int sbdsdc_(char *uplo, char *compq, int *n, float *d__, float *e, float *u, int *ldu, float *vt, int *ldvt, float *q, int *iq, float *work, int *iwork, int *info) { /* System generated locals */ int u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2; float r__1; /* Builtin functions */ double r_sign(float *, float *), log(double); /* Local variables */ int i__, j, k; float p, r__; int z__, ic, ii, kk; float cs; int is, iu; float sn; int nm1; float eps; int ivt, difl, difr, ierr, perm, mlvl, sqre; extern int lsame_(char *, char *); int poles; extern int slasr_(char *, char *, char *, int *, int *, float *, float *, float *, int *); int iuplo, nsize, start; extern int scopy_(int *, float *, int *, float *, int *), sswap_(int *, float *, int *, float *, int * ), slasd0_(int *, int *, float *, float *, float *, int * , float *, int *, int *, int *, float *, int *); extern double slamch_(char *); extern int slasda_(int *, int *, int *, int *, float *, float *, float *, int *, float *, int *, float *, float *, float *, float *, int *, int *, int *, int *, float *, float *, float *, float *, int *, int *), xerbla_(char *, int *); extern int ilaenv_(int *, char *, char *, int *, int *, int *, int *); extern int slascl_(char *, int *, int *, float *, float *, int *, int *, float *, int *, int *); int givcol; extern int slasdq_(char *, int *, int *, int *, int *, int *, float *, float *, float *, int *, float * , int *, float *, int *, float *, int *); int icompq; extern int slaset_(char *, int *, int *, float *, float *, float *, int *), slartg_(float *, float *, float * , float *, float *); float orgnrm; int givnum; extern double slanst_(char *, int *, float *, float *); int givptr, qstart, smlsiz, wstart, smlszp; /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* SBDSDC computes the singular value decomposition (SVD) of a float */ /* N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, */ /* using a divide and conquer method, where S is a diagonal matrix */ /* with non-negative diagonal elements (the singular values of B), and */ /* U and VT are orthogonal matrices of left and right singular vectors, */ /* respectively. SBDSDC can be used to compute all singular values, */ /* and optionally, singular vectors or singular vectors in compact form. */ /* 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2. */ /* It could conceivably fail on hexadecimal or decimal machines */ /* without guard digits, but we know of none. See SLASD3 for details. */ /* The code currently calls SLASDQ if singular values only are desired. */ /* However, it can be slightly modified to compute singular values */ /* using the divide and conquer method. */ /* Arguments */ /* ========= */ /* UPLO (input) CHARACTER*1 */ /* = 'U': B is upper bidiagonal. */ /* = 'L': B is lower bidiagonal. */ /* COMPQ (input) CHARACTER*1 */ /* Specifies whether singular vectors are to be computed */ /* as follows: */ /* = 'N': Compute singular values only; */ /* = 'P': Compute singular values and compute singular */ /* vectors in compact form; */ /* = 'I': Compute singular values and singular vectors. */ /* N (input) INTEGER */ /* The order of the matrix B. N >= 0. */ /* D (input/output) REAL array, dimension (N) */ /* On entry, the n diagonal elements of the bidiagonal matrix B. */ /* On exit, if INFO=0, the singular values of B. */ /* E (input/output) REAL array, dimension (N-1) */ /* On entry, the elements of E contain the offdiagonal */ /* elements of the bidiagonal matrix whose SVD is desired. */ /* On exit, E has been destroyed. */ /* U (output) REAL array, dimension (LDU,N) */ /* If COMPQ = 'I', then: */ /* On exit, if INFO = 0, U contains the left singular vectors */ /* of the bidiagonal matrix. */ /* For other values of COMPQ, U is not referenced. */ /* LDU (input) INTEGER */ /* The leading dimension of the array U. LDU >= 1. */ /* If singular vectors are desired, then LDU >= MAX( 1, N ). */ /* VT (output) REAL array, dimension (LDVT,N) */ /* If COMPQ = 'I', then: */ /* On exit, if INFO = 0, VT' contains the right singular */ /* vectors of the bidiagonal matrix. */ /* For other values of COMPQ, VT is not referenced. */ /* LDVT (input) INTEGER */ /* The leading dimension of the array VT. LDVT >= 1. */ /* If singular vectors are desired, then LDVT >= MAX( 1, N ). */ /* Q (output) REAL array, dimension (LDQ) */ /* If COMPQ = 'P', then: */ /* On exit, if INFO = 0, Q and IQ contain the left */ /* and right singular vectors in a compact form, */ /* requiring O(N log N) space instead of 2*N**2. */ /* In particular, Q contains all the REAL data in */ /* LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) */ /* words of memory, where SMLSIZ is returned by ILAENV and */ /* is equal to the maximum size of the subproblems at the */ /* bottom of the computation tree (usually about 25). */ /* For other values of COMPQ, Q is not referenced. */ /* IQ (output) INTEGER array, dimension (LDIQ) */ /* If COMPQ = 'P', then: */ /* On exit, if INFO = 0, Q and IQ contain the left */ /* and right singular vectors in a compact form, */ /* requiring O(N log N) space instead of 2*N**2. */ /* In particular, IQ contains all INTEGER data in */ /* LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) */ /* words of memory, where SMLSIZ is returned by ILAENV and */ /* is equal to the maximum size of the subproblems at the */ /* bottom of the computation tree (usually about 25). */ /* For other values of COMPQ, IQ is not referenced. */ /* WORK (workspace) REAL array, dimension (MAX(1,LWORK)) */ /* If COMPQ = 'N' then LWORK >= (4 * N). */ /* If COMPQ = 'P' then LWORK >= (6 * N). */ /* If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N). */ /* IWORK (workspace) INTEGER array, dimension (8*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. */ /* The update process of divide and conquer failed. */ /* Further Details */ /* =============== */ /* Based on contributions by */ /* Ming Gu and Huan Ren, Computer Science Division, University of */ /* California at Berkeley, USA */ /* ===================================================================== */ /* Changed dimension statement in comment describing E from (N) to */ /* (N-1). Sven, 17 Feb 05. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Test the input parameters. */ /* Parameter adjustments */ --d__; --e; u_dim1 = *ldu; u_offset = 1 + u_dim1; u -= u_offset; vt_dim1 = *ldvt; vt_offset = 1 + vt_dim1; vt -= vt_offset; --q; --iq; --work; --iwork; /* Function Body */ *info = 0; iuplo = 0; if (lsame_(uplo, "U")) { iuplo = 1; } if (lsame_(uplo, "L")) { iuplo = 2; } if (lsame_(compq, "N")) { icompq = 0; } else if (lsame_(compq, "P")) { icompq = 1; } else if (lsame_(compq, "I")) { icompq = 2; } else { icompq = -1; } if (iuplo == 0) { *info = -1; } else if (icompq < 0) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*ldu < 1 || icompq == 2 && *ldu < *n) { *info = -7; } else if (*ldvt < 1 || icompq == 2 && *ldvt < *n) { *info = -9; } if (*info != 0) { i__1 = -(*info); xerbla_("SBDSDC", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } smlsiz = ilaenv_(&c__9, "SBDSDC", " ", &c__0, &c__0, &c__0, &c__0); if (*n == 1) { if (icompq == 1) { q[1] = r_sign(&c_b15, &d__[1]); q[smlsiz * *n + 1] = 1.f; } else if (icompq == 2) { u[u_dim1 + 1] = r_sign(&c_b15, &d__[1]); vt[vt_dim1 + 1] = 1.f; } d__[1] = ABS(d__[1]); return 0; } nm1 = *n - 1; /* If matrix lower bidiagonal, rotate to be upper bidiagonal */ /* by applying Givens rotations on the left */ wstart = 1; qstart = 3; if (icompq == 1) { scopy_(n, &d__[1], &c__1, &q[1], &c__1); i__1 = *n - 1; scopy_(&i__1, &e[1], &c__1, &q[*n + 1], &c__1); } if (iuplo == 2) { qstart = 5; wstart = (*n << 1) - 1; 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 (icompq == 1) { q[i__ + (*n << 1)] = cs; q[i__ + *n * 3] = sn; } else if (icompq == 2) { work[i__] = cs; work[nm1 + i__] = -sn; } /* L10: */ } } /* If ICOMPQ = 0, use SLASDQ to compute the singular values. */ if (icompq == 0) { slasdq_("U", &c__0, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[ vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[ wstart], info); goto L40; } /* If N is smaller than the minimum divide size SMLSIZ, then solve */ /* the problem with another solver. */ if (*n <= smlsiz) { if (icompq == 2) { slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu); slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt); slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &vt[vt_offset] , ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[ wstart], info); } else if (icompq == 1) { iu = 1; ivt = iu + *n; slaset_("A", n, n, &c_b29, &c_b15, &q[iu + (qstart - 1) * *n], n); slaset_("A", n, n, &c_b29, &c_b15, &q[ivt + (qstart - 1) * *n], n); slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &q[ivt + ( qstart - 1) * *n], n, &q[iu + (qstart - 1) * *n], n, &q[ iu + (qstart - 1) * *n], n, &work[wstart], info); } goto L40; } if (icompq == 2) { slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu); slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt); } /* Scale. */ orgnrm = slanst_("M", n, &d__[1], &e[1]); if (orgnrm == 0.f) { return 0; } slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, n, &c__1, &d__[1], n, &ierr); slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, &nm1, &c__1, &e[1], &nm1, & ierr); eps = slamch_("Epsilon"); mlvl = (int) (log((float) (*n) / (float) (smlsiz + 1)) / log(2.f)) + 1; smlszp = smlsiz + 1; if (icompq == 1) { iu = 1; ivt = smlsiz + 1; difl = ivt + smlszp; difr = difl + mlvl; z__ = difr + (mlvl << 1); ic = z__ + mlvl; is = ic + 1; poles = is + 1; givnum = poles + (mlvl << 1); k = 1; givptr = 2; perm = 3; givcol = perm + mlvl; } i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if ((r__1 = d__[i__], ABS(r__1)) < eps) { d__[i__] = r_sign(&eps, &d__[i__]); } /* L20: */ } start = 1; sqre = 0; i__1 = nm1; for (i__ = 1; i__ <= i__1; ++i__) { if ((r__1 = e[i__], ABS(r__1)) < eps || i__ == nm1) { /* 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__ - start + 1; } else if ((r__1 = e[i__], ABS(r__1)) >= eps) { /* A subproblem with E(NM1) not too small but I = NM1. */ nsize = *n - start + 1; } else { /* A subproblem with E(NM1) small. This implies an */ /* 1-by-1 subproblem at D(N). Solve this 1-by-1 problem */ /* first. */ nsize = i__ - start + 1; if (icompq == 2) { u[*n + *n * u_dim1] = r_sign(&c_b15, &d__[*n]); vt[*n + *n * vt_dim1] = 1.f; } else if (icompq == 1) { q[*n + (qstart - 1) * *n] = r_sign(&c_b15, &d__[*n]); q[*n + (smlsiz + qstart - 1) * *n] = 1.f; } d__[*n] = (r__1 = d__[*n], ABS(r__1)); } if (icompq == 2) { slasd0_(&nsize, &sqre, &d__[start], &e[start], &u[start + start * u_dim1], ldu, &vt[start + start * vt_dim1], ldvt, &smlsiz, &iwork[1], &work[wstart], info); } else { slasda_(&icompq, &smlsiz, &nsize, &sqre, &d__[start], &e[ start], &q[start + (iu + qstart - 2) * *n], n, &q[ start + (ivt + qstart - 2) * *n], &iq[start + k * *n], &q[start + (difl + qstart - 2) * *n], &q[start + ( difr + qstart - 2) * *n], &q[start + (z__ + qstart - 2) * *n], &q[start + (poles + qstart - 2) * *n], &iq[ start + givptr * *n], &iq[start + givcol * *n], n, & iq[start + perm * *n], &q[start + (givnum + qstart - 2) * *n], &q[start + (ic + qstart - 2) * *n], &q[ start + (is + qstart - 2) * *n], &work[wstart], & iwork[1], info); if (*info != 0) { return 0; } } start = i__ + 1; } /* L30: */ } /* Unscale */ slascl_("G", &c__0, &c__0, &c_b15, &orgnrm, n, &c__1, &d__[1], n, &ierr); L40: /* Use Selection Sort to minimize swaps of singular vectors */ i__1 = *n; for (ii = 2; ii <= i__1; ++ii) { i__ = ii - 1; kk = i__; p = d__[i__]; i__2 = *n; for (j = ii; j <= i__2; ++j) { if (d__[j] > p) { kk = j; p = d__[j]; } /* L50: */ } if (kk != i__) { d__[kk] = d__[i__]; d__[i__] = p; if (icompq == 1) { iq[i__] = kk; } else if (icompq == 2) { sswap_(n, &u[i__ * u_dim1 + 1], &c__1, &u[kk * u_dim1 + 1], & c__1); sswap_(n, &vt[i__ + vt_dim1], ldvt, &vt[kk + vt_dim1], ldvt); } } else if (icompq == 1) { iq[i__] = i__; } /* L60: */ } /* If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO */ if (icompq == 1) { if (iuplo == 1) { iq[*n] = 1; } else { iq[*n] = 0; } } /* If B is lower bidiagonal, update U by those Givens rotations */ /* which rotated B to be upper bidiagonal */ if (iuplo == 2 && icompq == 2) { slasr_("L", "V", "B", n, n, &work[1], &work[*n], &u[u_offset], ldu); } return 0; /* End of SBDSDC */ } /* sbdsdc_ */
