#include #include #include #include #include #ifdef complex #undef complex #endif #ifdef I #undef I #endif #if defined(_WIN64) typedef long long BLASLONG; typedef unsigned long long BLASULONG; #else typedef long BLASLONG; typedef unsigned long BLASULONG; #endif #ifdef LAPACK_ILP64 typedef BLASLONG blasint; #if defined(_WIN64) #define blasabs(x) llabs(x) #else #define blasabs(x) labs(x) #endif #else typedef int blasint; #define blasabs(x) abs(x) #endif typedef blasint integer; typedef unsigned int uinteger; typedef char *address; typedef short int shortint; typedef float real; typedef double doublereal; typedef struct { real r, i; } complex; typedef struct { doublereal r, i; } doublecomplex; #ifdef _MSC_VER static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;} static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;} static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;} static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;} #else static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;} static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;} static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;} static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;} #endif #define pCf(z) (*_pCf(z)) #define pCd(z) (*_pCd(z)) typedef int logical; typedef short int shortlogical; typedef char logical1; typedef char integer1; #define TRUE_ (1) #define FALSE_ (0) /* Extern is for use with -E */ #ifndef Extern #define Extern extern #endif /* I/O stuff */ typedef int flag; typedef int ftnlen; typedef int ftnint; /*external read, write*/ typedef struct { flag cierr; ftnint ciunit; flag ciend; char *cifmt; ftnint cirec; } cilist; /*internal read, write*/ typedef struct { flag icierr; char *iciunit; flag iciend; char *icifmt; ftnint icirlen; ftnint icirnum; } icilist; /*open*/ typedef struct { flag oerr; ftnint ounit; char *ofnm; ftnlen ofnmlen; char *osta; char *oacc; char *ofm; ftnint orl; char *oblnk; } olist; /*close*/ typedef struct { flag cerr; ftnint cunit; char *csta; } cllist; /*rewind, backspace, endfile*/ typedef struct { flag aerr; ftnint aunit; } alist; /* inquire */ typedef struct { flag inerr; ftnint inunit; char *infile; ftnlen infilen; ftnint *inex; /*parameters in standard's order*/ ftnint *inopen; ftnint *innum; ftnint *innamed; char *inname; ftnlen innamlen; char *inacc; ftnlen inacclen; char *inseq; ftnlen inseqlen; char *indir; ftnlen indirlen; char *infmt; ftnlen infmtlen; char *inform; ftnint informlen; char *inunf; ftnlen inunflen; ftnint *inrecl; ftnint *innrec; char *inblank; ftnlen inblanklen; } inlist; #define VOID void union Multitype { /* for multiple entry points */ integer1 g; shortint h; integer i; /* longint j; */ real r; doublereal d; complex c; doublecomplex z; }; typedef union Multitype Multitype; struct Vardesc { /* for Namelist */ char *name; char *addr; ftnlen *dims; int type; }; typedef struct Vardesc Vardesc; struct Namelist { char *name; Vardesc **vars; int nvars; }; typedef struct Namelist Namelist; #define abs(x) ((x) >= 0 ? (x) : -(x)) #define dabs(x) (fabs(x)) #define f2cmin(a,b) ((a) <= (b) ? (a) : (b)) #define f2cmax(a,b) ((a) >= (b) ? (a) : (b)) #define dmin(a,b) (f2cmin(a,b)) #define dmax(a,b) (f2cmax(a,b)) #define bit_test(a,b) ((a) >> (b) & 1) #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b))) #define bit_set(a,b) ((a) | ((uinteger)1 << (b))) #define abort_() { sig_die("Fortran abort routine called", 1); } #define c_abs(z) (cabsf(Cf(z))) #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); } #ifdef _MSC_VER #define c_div(c, a, b) {Cf(c)._Val[0] = (Cf(a)._Val[0]/Cf(b)._Val[0]); Cf(c)._Val[1]=(Cf(a)._Val[1]/Cf(b)._Val[1]);} #define z_div(c, a, b) {Cd(c)._Val[0] = (Cd(a)._Val[0]/Cd(b)._Val[0]); Cd(c)._Val[1]=(Cd(a)._Val[1]/df(b)._Val[1]);} #else #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);} #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);} #endif #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));} #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));} #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));} //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));} #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));} #define d_abs(x) (fabs(*(x))) #define d_acos(x) (acos(*(x))) #define d_asin(x) (asin(*(x))) #define d_atan(x) (atan(*(x))) #define d_atn2(x, y) (atan2(*(x),*(y))) #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); } #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); } #define d_cos(x) (cos(*(x))) #define d_cosh(x) (cosh(*(x))) #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 ) #define d_exp(x) (exp(*(x))) #define d_imag(z) (cimag(Cd(z))) #define r_imag(z) (cimagf(Cf(z))) #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define d_log(x) (log(*(x))) #define d_mod(x, y) (fmod(*(x), *(y))) #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x))) #define d_nint(x) u_nint(*(x)) #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a))) #define d_sign(a,b) u_sign(*(a),*(b)) #define r_sign(a,b) u_sign(*(a),*(b)) #define d_sin(x) (sin(*(x))) #define d_sinh(x) (sinh(*(x))) #define d_sqrt(x) (sqrt(*(x))) #define d_tan(x) (tan(*(x))) #define d_tanh(x) (tanh(*(x))) #define i_abs(x) abs(*(x)) #define i_dnnt(x) ((integer)u_nint(*(x))) #define i_len(s, n) (n) #define i_nint(x) ((integer)u_nint(*(x))) #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b))) #define pow_dd(ap, bp) ( pow(*(ap), *(bp))) #define pow_si(B,E) spow_ui(*(B),*(E)) #define pow_ri(B,E) spow_ui(*(B),*(E)) #define pow_di(B,E) dpow_ui(*(B),*(E)) #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));} #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));} #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));} #define s_cat(lpp, rpp, rnp, np, llp) { ftnlen i, nc, ll; char *f__rp, *lp; ll = (llp); lp = (lpp); for(i=0; i < (int)*(np); ++i) { nc = ll; if((rnp)[i] < nc) nc = (rnp)[i]; ll -= nc; f__rp = (rpp)[i]; while(--nc >= 0) *lp++ = *(f__rp)++; } while(--ll >= 0) *lp++ = ' '; } #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d)))) #define s_copy(A,B,C,D) { int __i,__m; for (__i=0, __m=f2cmin((C),(D)); __i<__m && (B)[__i] != 0; ++__i) (A)[__i] = (B)[__i]; } #define sig_die(s, kill) { exit(1); } #define s_stop(s, n) {exit(0);} static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n"; #define z_abs(z) (cabs(Cd(z))) #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));} #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));} #define myexit_() break; #define mycycle() continue; #define myceiling(w) {ceil(w)} #define myhuge(w) {HUGE_VAL} //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);} #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)} /* procedure parameter types for -A and -C++ */ #define F2C_proc_par_types 1 #ifdef __cplusplus typedef logical (*L_fp)(...); #else typedef logical (*L_fp)(); #endif static float spow_ui(float x, integer n) { float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static double dpow_ui(double x, integer n) { double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #ifdef _MSC_VER static _Fcomplex cpow_ui(complex x, integer n) { complex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i; for(u = n; ; ) { if(u & 01) pow.r *= x.r, pow.i *= x.i; if(u >>= 1) x.r *= x.r, x.i *= x.i; else break; } } _Fcomplex p={pow.r, pow.i}; return p; } #else static _Complex float cpow_ui(_Complex float x, integer n) { _Complex float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif #ifdef _MSC_VER static _Dcomplex zpow_ui(_Dcomplex x, integer n) { _Dcomplex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1]; for(u = n; ; ) { if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1]; if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1]; else break; } } _Dcomplex p = {pow._Val[0], pow._Val[1]}; return p; } #else static _Complex double zpow_ui(_Complex double x, integer n) { _Complex double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif static integer pow_ii(integer x, integer n) { integer pow; unsigned long int u; if (n <= 0) { if (n == 0 || x == 1) pow = 1; else if (x != -1) pow = x == 0 ? 