is a small subproblem and is solved by SLASDQ. */ slaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[vt + st1], n); slaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[u + st1], n); slasdq_("U", &c__0, &nsize, &nsize, &nsize, &c__0, &d__[st], & e[st], &rwork[vt + st1], n, &rwork[u + st1], n, & rwork[nrwork], &c__1, &rwork[nrwork], info) ; if (*info != 0) { return 0; } /* In the real version, B is passed to SLASDQ and multiplied internally by Q'. Here B is complex and that product is computed below in two steps (real and imaginary parts). */ j = irwb - 1; i__2 = *nrhs; for (jcol = 1; jcol <= i__2; ++jcol) { i__3 = st + nsize - 1; for (jrow = st; jrow <= i__3; ++jrow) { ++j; i__4 = jrow + jcol * b_dim1; rwork[j] = b[i__4].r; /* L180: */ } /* L190: */ } sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1] , n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], & nsize); j = irwb - 1; i__2 = *nrhs; for (jcol = 1; jcol <= i__2; ++jcol) { i__3 = st + nsize - 1; for (jrow = st; jrow <= i__3; ++jrow) { ++j; rwork[j] = r_imag(&b[jrow + jcol * b_dim1]); /* L200: */ } /* L210: */ } sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1] , n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], & nsize); jreal = irwrb - 1; jimag = irwib - 1; i__2 = *nrhs; for (jcol = 1; jcol <= i__2; ++jcol) { i__3 = st + nsize - 1; for (jrow = st; jrow <= i__3; ++jrow) { ++jreal; ++jimag; i__4 = jrow + jcol * b_dim1; i__5 = jreal; i__6 = jimag; q__1.r = rwork[i__5], q__1.i = rwork[i__6]; b[i__4].r = q__1.r, b[i__4].i = q__1.i; /* L220: */ } /* L230: */ } clacpy_("A", &nsize, nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n); } else { /* A large problem. Solve it using divide and conquer. */ slasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], & rwork[u + st1], n, &rwork[vt + st1], &iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1], &rwork[z__ + st1], &rwork[poles + st1], &iwork[givptr + st1], & iwork[givcol + st1], n, &iwork[perm + st1], &rwork[ givnum + st1], &rwork[c__ + st1], &rwork[s + st1], & rwork[nrwork], &iwork[iwk], info); if (*info != 0) { return 0; } bxst = bx + st1; clalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b[st + b_dim1], ldb, & work[bxst], n, &rwork[u + st1], n, &rwork[vt + st1], & iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1] , &rwork[z__ + st1], &rwork[poles + st1], &iwork[ givptr + st1], &iwork[givcol + st1], n, &iwork[perm + st1], &rwork[givnum + st1], &rwork[c__ + st1], &rwork[ s + st1], &rwork[nrwork], &iwork[iwk], info); if (*info != 0) { return 0; } } st = i__ + 1; } /* L240: */ } /* Apply the singular values and treat the tiny ones as zero. */ tol = rcnd * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1)); i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { /* Some of the elements in D can be negative because 1-by-1 subproblems were not solved explicitly. */ if ((r__1 = d__[i__], dabs(r__1)) <= tol) { claset_("A", &c__1, nrhs, &c_b1, &c_b1, &work[bx + i__ - 1], n); } else { ++(*rank); clascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &work[ bx + i__ - 1], n, info); } d__[i__] = (r__1 = d__[i__], dabs(r__1)); /* L250: */ } /* Now apply back the right singular vectors. */ icmpq2 = 1; i__1 = nsub; for (i__ = 1; i__ <= i__1; ++i__) { st = iwork[i__]; st1 = st - 1; nsize = iwork[sizei + i__ - 1]; bxst = bx + st1; if (nsize == 1) { ccopy_(nrhs, &work[bxst], n, &b[st + b_dim1], ldb); } else if (nsize <= *smlsiz) { /* Since B and BX are complex, the following call to SGEMM is performed in two steps (real and imaginary parts). CALL SGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE, $ RWORK( VT+ST1 ), N, RWORK( BXST ), N, ZERO, $ B( ST, 1 ), LDB ) */ j = bxst - *n - 1; jreal = irwb - 1; i__2 = *nrhs; for (jcol = 1; jcol <= i__2; ++jcol) { j += *n; i__3 = nsize; for (jrow = 1; jrow <= i__3; ++jrow) { ++jreal; i__4 = j + jrow; rwork[jreal] = work[i__4].r; /* L260: */ } /* L270: */ } sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1], n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], &nsize); j = bxst - *n - 1; jimag = irwb - 1; i__2 = *nrhs; for (jcol = 1; jcol <= i__2; ++jcol) { j += *n; i__3 = nsize; for (jrow = 1; jrow <= i__3; ++jrow) { ++jimag; rwork[jimag] = r_imag(&work[j + jrow]); /* L280: */ } /* L290: */ } sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1], n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], &nsize); jreal = irwrb - 1; jimag = irwib - 1; i__2 = *nrhs; for (jcol = 1; jcol <= i__2; ++jcol) { i__3 = st + nsize - 1; for (jrow = st; jrow <= i__3; ++jrow) { ++jreal; ++jimag; i__4 = jrow + jcol * b_dim1; i__5 = jreal; i__6 = jimag; q__1.r = rwork[i__5], q__1.i = rwork[i__6]; b[i__4].r = q__1.r, b[i__4].i = q__1.i; /* L300: */ } /* L310: */ } } else { clalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b[st + b_dim1], ldb, &rwork[u + st1], n, &rwork[vt + st1], & iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1], & rwork[z__ + st1], &rwork[poles + st1], &iwork[givptr + st1], &iwork[givcol + st1], n, &iwork[perm + st1], &rwork[ givnum + st1], &rwork[c__ + st1], &rwork[s + st1], &rwork[ nrwork], &iwork[iwk], info); if (*info != 0) { return 0; } } /* L320: */ } /* Unscale and sort the singular values. */ slascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n, info); slasrt_("D", n, &d__[1], info); clascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset], ldb, info); return 0; /* End of CLALSD */ } /* clalsd_ */