/* 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) { /* 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 */ integer c__, i__, j, k; real r__; integer s, u, z__; real cs; integer bx; real sn; integer st, vt, nm1, st1; real eps; integer iwk; real tol; integer difl, difr; real rcnd; 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 *); integer irwib; extern /* Subroutine */ int ccopy_(integer *, complex *, integer *, complex *, integer *); integer poles, sizei, irwrb, nsize; extern /* Subroutine */ int csrot_(integer *, complex *, integer *, complex *, integer *, real *, real *); 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 *); 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 * ); real orgnrm; integer givnum; extern doublereal slanst_(char *, integer *, real *, real *); extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *); integer givptr, nrwork, irwwrk, smlszp; /* -- LAPACK routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* 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 */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Test the input parameters. */ /* Parameter adjustments */ --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_ */
/* Subroutine */ int slalsd_(char *uplo, integer *smlsiz, integer *n, integer *nrhs, real *d__, real *e, real *b, integer *ldb, real *rcond, integer *rank, real *work, integer *iwork, integer *info) { /* System generated locals */ integer b_dim1, b_offset, i__1, i__2; real r__1; /* Builtin functions */ double log(doublereal), r_sign(real *, real *); /* Local variables */ static integer difl, difr, perm, nsub, nlvl, sqre, bxst; extern /* Subroutine */ int srot_(integer *, real *, integer *, real *, integer *, real *, real *); static integer c__, i__, j, k; static real r__; static integer s, u, z__; extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); static integer poles, sizei, nsize; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); static integer nwork, icmpq1, icmpq2; extern doublereal sopbl3_(char *, integer *, integer *, integer *) ; static real cs; static integer bx; static real sn; static integer st; 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 *); extern doublereal slamch_(char *); static integer vt; extern /* Subroutine */ int xerbla_(char *, integer *), slalsa_( integer *, integer *, integer *, integer *, real *, integer *, real *, integer *, real *, integer *, real *, integer *, real *, real *, real *, real *, integer *, integer *, integer *, integer * , real *, real *, real *, real *, integer *, integer *), slascl_( char *, integer *, integer *, real *, real *, integer *, integer * , real *, integer *, integer *); static integer givcol; extern integer isamax_(integer *, real *, integer *); extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer *, integer *, integer *, real *, real *, real *, integer *, real * , integer *, real *, integer *, real *, integer *), slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *), slartg_(real *, real *, real *, real *, real * ), slaset_(char *, integer *, integer *, real *, real *, real *, integer *); static real orgnrm; static integer givnum; extern doublereal slanst_(char *, integer *, real *, real *); extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *); static integer givptr, nm1, smlszp, st1; static real eps; static integer iwk; static real tol; #define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1] /* -- LAPACK routine (instrumented to count ops, version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1999 Purpose ======= SLALSD 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) REAL array, dimension (N-1) Contains the super-diagonal entries of the bidiagonal matrix. On exit, E has been destroyed. B (input/output) REAL 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) REAL array, dimension at least (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1). IWORK (workspace) INTEGER array, dimension at least (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). ===================================================================== Test the input parameters. Parameter adjustments */ --d__; --e; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --work; --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_("SLALSD", &i__1); return 0; } eps = slamch_("Epsilon"); /* Set up the tolerance. */ if (*rcond <= 0.f || *rcond >= 1.f) { *rcond = eps; } *rank = 0; /* Quick return if possible. */ if (*n == 0) { return 0; } else if (*n == 1) { if (d__[1] == 0.f) { slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[b_offset], ldb); } else { *rank = 1; latime_1.ops += (real) (*nrhs << 1); slascl_("G", &c__0, &c__0, &d__[1], &c_b11, &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') { latime_1.ops += (real) ((*n - 1) * 6); 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) { latime_1.ops += 6.f; srot_(&c__1, &b_ref(i__, 1), &c__1, &b_ref(i__ + 1, 1), &c__1, &cs, &sn); } else { work[(i__ << 1) - 1] = cs; work[i__ * 2] = sn; } /* L10: */ } if (*nrhs > 1) { latime_1.ops += (real) ((*n - 1) * 6 * *nrhs); i__1 = *nrhs; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *n - 1; for (j = 1; j <= i__2; ++j) { cs = work[(j << 1) - 1]; sn = work[j * 2]; srot_(&c__1, &b_ref(j, i__), &c__1, &b_ref(j + 1, i__), & c__1, &cs, &sn); /* L20: */ } /* L30: */ } } } /* Scale. */ nm1 = *n - 1; orgnrm = slanst_("M", n, &d__[1], &e[1]); if (orgnrm == 0.f) { slaset_("A", n, nrhs, &c_b6, &c_b6, &b[b_offset], ldb); return 0; } latime_1.ops += (real) (*n + nm1); slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, &c__1, &d__[1], n, info); slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, &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) { nwork = *n * *n + 1; slaset_("A", n, n, &c_b6, &c_b11, &work[1], n); slasdq_("U", &c__0, n, n, &c__0, nrhs, &d__[1], &e[1], &work[1], n, & work[1], n, &b[b_offset], ldb, &work[nwork], info); if (*info != 0) { return 0; } latime_1.ops += 1.f; tol = *rcond * (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) { slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b_ref(i__, 1), ldb); } else { latime_1.ops += (real) (*nrhs); slascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, & b_ref(i__, 1), ldb, info); ++(*rank); } /* L40: */ } latime_1.ops += sopbl3_("SGEMM ", n, nrhs, n); sgemm_("T", "N", n, nrhs, n, &c_b11, &work[1], n, &b[b_offset], ldb, & c_b6, &work[nwork], n); slacpy_("A", n, nrhs, &work[nwork], n, &b[b_offset], ldb); /* Unscale. */ latime_1.ops += (real) (*n + *n * *nrhs); slascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info); slasrt_("D", n, &d__[1], info); slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, 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; bx = givnum + (nlvl << 1) * *n; nwork = bx + *n * *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__]); } /* L50: */ } 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; scopy_(nrhs, &b_ref(*n, 1), ldb, &work[bx + nm1], n); } st1 = st - 1; if (nsize == 1) { /* This is a 1-by-1 subproblem and is not solved explicitly. */ scopy_(nrhs, &b_ref(st, 1), ldb, &work[bx + st1], n); } else if (nsize <= *smlsiz) { /* This is a small subproblem and is solved by SLASDQ. */ slaset_("A", &nsize, &nsize, &c_b6, &c_b11, &work[vt + st1], n); slasdq_("U", &c__0, &nsize, &nsize, &c__0, nrhs, &d__[st], &e[ st], &work[vt + st1], n, &work[nwork], n, &b_ref(st, 1), ldb, &work[nwork], info); if (*info != 0) { return 0; } slacpy_("A", &nsize, nrhs, &b_ref(st, 1), ldb, &work[bx + st1] , n); } else { /* A large problem. Solve it using divide and conquer. */ slasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], & work[u + st1], n, &work[vt + st1], &iwork[k + st1], & work[difl + st1], &work[difr + st1], &work[z__ + st1], &work[poles + st1], &iwork[givptr + st1], &iwork[ givcol + st1], n, &iwork[perm + st1], &work[givnum + st1], &work[c__ + st1], &work[s + st1], &work[nwork], &iwork[iwk], info); if (*info != 0) { return 0; } bxst = bx + st1; slalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b_ref(st, 1), ldb, & work[bxst], n, &work[u + st1], n, &work[vt + st1], & iwork[k + st1], &work[difl + st1], &work[difr + st1], &work[z__ + st1], &work[poles + st1], &iwork[givptr + st1], &iwork[givcol + st1], n, &iwork[perm + st1], & work[givnum + st1], &work[c__ + st1], &work[s + st1], &work[nwork], &iwork[iwk], info); if (*info != 0) { return 0; } } st = i__ + 1; } /* L60: */ } /* Apply the singular values and treat the tiny ones as