1/x : 0; else n = -n; } if ((n > 0) || !(n == 0 || x == 1 || x != -1)) { u = n; for(pow = 1; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static integer dmaxloc_(double *w, integer s, integer e, integer *n) { double m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static integer smaxloc_(float *w, integer s, integer e, integer *n) { float m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) { integer n = *n_, incx = *incx_, incy = *incy_, i; #ifdef _MSC_VER _Fcomplex zdotc = {0.0, 0.0}; if (incx == 1 && incy == 1) { for (i=0;i \brief \b SGEJSV */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SGEJSV + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, */ /* M, N, A, LDA, SVA, U, LDU, V, LDV, */ /* WORK, LWORK, IWORK, INFO ) */ /* IMPLICIT NONE */ /* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N */ /* REAL A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ), */ /* $ WORK( LWORK ) */ /* INTEGER IWORK( * ) */ /* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > SGEJSV computes the singular value decomposition (SVD) of a real M-by-N */ /* > matrix [A], where M >= N. The SVD of [A] is written as */ /* > */ /* > [A] = [U] * [SIGMA] * [V]^t, */ /* > */ /* > where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N */ /* > diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and */ /* > [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are */ /* > the singular values of [A]. The columns of [U] and [V] are the left and */ /* > the right singular vectors of [A], respectively. The matrices [U] and [V] */ /* > are computed and stored in the arrays U and V, respectively. The diagonal */ /* > of [SIGMA] is computed and stored in the array SVA. */ /* > SGEJSV can sometimes compute tiny singular values and their singular vectors much */ /* > more accurately than other SVD routines, see below under Further Details. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] JOBA */ /* > \verbatim */ /* > JOBA is CHARACTER*1 */ /* > Specifies the level of accuracy: */ /* > = 'C': This option works well (high relative accuracy) if A = B * D, */ /* > with well-conditioned B and arbitrary diagonal matrix D. */ /* > The accuracy cannot be spoiled by COLUMN scaling. The */ /* > accuracy of the computed output depends on the condition of */ /* > B, and the procedure aims at the best theoretical accuracy. */ /* > The relative error max_{i=1:N}|d sigma_i| / sigma_i is */ /* > bounded by f(M,N)*epsilon* cond(B), independent of D. */ /* > The input matrix is preprocessed with the QRF with column */ /* > pivoting. This initial preprocessing and preconditioning by */ /* > a rank revealing QR factorization is common for all values of */ /* > JOBA. Additional actions are specified as follows: */ /* > = 'E': Computation as with 'C' with an additional estimate of the */ /* > condition number of B. It provides a realistic error bound. */ /* > = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings */ /* > D1, D2, and well-conditioned matrix C, this option gives */ /* > higher accuracy than the 'C' option. If the structure of the */ /* > input matrix is not known, and relative accuracy is */ /* > desirable, then this option is advisable. The input matrix A */ /* > is preprocessed with QR factorization with FULL (row and */ /* > column) pivoting. */ /* > = 'G': Computation as with 'F' with an additional estimate of the */ /* > condition number of B, where A=D*B. If A has heavily weighted */ /* > rows, then using this condition number gives too pessimistic */ /* > error bound. */ /* > = 'A': Small singular values are the noise and the matrix is treated */ /* > as numerically rank deficient. The error in the computed */ /* > singular values is bounded by f(m,n)*epsilon*||A||. */ /* > The computed SVD A = U * S * V^t restores A up to */ /* > f(m,n)*epsilon*||A||. */ /* > This gives the procedure the licence to discard (set to zero) */ /* > all singular values below N*epsilon*||A||. */ /* > = 'R': Similar as in 'A'. Rank revealing property of the initial */ /* > QR factorization is used do reveal (using triangular factor) */ /* > a gap sigma_{r+1} < epsilon * sigma_r in which case the */ /* > numerical RANK is declared to be r. The SVD is computed with */ /* > absolute error bounds, but more accurately than with 'A'. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBU */ /* > \verbatim */ /* > JOBU is CHARACTER*1 */ /* > Specifies whether to compute the columns of U: */ /* > = 'U': N columns of U are returned in the array U. */ /* > = 'F': full set of M left sing. vectors is returned in the array U. */ /* > = 'W': U may be used as workspace of length M*N. See the description */ /* > of U. */ /* > = 'N': U is not computed. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBV */ /* > \verbatim */ /* > JOBV is CHARACTER*1 */ /* > Specifies whether to compute the matrix V: */ /* > = 'V': N columns of V are returned in the array V; Jacobi rotations */ /* > are not explicitly accumulated. */ /* > = 'J': N columns of V are returned in the array V, but they are */ /* > computed as the product of Jacobi rotations. This option is */ /* > allowed only if JOBU .NE. 'N', i.e. in computing the full SVD. */ /* > = 'W': V may be used as workspace of length N*N. See the description */ /* > of V. */ /* > = 'N': V is not computed. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBR */ /* > \verbatim */ /* > JOBR is CHARACTER*1 */ /* > Specifies the RANGE for the singular values. Issues the licence to */ /* > set to zero small positive singular values if they are outside */ /* > specified range. If A .NE. 0 is scaled so that the largest singular */ /* > value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues */ /* > the licence to kill columns of A whose norm in c*A is less than */ /* > SQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN, */ /* > where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E'). */ /* > = 'N': Do not kill small columns of c*A. This option assumes that */ /* > BLAS and QR factorizations and triangular solvers are */ /* > implemented to work in that range. If the condition of A */ /* > is greater than BIG, use SGESVJ. */ /* > = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)] */ /* > (roughly, as described above). This option is recommended. */ /* > =========================== */ /* > For computing the singular values in the FULL range [SFMIN,BIG] */ /* > use SGESVJ. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBT */ /* > \verbatim */ /* > JOBT is CHARACTER*1 */ /* > If the matrix is square then the procedure may determine to use */ /* > transposed A if A^t seems to be better with respect to convergence. */ /* > If the matrix is not square, JOBT is ignored. This is subject to */ /* > changes in the future. */ /* > The decision is based on two values of entropy over the adjoint */ /* > orbit of A^t * A. See the descriptions of WORK(6) and WORK(7). */ /* > = 'T': transpose if entropy test indicates possibly faster */ /* > convergence of Jacobi process if A^t is taken as input. If A is */ /* > replaced with A^t, then the row pivoting is included automatically. */ /* > = 'N': do not speculate. */ /* > This option can be used to compute only the singular values, or the */ /* > full SVD (U, SIGMA and V). For only one set of singular vectors */ /* > (U or V), the caller should provide both U and V, as one of the */ /* > matrices is used as workspace if the matrix A is transposed. */ /* > The implementer can easily remove this constraint and make the */ /* > code more complicated. See the descriptions of U and V. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBP */ /* > \verbatim */ /* > JOBP is CHARACTER*1 */ /* > Issues the licence to introduce structured perturbations to drown */ /* > denormalized numbers. This licence should be active if the */ /* > denormals are poorly implemented, causing slow computation, */ /* > especially in cases of fast convergence (!). For details see [1,2]. */ /* > For the sake of simplicity, this perturbations are included only */ /* > when the full SVD or only the singular values are requested. The */ /* > implementer/user can easily add the perturbation for the cases of */ /* > computing one set of singular vectors. */ /* > = 'P': introduce perturbation */ /* > = 'N': do not perturb */ /* > \endverbatim */ /* > */ /* > \param[in] M */ /* > \verbatim */ /* > M is INTEGER */ /* > The number of rows of the input matrix A. M >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The number of columns of the input matrix A. M >= N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] A */ /* > \verbatim */ /* > A is REAL array, dimension (LDA,N) */ /* > On entry, the M-by-N matrix A. */ /* > \endverbatim */ /* > */ /* > \param[in] LDA */ /* > \verbatim */ /* > LDA is INTEGER */ /* > The leading dimension of the array A. LDA >= f2cmax(1,M). */ /* > \endverbatim */ /* > */ /* > \param[out] SVA */ /* > \verbatim */ /* > SVA is REAL array, dimension (N) */ /* > On exit, */ /* > - For WORK(1)/WORK(2) = ONE: The singular values of A. During the */ /* > computation SVA contains Euclidean column norms of the */ /* > iterated matrices in the array A. */ /* > - For WORK(1) .NE. WORK(2): The singular values of A are */ /* > (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if */ /* > sigma_max(A) overflows or if small singular values have been */ /* > saved from underflow by scaling the input matrix A. */ /* > - If JOBR='R' then some of the singular values may be returned */ /* > as exact zeros obtained by "set to zero" because they are */ /* > below the numerical rank threshold or are denormalized numbers. */ /* > \endverbatim */ /* > */ /* > \param[out] U */ /* > \verbatim */ /* > U is REAL array, dimension ( LDU, N ) */ /* > If JOBU = 'U', then U contains on exit the M-by-N matrix of */ /* > the left singular vectors. */ /* > If JOBU = 'F', then U contains on exit the M-by-M matrix of */ /* > the left singular vectors, including an ONB */ /* > of the orthogonal complement of the Range(A). */ /* > If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N), */ /* > then U is used as workspace if the procedure */ /* > replaces A with A^t. In that case, [V] is computed */ /* > in U as left singular vectors of A^t and then */ /* > copied back to the V array. This 'W' option is just */ /* > a reminder to the caller that in this case U is */ /* > reserved as workspace of length N*N. */ /* > If JOBU = 'N' U is not referenced, unless JOBT='T'. */ /* > \endverbatim */ /* > */ /* > \param[in] LDU */ /* > \verbatim */ /* > LDU is INTEGER */ /* > The leading dimension of the array U, LDU >= 1. */ /* > IF JOBU = 'U' or 'F' or 'W', then LDU >= M. */ /* > \endverbatim */ /* > */ /* > \param[out] V */ /* > \verbatim */ /* > V is REAL array, dimension ( LDV, N ) */ /* > If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of */ /* > the right singular vectors; */ /* > If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N), */ /* > then V is used as workspace if the pprocedure */ /* > replaces A with A^t. In that case, [U] is computed */ /* > in V as right singular vectors of A^t and then */ /* > copied back to the U array. This 'W' option is just */ /* > a reminder to the caller that in this case V is */ /* > reserved as workspace of length N*N. */ /* > If JOBV = 'N' V is not referenced, unless JOBT='T'. */ /* > \endverbatim */ /* > */ /* > \param[in] LDV */ /* > \verbatim */ /* > LDV is INTEGER */ /* > The leading dimension of the array V, LDV >= 1. */ /* > If JOBV = 'V' or 'J' or 'W', then LDV >= N. */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is REAL array, dimension (LWORK) */ /* > On exit, */ /* > WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such */ /* > that SCALE*SVA(1:N) are the computed singular values */ /* > of A. (See the description of SVA().) */ /* > WORK(2) = See the description of WORK(1). */ /* > WORK(3) = SCONDA is an estimate for the condition number of */ /* > column equilibrated A. (If JOBA = 'E' or 'G') */ /* > SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1). */ /* > It is computed using SPOCON. It holds */ /* > N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA */ /* > where R is the triangular factor from the QRF of A. */ /* > However, if R is truncated and the numerical rank is */ /* > determined to be strictly smaller than N, SCONDA is */ /* > returned as -1, thus indicating that the smallest */ /* > singular values might be lost. */ /* > */ /* > If full SVD is needed, the following two condition numbers are */ /* > useful for the analysis of the algorithm. They are provied for */ /* > a developer/implementer who is familiar with the details of */ /* > the method. */ /* > */ /* > WORK(4) = an estimate of the scaled condition number of the */ /* > triangular factor in the first QR factorization. */ /* > WORK(5) = an estimate of the scaled condition number of the */ /* > triangular factor in the second QR factorization. */ /* > The following two parameters are computed if JOBT = 'T'. */ /* > They are provided for a developer/implementer who is familiar */ /* > with the details of the method. */ /* > */ /* > WORK(6) = the entropy of A^t*A :: this is the Shannon entropy */ /* > of diag(A^t*A) / Trace(A^t*A) taken as point in the */ /* > probability simplex. */ /* > WORK(7) = the entropy of A*A^t. */ /* > \endverbatim */ /* > */ /* > \param[in] LWORK */ /* > \verbatim */ /* > LWORK is INTEGER */ /* > Length of WORK to confirm proper allocation of work space. */ /* > LWORK depends on the job: */ /* > */ /* > If only SIGMA is needed ( JOBU = 'N', JOBV = 'N' ) and */ /* > -> .. no scaled condition estimate required (JOBE = 'N'): */ /* > LWORK >= f2cmax(2*M+N,4*N+1,7). This is the minimal requirement. */ /* > ->> For optimal performance (blocked code) the optimal value */ /* > is LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal */ /* > block size for DGEQP3 and DGEQRF. */ /* > In general, optimal LWORK is computed as */ /* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7). */ /* > -> .. an estimate of the scaled condition number of A is */ /* > required (JOBA='E', 'G'). In this case, LWORK is the maximum */ /* > of the above and N*N+4*N, i.e. LWORK >= f2cmax(2*M+N,N*N+4*N,7). */ /* > ->> For optimal performance (blocked code) the optimal value */ /* > is LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7). */ /* > In general, the optimal length LWORK is computed as */ /* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), */ /* > N+N*N+LWORK(DPOCON),7). */ /* > */ /* > If SIGMA and the right singular vectors are needed (JOBV = 'V'), */ /* > -> the minimal requirement is LWORK >= f2cmax(2*M+N,4*N+1,7). */ /* > -> For optimal performance, LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,7), */ /* > where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ, */ /* > DORMLQ. In general, the optimal length LWORK is computed as */ /* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), */ /* > N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)). */ /* > */ /* > If SIGMA and the left singular vectors are needed */ /* > -> the minimal requirement is LWORK >= f2cmax(2*M+N,4*N+1,7). */ /* > -> For optimal performance: */ /* > if JOBU = 'U' :: LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,7), */ /* > if JOBU = 'F' :: LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,N+M*NB,7), */ /* > where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR. */ /* > In general, the optimal length LWORK is computed as */ /* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON), */ /* > 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)). */ /* > Here LWORK(DORMQR) equals N*NB (for JOBU = 'U') or */ /* > M*NB (for JOBU = 'F'). */ /* > */ /* > If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and */ /* > -> if JOBV = 'V' */ /* > the minimal requirement is LWORK >= f2cmax(2*M+N,6*N+2*N*N). */ /* > -> if JOBV = 'J' the minimal requirement is */ /* > LWORK >= f2cmax(2*M+N, 4*N+N*N,2*N+N*N+6). */ /* > -> For optimal performance, LWORK should be additionally */ /* > larger than N+M*NB, where NB is the optimal block size */ /* > for DORMQR. */ /* > \endverbatim */ /* > */ /* > \param[out] IWORK */ /* > \verbatim */ /* > IWORK is INTEGER array, dimension (M+3*N). */ /* > On exit, */ /* > IWORK(1) = the numerical rank determined after the initial */ /* > QR factorization with pivoting. See the descriptions */ /* > of JOBA and JOBR. */ /* > IWORK(2) = the number of the computed nonzero singular values */ /* > IWORK(3) = if nonzero, a warning message: */ /* > If IWORK(3) = 1 then some of the column norms of A */ /* > were denormalized floats. The requested high accuracy */ /* > is not warranted by the data. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > < 0: if INFO = -i, then the i-th argument had an illegal value. */ /* > = 0: successful exit; */ /* > > 0: SGEJSV did not converge in the maximal allowed number */ /* > of sweeps. The computed values may be inaccurate. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date June 2016 */ /* > \ingroup realGEsing */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > */ /* > SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3, */ /* > SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an */ /* > additional row pivoting can be used as a preprocessor, which in some */ /* > cases results in much higher accuracy. An example is matrix A with the */ /* > structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned */ /* > diagonal matrices and C is well-conditioned matrix. In that case, complete */ /* > pivoting in the first QR factorizations provides accuracy dependent on the */ /* > condition number of C, and independent of D1, D2. Such higher accuracy is */ /* > not completely understood theoretically, but it works well in practice. */ /* > Further, if A can be written as A = B*D, with well-conditioned B and some */ /* > diagonal D, then the high accuracy is guaranteed, both theoretically and */ /* > in software, independent of D. For more details see [1], [2]. */ /* > The computational range for the singular values can be the full range */ /* > ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS */ /* > & LAPACK routines called by SGEJSV are implemented to work in that range. */ /* > If that is not the case, then the restriction for safe computation with */ /* > the singular values in the range of normalized IEEE numbers is that the */ /* > spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not */ /* > overflow. This code (SGEJSV) is best used in this restricted range, */ /* > meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are */ /* > returned as zeros. See JOBR for details on this. */ /* > Further, this implementation is somewhat slower than the one described */ /* > in [1,2] due to replacement of some non-LAPACK components, and because */ /* > the choice of some tuning parameters in the iterative part (SGESVJ) is */ /* > left to the implementer on a particular machine. */ /* > The rank revealing QR factorization (in this code: SGEQP3) should be */ /* > implemented as in [3]. We have a new version of SGEQP3 under development */ /* > that is more robust than the current one in LAPACK, with a cleaner cut in */ /* > rank deficient cases. It will be available in the SIGMA library [4]. */ /* > If M is much larger than N, it is obvious that the initial QRF with */ /* > column pivoting can be preprocessed by the QRF without pivoting. That */ /* > well known trick is not used in SGEJSV because in some cases heavy row */ /* > weighting can be treated with complete pivoting. The overhead in cases */ /* > M much larger than N is then only due to pivoting, but the benefits in */ /* > terms of accuracy have prevailed. The implementer/user can incorporate */ /* > this extra QRF step easily. The implementer can also improve data movement */ /* > (matrix transpose, matrix copy, matrix transposed copy) - this */ /* > implementation of SGEJSV uses only the simplest, naive data movement. */ /* > \endverbatim */ /* > \par Contributors: */ /* ================== */ /* > */ /* > Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) */ /* > \par References: */ /* ================ */ /* > */ /* > \verbatim */ /* > */ /* > [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. */ /* > SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. */ /* > LAPACK Working note 169. */ /* > [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. */ /* > SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. */ /* > LAPACK Working note 170. */ /* > [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR */ /* > factorization software - a case study. */ /* > ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. */ /* > LAPACK Working note 176. */ /* > [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, */ /* > QSVD, (H,K)-SVD computations. */ /* > Department of Mathematics, University of Zagreb, 2008. */ /* > \endverbatim */ /* > \par Bugs, examples and comments: */ /* ================================= */ /* > */ /* > Please report all bugs and send interesting examples and/or comments to */ /* > drmac@math.hr. Thank you. */ /* > */ /* ===================================================================== */ /* Subroutine */ int sgejsv_(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, integer *m, integer *n, real *a, integer *lda, real *sva, real *u, integer *ldu, real *v, integer *ldv, real *work, integer *lwork, integer *iwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, u_dim1, u_offset, v_dim1, v_offset, i__1, i__2, i__3, i__4, i__5, i__6, i__7, i__8, i__9, i__10, i__11, i__12; real