zero. */ tol = *rcond * (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) { slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &work[bx + i__ - 1], n); } else { ++(*rank); latime_1.ops += (real) (*nrhs); slascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &work[ bx + i__ - 1], n, info); } d__[i__] = (r__1 = d__[i__], dabs(r__1)); /* L70: */ } /* 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) { scopy_(nrhs, &work[bxst], n, &b_ref(st, 1), ldb); } else if (nsize <= *smlsiz) { latime_1.ops += sopbl3_("SGEMM ", &nsize, nrhs, &nsize) ; sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b11, &work[vt + st1], n, &work[bxst], n, &c_b6, &b_ref(st, 1), ldb); } else { slalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b_ref(st, 1), ldb, &work[u + st1], n, &work[vt + st1], &iwork[k + st1], &work[difl + st1], &work[difr + st1], &work[z__ + st1], &work[poles + st1], &iwork[givptr + st1], &iwork[ givcol + st1], n, &iwork[perm + st1], &work[givnum + st1], &work[c__ + st1], &work[s + st1], &work[nwork], &iwork[ iwk], info); if (*info != 0) { return 0; } } /* L80: */ } /* Unscale and sort the singular values. */ latime_1.ops += (real) (*n + *n * *nrhs); slascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info); slasrt_("D", n, &d__[1], info); slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb, info); return 0; /* End of SLALSD */ } /* slalsd_ */
/* Subroutine */ int sbdsdc_(char *uplo, char *compq, integer *n, real *d__, real *e, real *u, integer *ldu, real *vt, integer *ldvt, real *q, integer *iq, real *work, integer *iwork, integer *info) { /* System generated locals */ integer u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2; real r__1; /* Builtin functions */ double r_sign(real *, real *), log(doublereal); /* Local variables */ static integer difl, difr, ierr, perm, mlvl, sqre, i__, j, k; static real p, r__; static integer z__; extern logical lsame_(char *, char *); static integer poles; extern /* Subroutine */ int slasr_(char *, char *, char *, integer *, integer *, real *, real *, real *, integer *); static integer iuplo, nsize, start; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *), sswap_(integer *, real *, integer *, real *, integer * ), slasd0_(integer *, integer *, real *, real *, real *, integer * , real *, integer *, integer *, integer *, real *, integer *); static integer ic, ii, kk; static real cs; static integer is, iu; static real sn; 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 *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, real *, integer *, integer *, real *, integer *, integer *); static integer givcol; extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer *, integer *, integer *, real *, real *, real *, integer *, real * , integer *, real *, integer *, real *, integer *); static integer icompq; extern /* Subroutine */ int 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 *); static integer givptr, nm1, qstart, smlsiz, wstart, smlszp; static real eps; static integer ivt; #define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1] #define vt_ref(a_1,a_2) vt[(a_2)*vt_dim1 + a_1] /* -- LAPACK routine (instrumented to count ops, version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1999 Purpose ======= SBDSDC computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, using a divide and conquer method, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively. SBDSDC can be used to compute all singular values, and optionally, singular vectors or singular vectors in compact form. 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See SLASD3 for details. The code currently call SLASDQ if singular values only are desired. However, it can be slightly modified to compute singular values using the divide and conquer method. Arguments ========= UPLO (input) CHARACTER*1 = 'U': B is upper bidiagonal. = 'L': B is lower bidiagonal. COMPQ (input) CHARACTER*1 Specifies whether singular vectors are to be computed as follows: = 'N': Compute singular values only; = 'P': Compute singular values and compute singular vectors in compact form; = 'I': Compute singular values and singular vectors. N (input) INTEGER The order of the matrix B. N >= 0. D (input/output) REAL array, dimension (N) On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B. E (input/output) REAL array, dimension (N) On entry, the elements of E contain the offdiagonal elements of the bidiagonal matrix whose SVD is desired. On exit, E has been destroyed. U (output) REAL array, dimension (LDU,N) If COMPQ = 'I', then: On exit, if INFO = 0, U contains the left singular vectors of the bidiagonal matrix. For other values of COMPQ, U is not referenced. LDU (input) INTEGER The leading dimension of the array U. LDU >= 1. If singular vectors are desired, then LDU >= max( 1, N ). VT (output) REAL array, dimension (LDVT,N) If COMPQ = 'I', then: On exit, if INFO = 0, VT' contains the right singular vectors of the bidiagonal matrix. For other values of COMPQ, VT is not referenced. LDVT (input) INTEGER The leading dimension of the array VT. LDVT >= 1. If singular vectors are desired, then LDVT >= max( 1, N ). Q (output) REAL array, dimension (LDQ) If COMPQ = 'P', then: On exit, if INFO = 0, Q and IQ contain the left and right singular vectors in a compact form, requiring O(N log N) space instead of 2*N**2. In particular, Q contains all the REAL data in LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25). For other values of COMPQ, Q is not referenced. IQ (output) INTEGER array, dimension (LDIQ) If COMPQ = 'P', then: On exit, if INFO = 0, Q and IQ contain the left and right singular vectors in a compact form, requiring O(N log N) space instead of 2*N**2. In particular, IQ contains all INTEGER data in LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25). For other values of COMPQ, IQ is not referenced. WORK (workspace) REAL array, dimension (LWORK) If COMPQ = 'N' then LWORK >= (4 * N). If COMPQ = 'P' then LWORK >= (6 * N). If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N). IWORK (workspace) INTEGER array, dimension (7*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. The update process of divide and conquer failed. Further Details =============== Based on contributions by Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA ===================================================================== Test the input parameters. Parameter adjustments */ --d__; --e; u_dim1 = *ldu; u_offset = 1 + u_dim1 * 1; u -= u_offset; vt_dim1 = *ldvt; vt_offset = 1 + vt_dim1 * 1; vt -= vt_offset; --q; --iq; --work; --iwork; /* Function Body */ *info = 0; iuplo = 0; if (lsame_(uplo, "U")) { iuplo = 1; } if (lsame_(uplo, "L")) { iuplo = 2; } if (lsame_(compq, "N")) { icompq = 0; } else if (lsame_(compq, "P")) { icompq = 1; } else if (lsame_(compq, "I")) { icompq = 2; } else { icompq = -1; } if (iuplo == 0) { *info = -1; } else if (icompq < 0) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*ldu < 1 || icompq == 2 && *ldu < *n) { *info = -7; } else if (*ldvt < 1 || icompq == 2 && *ldvt < *n) { *info = -9; } if (*info != 0) { i__1 = -(*info); xerbla_("SBDSDC", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } smlsiz = ilaenv_(&c__9, "SBDSDC", " ", &c__0, &c__0, &c__0, &c__0, ( ftnlen)6, (ftnlen)1); if (*n == 1) { if (icompq == 1) { q[1] = r_sign(&c_b15, &d__[1]); q[smlsiz * *n + 1] = 1.f; } else if (icompq == 2) { u_ref(1, 1) = r_sign(&c_b15, &d__[1]); vt_ref(1, 1) = 1.f; } d__[1] = dabs(d__[1]); return 0; } nm1 = *n - 1; /* If matrix lower bidiagonal, rotate to be upper bidiagonal by applying Givens rotations on the left */ wstart = 1; qstart = 3; if (icompq == 1) { scopy_(n, &d__[1], &c__1, &q[1], &c__1); i__1 = *n - 1; scopy_(&i__1, &e[1], &c__1, &q[*n + 1], &c__1); } if (iuplo == 2) { qstart = 5; wstart = (*n << 1) - 1; latime_1.ops += (real) (*n - 1 << 3); 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 (icompq == 1) { q[i__ + (*n << 1)] = cs; q[i__ + *n * 3] = sn; } else if (icompq == 2) { work[i__] = cs; work[nm1 + i__] = -sn; } /* L10: */ } } /* If ICOMPQ = 0, use SLASDQ to compute the singular values. */ if (icompq == 0) { slasdq_("U", &c__0, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[ vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[ wstart], info); goto L40; } /* If N is smaller than the minimum divide size SMLSIZ, then solve