r__1, r__2, r__3, r__4; /* Local variables */ logical defr; real aapp, aaqq; logical kill; integer ierr; real temp1; extern real snrm2_(integer *, real *, integer *); integer p, q; logical jracc; extern logical lsame_(char *, char *); extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); real small, entra, sfmin; logical lsvec; real epsln; logical rsvec; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); integer n1; extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *, integer *); logical l2aber; extern /* Subroutine */ int strsm_(char *, char *, char *, char *, integer *, integer *, real *, real *, integer *, real *, integer * ); real condr1, condr2, uscal1, uscal2; logical l2kill, l2rank, l2tran; extern /* Subroutine */ int sgeqp3_(integer *, integer *, real *, integer *, integer *, real *, real *, integer *, integer *); logical l2pert; integer nr; real scalem, sconda; logical goscal; real aatmin; extern real slamch_(char *); real aatmax; extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen); logical noscal; extern /* Subroutine */ int sgelqf_(integer *, integer *, real *, integer *, real *, real *, integer *, integer *); extern integer isamax_(integer *, real *, integer *); extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, real *, integer *, integer *, real *, integer *, integer *), sgeqrf_(integer *, integer *, real *, integer *, real *, real *, integer *, integer *), slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *), slaset_(char *, integer *, integer *, real *, real *, real *, integer *); real entrat; logical almort; real maxprj; extern /* Subroutine */ int spocon_(char *, integer *, real *, integer *, real *, real *, real *, integer *, integer *); logical errest; extern /* Subroutine */ int sgesvj_(char *, char *, char *, integer *, integer *, real *, integer *, real *, integer *, real *, integer * , real *, integer *, integer *), slassq_( integer *, real *, integer *, real *, real *); logical transp; extern /* Subroutine */ int slaswp_(integer *, real *, integer *, integer *, integer *, integer *, integer *), sorgqr_(integer *, integer *, integer *, real *, integer *, real *, real *, integer *, integer *), sormlq_(char *, char *, integer *, integer *, integer *, real *, integer *, real *, real *, integer *, real *, integer *, integer *), sormqr_(char *, char *, integer *, integer *, integer *, real *, integer *, real *, real *, integer * , real *, integer *, integer *); logical rowpiv; real big, cond_ok__, xsc, big1; integer warning, numrank; /* -- LAPACK computational routine (version 3.7.1) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* June 2016 */ /* =========================================================================== */ /* Test the input arguments */ /* Parameter adjustments */ --sva; a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; u_dim1 = *ldu; u_offset = 1 + u_dim1 * 1; u -= u_offset; v_dim1 = *ldv; v_offset = 1 + v_dim1 * 1; v -= v_offset; --work; --iwork; /* Function Body */ lsvec = lsame_(jobu, "U") || lsame_(jobu, "F"); jracc = lsame_(jobv, "J"); rsvec = lsame_(jobv, "V") || jracc; rowpiv = lsame_(joba, "F") || lsame_(joba, "G"); l2rank = lsame_(joba, "R"); l2aber = lsame_(joba, "A"); errest = lsame_(joba, "E") || lsame_(joba, "G"); l2tran = lsame_(jobt, "T"); l2kill = lsame_(jobr, "R"); defr = lsame_(jobr, "N"); l2pert = lsame_(jobp, "P"); if (! (rowpiv || l2rank || l2aber || errest || lsame_(joba, "C"))) { *info = -1; } else if (! (lsvec || lsame_(jobu, "N") || lsame_( jobu, "W"))) { *info = -2; } else if (! (rsvec || lsame_(jobv, "N") || lsame_( jobv, "W")) || jracc && ! lsvec) { *info = -3; } else if (! (l2kill || defr)) { *info = -4; } else if (! (l2tran || lsame_(jobt, "N"))) { *info = -5; } else if (! (l2pert || lsame_(jobp, "N"))) { *info = -6; } else if (*m < 0) { *info = -7; } else if (*n < 0 || *n > *m) { *info = -8; } else if (*lda < *m) { *info = -10; } else if (lsvec && *ldu < *m) { *info = -13; } else if (rsvec && *ldv < *n) { *info = -15; } else /* if(complicated condition) */ { /* Computing MAX */ i__1 = 7, i__2 = (*n << 2) + 1, i__1 = f2cmax(i__1,i__2), i__2 = (*m << 1) + *n; /* Computing MAX */ i__3 = 7, i__4 = (*n << 2) + *n * *n, i__3 = f2cmax(i__3,i__4), i__4 = (* m << 1) + *n; /* Computing MAX */ i__5 = 7, i__6 = (*m << 1) + *n, i__5 = f2cmax(i__5,i__6), i__6 = (*n << 2) + 1; /* Computing MAX */ i__7 = 7, i__8 = (*m << 1) + *n, i__7 = f2cmax(i__7,i__8), i__8 = (*n << 2) + 1; /* Computing MAX */ i__9 = (*m << 1) + *n, i__10 = *n * 6 + (*n << 1) * *n; /* Computing MAX */ i__11 = (*m << 1) + *n, i__12 = (*n << 2) + *n * *n, i__11 = f2cmax( i__11,i__12), i__12 = (*n << 1) + *n * *n + 6; if (! (lsvec || rsvec || errest) && *lwork < f2cmax(i__1,i__2) || ! ( lsvec || rsvec) && errest && *lwork < f2cmax(i__3,i__4) || lsvec && ! rsvec && *lwork < f2cmax(i__5,i__6) || rsvec && ! lsvec && * lwork < f2cmax(i__7,i__8) || lsvec && rsvec && ! jracc && *lwork < f2cmax(i__9,i__10) || lsvec && rsvec && jracc && *lwork < f2cmax( i__11,i__12)) { *info = -17; } else { /* #:) */ *info = 0; } } if (*info != 0) { /* #:( */ i__1 = -(*info); xerbla_("SGEJSV", &i__1, (ftnlen)6); return 0; } /* Quick return for void matrix (Y3K safe) */ /* #:) */ if (*m == 0 || *n == 0) { iwork[1] = 0; iwork[2] = 0; iwork[3] = 0; work[1] = 0.f; work[2] = 0.f; work[3] = 0.f; work[4] = 0.f; work[5] = 0.f; work[6] = 0.f; work[7] = 0.f; return 0; } /* Determine whether the matrix U should be M x N or M x M */ if (lsvec) { n1 = *n; if (lsame_(jobu, "F")) { n1 = *m; } } /* Set numerical parameters */ /* ! NOTE: Make sure SLAMCH() does not fail on the target architecture. */ epsln = slamch_("Epsilon"); sfmin = slamch_("SafeMinimum"); small = sfmin / epsln; big = slamch_("O"); /* BIG = ONE / SFMIN */ /* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N */ /* (!) If necessary, scale SVA() to protect the largest norm from */ /* overflow. It is possible that this scaling pushes the smallest */ /* column norm left from the underflow threshold (extreme case). */ scalem = 1.f / sqrt((real) (*m) * (real) (*n)); noscal = TRUE_; goscal = TRUE_; i__1 = *n; for (p = 1; p <= i__1; ++p) { aapp = 0.f; aaqq = 1.f; slassq_(m, &a[p * a_dim1 + 1], &c__1, &aapp, &aaqq); if (aapp > big) { *info = -9; i__2 = -(*info); xerbla_("SGEJSV", &i__2, (ftnlen)6); return 0; } aaqq = sqrt(aaqq); if (aapp < big / aaqq && noscal) { sva[p] = aapp * aaqq; } else { noscal = FALSE_; sva[p] = aapp * (aaqq * scalem); if (goscal) { goscal = FALSE_; i__2 = p - 1; sscal_(&i__2, &scalem, &sva[1], &c__1); } } /* L1874: */ } if (noscal) { scalem = 1.f; } aapp = 0.f; aaqq = big; i__1 = *n; for (p = 1; p <= i__1; ++p) { /* Computing MAX */ r__1 = aapp, r__2 = sva[p]; aapp = f2cmax(r__1,r__2); if (sva[p] != 0.f) { /* Computing MIN */ r__1 = aaqq, r__2 = sva[p]; aaqq = f2cmin(r__1,r__2); } /* L4781: */ } /* Quick return for zero M x N matrix */ /* #:) */ if (aapp == 0.f) { if (lsvec) { slaset_("G", m, &n1, &c_b34, &c_b35, &u[u_offset], ldu) ; } if (rsvec) { slaset_("G", n, n, &c_b34, &c_b35, &v[v_offset], ldv); } work[1] = 1.f; work[2] = 1.f; if (errest) { work[3] = 1.f; } if (lsvec && rsvec) { work[4] = 1.f; work[5] = 1.f; } if (l2tran) { work[6] = 0.f; work[7] = 0.f; } iwork[1] = 0; iwork[2] = 0; iwork[3] = 0; return 0; } /* Issue warning if denormalized column norms detected. Override the */ /* high relative accuracy request. Issue licence to kill columns */ /* (set them to zero) whose norm is less than sigma_max / BIG (roughly). */ /* #:( */ warning = 0; if (aaqq <= sfmin) { l2rank = TRUE_; l2kill = TRUE_; warning = 1; } /* Quick return for one-column matrix */ /* #:) */ if (*n == 1) { if (lsvec) { slascl_("G", &c__0, &c__0, &sva[1], &scalem, m, &c__1, &a[a_dim1 + 1], lda, &ierr); slacpy_("A", m, &c__1, &a[a_offset], lda, &u[u_offset], ldu); /* computing all M left singular vectors of the M x 1 matrix */ if (n1 != *n) { i__1 = *lwork - *n; sgeqrf_(m, n, &u[u_offset], ldu, &work[1], &work[*n + 1], & i__1, &ierr); i__1 = *lwork - *n; sorgqr_(m, &n1, &c__1, &u[u_offset], ldu, &work[1], &work[*n + 1], &i__1, &ierr); scopy_(m, &a[a_dim1 + 1], &c__1, &u[u_dim1 + 1], &c__1); } } if (rsvec) { v[v_dim1 + 1] = 1.f; } if (sva[1] < big * scalem) { sva[1] /= scalem; scalem = 1.f; } work[1] = 1.f / scalem; work[2] = 1.f; if (sva[1] != 0.f) { iwork[1] = 1; if (sva[1] / scalem >= sfmin) { iwork[2] = 1; } else { iwork[2] = 0; } } else { iwork[1] = 0; iwork[2] = 0; } iwork[3] = 0; if (errest) { work[3] = 1.f; } if (lsvec && rsvec) { work[4] = 1.f; work[5] = 1.f; } if (l2tran) { work[6] = 0.f; work[7] = 0.f; } return 0; } transp = FALSE_; l2tran = l2tran && *m == *n; aatmax = -1.f; aatmin = big; if (rowpiv || l2tran) { /* Compute the row norms, needed to determine row pivoting sequence */ /* (in the case of heavily row weighted A, row pivoting is strongly */ /* advised) and to collect information needed to compare the */ /* structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.). */ if (l2tran) { i__1 = *m; for (p = 1; p <= i__1; ++p) { xsc = 0.f; temp1 = 1.f; slassq_(n, &a[p + a_dim1], lda, &xsc, &temp1); /* SLASSQ gets both the ell_2 and the ell_infinity norm */ /* in one pass through the vector */ work[*m + *n + p] = xsc * scalem; work[*n + p] = xsc * (scalem * sqrt(temp1)); /* Computing MAX */ r__1 = aatmax, r__2 = work[*n + p]; aatmax = f2cmax(r__1,r__2); if (work[*n + p] != 0.f) { /* Computing MIN */ r__1 = aatmin, r__2 = work[*n + p]; aatmin = f2cmin(r__1,r__2); } /* L1950: */ } } else { i__1 = *m; for (p = 1; p <= i__1; ++p) { work[*m + *n + p] = scalem * (r__1 = a[p + isamax_(n, &a[p + a_dim1], lda) * a_dim1], abs(r__1)); /* Computing MAX */ r__1 = aatmax, r__2 = work[*m + *n + p]; aatmax = f2cmax(r__1,r__2); /* Computing MIN */ r__1 = aatmin, r__2 = work[*m + *n + p]; aatmin = f2cmin(r__1,r__2); /* L1904: */ } } } /* For square matrix A try to determine whether A^t would be better */ /* input for the preconditioned Jacobi SVD, with faster convergence. */ /* The decision is based on an O(N) function of the vector of column */ /* and row norms of A, based on the Shannon entropy. This should give */ /* the right choice in most cases when the difference actually matters. */ /* It may fail and pick the slower converging side. */ entra = 0.f; entrat = 0.f; if (l2tran) { xsc = 0.f; temp1 = 1.f; slassq_(n, &sva[1], &c__1, &xsc, &temp1); temp1 = 1.f / temp1; entra = 0.f; i__1 = *n; for (p = 1; p <= i__1; ++p) { /* Computing 2nd power */ r__1 = sva[p] / xsc; big1 = r__1 * r__1 * temp1; if (big1 != 0.f) { entra += big1 * log(big1); } /* L1113: */ } entra = -entra / log((real) (*n)); /* Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex. */ /* It is derived from the diagonal of A^t * A. Do the same with the */ /* diagonal of A * A^t, compute the entropy of the corresponding */ /* probability distribution. Note that A * A^t and A^t * A have the */ /* same trace. */ entrat = 0.f; i__1 = *n + *m; for (p = *n + 1; p <= i__1; ++p) { /* Computing 2nd power */ r__1 = work[p] / xsc; big1 = r__1 * r__1 * temp1; if (big1 != 0.f) { entrat += big1 * log(big1); } /* L1114: */ } entrat = -entrat / log((real) (*m)); /* Analyze the entropies and decide A or A^t. Smaller entropy */ /* usually means better input for the algorithm. */ transp = entrat < entra; /* If A^t is better than A, transpose A. */ if (transp) { /* In an optimal implementation, this trivial transpose */ /* should be replaced with faster transpose. */ i__1 = *n - 1; for (p = 1; p <= i__1; ++p) { i__2 = *n; for (q = p + 1; q <= i__2; ++q) { temp1 = a[q + p * a_dim1]; a[q + p * a_dim1] = a[p + q * a_dim1]; a[p + q * a_dim1] = temp1; /* L1116: */ } /* L1115: */ } i__1 = *n; for (p = 1; p <= i__1; ++p) { work[*m + *n + p] = sva[p]; sva[p] = work[*n + p]; /* L1117: */ } temp1 = aapp; aapp = aatmax; aatmax = temp1; temp1 = aaqq; aaqq = aatmin; aatmin = temp1; kill = lsvec; lsvec = rsvec; rsvec = kill; if (lsvec) { n1 = *n; } rowpiv = TRUE_; } } /* END IF L2TRAN */ /* Scale the matrix so that its maximal singular value remains less */ /* than SQRT(BIG) -- the matrix is scaled so that its maximal column */ /* has Euclidean norm equal to SQRT(BIG/N). The only reason to keep */ /* SQRT(BIG) instead of BIG is the fact that SGEJSV uses LAPACK and */ /* BLAS routines that, in some implementations, are not capable of */ /* working in the full interval [SFMIN,BIG] and that they may provoke */ /* overflows in the intermediate results. If the singular values spread */ /* from SFMIN to BIG, then SGESVJ will compute them. So, in that case, */ /* one should use SGESVJ instead of SGEJSV. */ big1 = sqrt(big); temp1 = sqrt(big / (real) (*n)); slascl_("G", &c__0, &c__0, &aapp, &temp1, n, &c__1, &sva[1], n, &ierr); if (aaqq > aapp * sfmin) { aaqq = aaqq / aapp * temp1; } else { aaqq = aaqq * temp1 / aapp; } temp1 *= scalem; slascl_("G", &c__0, &c__0, &aapp, &temp1, m, n, &a[a_offset], lda, &ierr); /* To undo scaling at the end of this procedure, multiply the */ /* computed singular values with USCAL2 / USCAL1. */ uscal1 = temp1; uscal2 = aapp; if (l2kill) { /* L2KILL enforces computation of nonzero singular values in */ /* the restricted range of condition number of the initial A, */ /* sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN). */ xsc = sqrt(sfmin); } else { xsc = small; /* Now, if the condition number of A is too big, */ /* sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN, */ /* as a precaution measure, the full SVD is computed using SGESVJ */ /* with accumulated Jacobi rotations. This provides numerically */ /* more robust computation, at the cost of slightly increased run */ /* time. Depending on the concrete implementation of BLAS and LAPACK */ /* (i.e. how they behave in presence of extreme ill-conditioning) the */ /* implementor may decide to remove this switch. */ if (aaqq < sqrt(sfmin) && lsvec && rsvec) { jracc = TRUE_; } } if (aaqq < xsc) { i__1 = *n; for (p = 1; p <= i__1; ++p) { if (sva[p] < xsc) { slaset_("A", m, &c__1, &c_b34, &c_b34, &a[p * a_dim1 + 1], lda); sva[p] = 0.f; } /* L700: */ } } /* Preconditioning using QR factorization with pivoting */ if (rowpiv) { /* Optional row permutation (Bjoerck row pivoting): */ /* A result by Cox and Higham shows that the Bjoerck's */ /* row pivoting combined with standard column pivoting */ /* has similar effect as Powell-Reid complete pivoting. */ /* The ell-infinity norms of A are made nonincreasing. */ i__1 = *m - 1; for (p = 1; p <= i__1; ++p) { i__2 = *m - p + 1; q = isamax_(&i__2, &work[*m + *n + p], &c__1) + p - 1; iwork[(*n << 1) + p] = q; if (p != q) { temp1 = work[*m + *n + p]; work[*m + *n + p] = work[*m + *n + q]; work[*m + *n + q] = temp1; } /* L1952: */ } i__1 = *m - 1; slaswp_(n, &a[a_offset], lda, &c__1, &i__1, &iwork[(*n << 1) + 1], & c__1); } /* End of the preparation phase (scaling, optional sorting and */ /* transposing, optional