the problem with another solver. */ if (*n <= smlsiz) { if (icompq == 2) { slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu); slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt); slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &vt[vt_offset] , ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[ wstart], info); } else if (icompq == 1) { iu = 1; ivt = iu + *n; slaset_("A", n, n, &c_b29, &c_b15, &q[iu + (qstart - 1) * *n], n); slaset_("A", n, n, &c_b29, &c_b15, &q[ivt + (qstart - 1) * *n], n); slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &q[ivt + ( qstart - 1) * *n], n, &q[iu + (qstart - 1) * *n], n, &q[ iu + (qstart - 1) * *n], n, &work[wstart], info); } goto L40; } if (icompq == 2) { slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu); slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt); } /* Scale. */ orgnrm = slanst_("M", n, &d__[1], &e[1]); if (orgnrm == 0.f) { return 0; } latime_1.ops += (real) (*n + nm1); slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, n, &c__1, &d__[1], n, &ierr); slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, &nm1, &c__1, &e[1], &nm1, & ierr); eps = slamch_("Epsilon"); mlvl = (integer) (log((real) (*n) / (real) (smlsiz + 1)) / log(2.f)) + 1; smlszp = smlsiz + 1; if (icompq == 1) { iu = 1; ivt = smlsiz + 1; difl = ivt + smlszp; difr = difl + mlvl; z__ = difr + (mlvl << 1); ic = z__ + mlvl; is = ic + 1; poles = is + 1; givnum = poles + (mlvl << 1); k = 1; givptr = 2; perm = 3; givcol = perm + mlvl; } 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__]); } /* L20: */ } start = 1; sqre = 0; i__1 = nm1; for (i__ = 1; i__ <= i__1; ++i__) { if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) { /* 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__ - start + 1; } else if ((r__1 = e[i__], dabs(r__1)) >= eps) { /* A subproblem with E(NM1) not too small but I = NM1. */ nsize = *n - start + 1; } else { /* A subproblem with E(NM1) small. This implies an 1-by-1 subproblem at D(N). Solve this 1-by-1 problem first. */ nsize = i__ - start + 1; if (icompq == 2) { u_ref(*n, *n) = r_sign(&c_b15, &d__[*n]); vt_ref(*n, *n) = 1.f; } else if (icompq == 1) { q[*n + (qstart - 1) * *n] = r_sign(&c_b15, &d__[*n]); q[*n + (smlsiz + qstart - 1) * *n] = 1.f; } d__[*n] = (r__1 = d__[*n], dabs(r__1)); } if (icompq == 2) { slasd0_(&nsize, &sqre, &d__[start], &e[start], &u_ref(start, start), ldu, &vt_ref(start, start), ldvt, &smlsiz, & iwork[1], &work[wstart], info); } else { slasda_(&icompq, &smlsiz, &nsize, &sqre, &d__[start], &e[ start], &q[start + (iu + qstart - 2) * *n], n, &q[ start + (ivt + qstart - 2) * *n], &iq[start + k * *n], &q[start + (difl + qstart - 2) * *n], &q[start + ( difr + qstart - 2) * *n], &q[start + (z__ + qstart - 2) * *n], &q[start + (poles + qstart - 2) * *n], &iq[ start + givptr * *n], &iq[start + givcol * *n], n, & iq[start + perm * *n], &q[start + (givnum + qstart - 2) * *n], &q[start + (ic + qstart - 2) * *n], &q[ start + (is + qstart - 2) * *n], &work[wstart], & iwork[1], info); if (*info != 0) { return 0; } } start = i__ + 1; } /* L30: */ } /* Unscale */ latime_1.ops += (real) (*n); slascl_("G", &c__0, &c__0, &c_b15, &orgnrm, n, &c__1, &d__[1], n, &ierr); L40: /* Use Selection Sort to minimize swaps of singular vectors */ i__1 = *n; for (ii = 2; ii <= i__1; ++ii) { i__ = ii - 1; kk = i__; p = d__[i__]; i__2 = *n; for (j = ii; j <= i__2; ++j) { if (d__[j] > p) { kk = j; p = d__[j]; } /* L50: */ } if (kk != i__) { d__[kk] = d__[i__]; d__[i__] = p; if (icompq == 1) { iq[i__] = kk; } else if (icompq == 2) { sswap_(n, &u_ref(1, i__), &c__1, &u_ref(1, kk), &c__1); sswap_(n, &vt_ref(i__, 1), ldvt, &vt_ref(kk, 1), ldvt); } } else if (icompq == 1) { iq[i__] = i__; } /* L60: */ } /* If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO */ if (icompq == 1) { if (iuplo == 1) { iq[*n] = 1; } else { iq[*n] = 0; } } /* If B is lower bidiagonal, update U by those Givens rotations which rotated B to be upper bidiagonal */ if (iuplo == 2 && icompq == 2) { latime_1.ops += (real) ((*n - 1) * 6 * *n); slasr_("L", "V", "B", n, n, &work[1], &work[*n], &u[u_offset], ldu); } return 0; /* End of SBDSDC */ } /* sbdsdc_ */