flushing of small columns). */ /* Preconditioning */ /* If the full SVD is needed, the right singular vectors are computed */ /* from a matrix equation, and for that we need theoretical analysis */ /* of the Businger-Golub pivoting. So we use SGEQP3 as the first RR QRF. */ /* In all other cases the first RR QRF can be chosen by other criteria */ /* (eg speed by replacing global with restricted window pivoting, such */ /* as in SGEQPX from TOMS # 782). Good results will be obtained using */ /* SGEQPX with properly (!) chosen numerical parameters. */ /* Any improvement of SGEQP3 improves overal performance of SGEJSV. */ /* A * P1 = Q1 * [ R1^t 0]^t: */ i__1 = *n; for (p = 1; p <= i__1; ++p) { iwork[p] = 0; /* L1963: */ } i__1 = *lwork - *n; sgeqp3_(m, n, &a[a_offset], lda, &iwork[1], &work[1], &work[*n + 1], & i__1, &ierr); /* The upper triangular matrix R1 from the first QRF is inspected for */ /* rank deficiency and possibilities for deflation, or possible */ /* ill-conditioning. Depending on the user specified flag L2RANK, */ /* the procedure explores possibilities to reduce the numerical */ /* rank by inspecting the computed upper triangular factor. If */ /* L2RANK or L2ABER are up, then SGEJSV will compute the SVD of */ /* A + dA, where ||dA|| <= f(M,N)*EPSLN. */ nr = 1; if (l2aber) { /* Standard absolute error bound suffices. All sigma_i with */ /* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an */ /* aggressive enforcement of lower numerical rank by introducing a */ /* backward error of the order of N*EPSLN*||A||. */ temp1 = sqrt((real) (*n)) * epsln; i__1 = *n; for (p = 2; p <= i__1; ++p) { if ((r__2 = a[p + p * a_dim1], abs(r__2)) >= temp1 * (r__1 = a[ a_dim1 + 1], abs(r__1))) { ++nr; } else { goto L3002; } /* L3001: */ } L3002: ; } else if (l2rank) { /* Sudden drop on the diagonal of R1 is used as the criterion for */ /* close-to-rank-deficient. */ temp1 = sqrt(sfmin); i__1 = *n; for (p = 2; p <= i__1; ++p) { if ((r__2 = a[p + p * a_dim1], abs(r__2)) < epsln * (r__1 = a[p - 1 + (p - 1) * a_dim1], abs(r__1)) || (r__3 = a[p + p * a_dim1], abs(r__3)) < small || l2kill && (r__4 = a[p + p * a_dim1], abs(r__4)) < temp1) { goto L3402; } ++nr; /* L3401: */ } L3402: ; } else { /* The goal is high relative accuracy. However, if the matrix */ /* has high scaled condition number the relative accuracy is in */ /* general not feasible. Later on, a condition number estimator */ /* will be deployed to estimate the scaled condition number. */ /* Here we just remove the underflowed part of the triangular */ /* factor. This prevents the situation in which the code is */ /* working hard to get the accuracy not warranted by the data. */ temp1 = sqrt(sfmin); i__1 = *n; for (p = 2; p <= i__1; ++p) { if ((r__1 = a[p + p * a_dim1], abs(r__1)) < small || l2kill && ( r__2 = a[p + p * a_dim1], abs(r__2)) < temp1) { goto L3302; } ++nr; /* L3301: */ } L3302: ; } almort = FALSE_; if (nr == *n) { maxprj = 1.f; i__1 = *n; for (p = 2; p <= i__1; ++p) { temp1 = (r__1 = a[p + p * a_dim1], abs(r__1)) / sva[iwork[p]]; maxprj = f2cmin(maxprj,temp1); /* L3051: */ } /* Computing 2nd power */ r__1 = maxprj; if (r__1 * r__1 >= 1.f - (real) (*n) * epsln) { almort = TRUE_; } } sconda = -1.f; condr1 = -1.f; condr2 = -1.f; if (errest) { if (*n == nr) { if (rsvec) { slacpy_("U", n, n, &a[a_offset], lda, &v[v_offset], ldv); i__1 = *n; for (p = 1; p <= i__1; ++p) { temp1 = sva[iwork[p]]; r__1 = 1.f / temp1; sscal_(&p, &r__1, &v[p * v_dim1 + 1], &c__1); /* L3053: */ } spocon_("U", n, &v[v_offset], ldv, &c_b35, &temp1, &work[*n + 1], &iwork[(*n << 1) + *m + 1], &ierr); } else if (lsvec) { slacpy_("U", n, n, &a[a_offset], lda, &u[u_offset], ldu); i__1 = *n; for (p = 1; p <= i__1; ++p) { temp1 = sva[iwork[p]]; r__1 = 1.f / temp1; sscal_(&p, &r__1, &u[p * u_dim1 + 1], &c__1); /* L3054: */ } spocon_("U", n, &u[u_offset], ldu, &c_b35, &temp1, &work[*n + 1], &iwork[(*n << 1) + *m + 1], &ierr); } else { slacpy_("U", n, n, &a[a_offset], lda, &work[*n + 1], n); i__1 = *n; for (p = 1; p <= i__1; ++p) { temp1 = sva[iwork[p]]; r__1 = 1.f / temp1; sscal_(&p, &r__1, &work[*n + (p - 1) * *n + 1], &c__1); /* L3052: */ } spocon_("U", n, &work[*n + 1], n, &c_b35, &temp1, &work[*n + * n * *n + 1], &iwork[(*n << 1) + *m + 1], &ierr); } sconda = 1.f / sqrt(temp1); /* SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1). */ /* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA */ } else { sconda = -1.f; } } l2pert = l2pert && (r__1 = a[a_dim1 + 1] / a[nr + nr * a_dim1], abs(r__1)) > sqrt(big1); /* If there is no violent scaling, artificial perturbation is not needed. */ /* Phase 3: */ if (! (rsvec || lsvec)) { /* Singular Values only */ /* Computing MIN */ i__2 = *n - 1; i__1 = f2cmin(i__2,nr); for (p = 1; p <= i__1; ++p) { i__2 = *n - p; scopy_(&i__2, &a[p + (p + 1) * a_dim1], lda, &a[p + 1 + p * a_dim1], &c__1); /* L1946: */ } /* The following two DO-loops introduce small relative perturbation */ /* into the strict upper triangle of the lower triangular matrix. */ /* Small entries below the main diagonal are also changed. */ /* This modification is useful if the computing environment does not */ /* provide/allow FLUSH TO ZERO underflow, for it prevents many */ /* annoying denormalized numbers in case of strongly scaled matrices. */ /* The perturbation is structured so that it does not introduce any */ /* new perturbation of the singular values, and it does not destroy */ /* the job done by the preconditioner. */ /* The licence for this perturbation is in the variable L2PERT, which */ /* should be .FALSE. if FLUSH TO ZERO underflow is active. */ if (! almort) { if (l2pert) { /* XSC = SQRT(SMALL) */ xsc = epsln / (real) (*n); i__1 = nr; for (q = 1; q <= i__1; ++q) { temp1 = xsc * (r__1 = a[q + q * a_dim1], abs(r__1)); i__2 = *n; for (p = 1; p <= i__2; ++p) { if (p > q && (r__1 = a[p + q * a_dim1], abs(r__1)) <= temp1 || p < q) { a[p + q * a_dim1] = r_sign(&temp1, &a[p + q * a_dim1]); } /* L4949: */ } /* L4947: */ } } else { i__1 = nr - 1; i__2 = nr - 1; slaset_("U", &i__1, &i__2, &c_b34, &c_b34, &a[(a_dim1 << 1) + 1], lda); } i__1 = *lwork - *n; sgeqrf_(n, &nr, &a[a_offset], lda, &work[1], &work[*n + 1], &i__1, &ierr); i__1 = nr - 1; for (p = 1; p <= i__1; ++p) { i__2 = nr - p; scopy_(&i__2, &a[p + (p + 1) * a_dim1], lda, &a[p + 1 + p * a_dim1], &c__1); /* L1948: */ } } /* Row-cyclic Jacobi SVD algorithm with column pivoting */ /* to drown denormals */ if (l2pert) { /* XSC = SQRT(SMALL) */ xsc = epsln / (real) (*n); i__1 = nr; for (q = 1; q <= i__1; ++q) { temp1 = xsc * (r__1 = a[q + q * a_dim1], abs(r__1)); i__2 = nr; for (p = 1; p <= i__2; ++p) { if (p > q && (r__1 = a[p + q * a_dim1], abs(r__1)) <= temp1 || p < q) { a[p + q * a_dim1] = r_sign(&temp1, &a[p + q * a_dim1]) ; } /* L1949: */ } /* L1947: */ } } else { i__1 = nr - 1; i__2 = nr - 1; slaset_("U", &i__1, &i__2, &c_b34, &c_b34, &a[(a_dim1 << 1) + 1], lda); } /* triangular matrix (plus perturbation which is ignored in */ /* the part which destroys triangular form (confusing?!)) */ sgesvj_("L", "NoU", "NoV", &nr, &nr, &a[a_offset], lda, &sva[1], n, & v[v_offset], ldv, &work[1], lwork, info); scalem = work[1]; numrank = i_nint(&work[2]); } else if (rsvec && ! lsvec) { /* -> Singular Values and Right Singular Vectors <- */ if (almort) { i__1 = nr; for (p = 1; p <= i__1; ++p) { i__2 = *n - p + 1; scopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], & c__1); /* L1998: */ } i__1 = nr - 1; i__2 = nr - 1; slaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 1], ldv); sgesvj_("L", "U", "N", n, &nr, &v[v_offset], ldv, &sva[1], &nr, & a[a_offset], lda, &work[1], lwork, info); scalem = work[1]; numrank = i_nint(&work[2]); } else { /* accumulated product of Jacobi rotations, three are perfect ) */ i__1 = nr - 1; i__2 = nr - 1; slaset_("Lower", &i__1, &i__2, &c_b34, &c_b34, &a[a_dim1 + 2], lda); i__1 = *lwork - *n; sgelqf_(&nr, n, &a[a_offset], lda, &work[1], &work[*n + 1], &i__1, &ierr); slacpy_("Lower", &nr, &nr, &a[a_offset], lda, &v[v_offset], ldv); i__1 = nr - 1; i__2 = nr - 1; slaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 1], ldv); i__1 = *lwork - (*n << 1); sgeqrf_(&nr, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*n << 1) + 1], &i__1, &ierr); i__1 = nr; for (p = 1; p <= i__1; ++p) { i__2 = nr - p + 1; scopy_(&i__2, &v[p + p * v_dim1], ldv, &v[p + p * v_dim1], & c__1); /* L8998: */ } i__1 = nr - 1; i__2 = nr - 1; slaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 1], ldv); i__1 = *lwork - *n; sgesvj_("Lower", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[1], & nr, &u[u_offset], ldu, &work[*n + 1], &i__1, info); scalem = work[*n + 1]; numrank = i_nint(&work[*n + 2]); if (nr < *n) { i__1 = *n - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1], ldv); i__1 = *n - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1 + 1], ldv); i__1 = *n - nr; i__2 = *n - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr + 1) * v_dim1], ldv); } i__1 = *lwork - *n; sormlq_("Left", "Transpose", n, n, &nr, &a[a_offset], lda, &work[ 1], &v[v_offset], ldv, &work[*n + 1], &i__1, &ierr); } i__1 = *n; for (p = 1; p <= i__1; ++p) { scopy_(n, &v[p + v_dim1], ldv, &a[iwork[p] + a_dim1], lda); /* L8991: */ } slacpy_("All", n, n, &a[a_offset], lda, &v[v_offset], ldv); if (transp) { slacpy_("All", n, n, &v[v_offset], ldv, &u[u_offset], ldu); } } else if (lsvec && ! rsvec) { /* Jacobi rotations in the Jacobi iterations. */ i__1 = nr; for (p = 1; p <= i__1; ++p) { i__2 = *n - p + 1; scopy_(&i__2, &a[p + p * a_dim1], lda, &u[p + p * u_dim1], &c__1); /* L1965: */ } i__1 = nr - 1; i__2 = nr - 1; slaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 1], ldu); i__1 = *lwork - (*n << 1); sgeqrf_(n, &nr, &u[u_offset], ldu, &work[*n + 1], &work[(*n << 1) + 1] , &i__1, &ierr); i__1 = nr - 1; for (p = 1; p <= i__1; ++p) { i__2 = nr - p; scopy_(&i__2, &u[p + (p + 1) * u_dim1], ldu, &u[p + 1 + p * u_dim1], &c__1); /* L1967: */ } i__1 = nr - 1; i__2 = nr - 1; slaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 1], ldu); i__1 = *lwork - *n; sgesvj_("Lower", "U", "N", &nr, &nr, &u[u_offset], ldu, &sva[1], &nr, &a[a_offset], lda, &work[*n + 1], &i__1, info); scalem = work[*n + 1]; numrank = i_nint(&work[*n + 2]); if (nr < *m) { i__1 = *m - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 + u_dim1], ldu); if (nr < n1) { i__1 = n1 - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) * u_dim1 + 1], ldu); i__1 = *m - nr; i__2 = n1 - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 + (nr + 1) * u_dim1], ldu); } } i__1 = *lwork - *n; sormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[1], &u[ u_offset], ldu, &work[*n + 1], &i__1, &ierr); if (rowpiv) { i__1 = *m - 1; slaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1) + 1], &c_n1); } i__1 = n1; for (p = 1; p <= i__1; ++p) { xsc = 1.f / snrm2_(m, &u[p * u_dim1 + 1], &c__1); sscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1); /* L1974: */ } if (transp) { slacpy_("All", n, n, &u[u_offset], ldu, &v[v_offset], ldv); } } else { if (! jracc) { if (! almort) { /* Second Preconditioning Step (QRF [with pivoting]) */ /* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is */ /* equivalent to an LQF CALL. Since in many libraries the QRF */ /* seems to be better optimized than the LQF, we do explicit */ /* transpose and use the QRF. This is subject to changes in an */ /* optimized implementation of SGEJSV. */ i__1 = nr; for (p = 1; p <= i__1; ++p) { i__2 = *n - p + 1; scopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], &c__1); /* L1968: */ } /* denormals in the second QR factorization, where they are */ /* as good as zeros. This is done to avoid painfully slow */ /* computation with denormals. The relative size of the perturbation */ /* is a parameter that can be changed by the implementer. */ /* This perturbation device will be obsolete on machines with */ /* properly implemented arithmetic. */ /* To switch it off, set L2PERT=.FALSE. To remove it from the */ /* code, remove the action under L2PERT=.TRUE., leave the ELSE part. */ /* The following two loops should be blocked and fused with the */ /* transposed copy above. */ if (l2pert) { xsc = sqrt(small); i__1 = nr; for (q = 1; q <= i__1; ++q) { temp1 = xsc * (r__1 = v[q + q * v_dim1], abs(r__1)); i__2 = *n; for (p = 1; p <= i__2; ++p) { if (p > q && (r__1 = v[p + q * v_dim1], abs(r__1)) <= temp1 || p < q) { v[p + q * v_dim1] = r_sign(&temp1, &v[p + q * v_dim1]); } if (p < q) { v[p + q * v_dim1] = -v[p + q * v_dim1]; } /* L2968: */ } /* L2969: */ } } else { i__1 = nr - 1; i__2 = nr - 1; slaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 1], ldv); } /* Estimate the row scaled condition number of R1 */ /* (If R1 is rectangular, N > NR, then the condition number */ /* of the leading NR x NR submatrix is estimated.) */ slacpy_("L", &nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1] , &nr); i__1 = nr; for (p = 1; p <= i__1; ++p) { i__2 = nr - p + 1; temp1 = snrm2_(&i__2, &work[(*n << 1) + (p - 1) * nr + p], &c__1); i__2 = nr - p + 1; r__1 = 1.f / temp1; sscal_(&i__2, &r__1, &work[(*n << 1) + (p - 1) * nr + p], &c__1); /* L3950: */ } spocon_("Lower", &nr, &work[(*n << 1) + 1], &nr, &c_b35, & temp1, &work[(*n << 1) + nr * nr + 1], &iwork[*m + (* n << 1) + 1], &ierr); condr1 = 1.f / sqrt(temp1); /* R1 is OK for inverse <=> CONDR1 .LT. FLOAT(N) */ /* more conservative <=> CONDR1 .LT. SQRT(FLOAT(N)) */ cond_ok__ = sqrt((real) nr); /* [TP] COND_OK is a tuning parameter. */ if (condr1 < cond_ok__) { /* implementation, this QRF should be implemented as the QRF */ /* of a lower triangular matrix. */ /* R1^t = Q2 * R2 */ i__1 = *lwork - (*n << 1); sgeqrf_(n, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(* n << 1) + 1], &i__1, &ierr); if (l2pert) { xsc = sqrt(small) / epsln; i__1 = nr; for (p = 2; p <= i__1; ++p) { i__2 = p - 1; for (q = 1; q <= i__2; ++q) { /* Computing MIN */ r__3 = (r__1 = v[p + p * v_dim1], abs(r__1)), r__4 = (r__2 = v[q + q * v_dim1], abs( r__2)); temp1 = xsc * f2cmin(r__3,r__4); if ((r__1 = v[q + p * v_dim1], abs(r__1)) <= temp1) { v[q + p * v_dim1] = r_sign(&temp1, &v[q + p * v_dim1]); } /* L3958: */ } /* L3959: */ } } if (nr != *n) { slacpy_("A", n, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1], n); } i__1 = nr - 1; for (p = 1; p <= i__1; ++p) { i__2 = nr - p; scopy_(&i__2, &v[p + (p + 1) * v_dim1], ldv, &v[p + 1 + p * v_dim1], &c__1); /* L1969: */ } condr2 = condr1; } else { /* Note that windowed pivoting would be equally good */ /* numerically, and more run-time efficient. So, in */ /* an optimal implementation, the next call to SGEQP3 */ /* should be replaced with eg. CALL SGEQPX (ACM TOMS #782) */ /* with properly (carefully) chosen parameters. */ /* R1^t * P2 = Q2 * R2 */ i__1 = nr; for (p = 1; p <= i__1; ++p) { iwork[*n + p] = 0; /* L3003: */ } i__1 = *lwork - (*n << 1); sgeqp3_(n, &nr, &v[v_offset], ldv, &iwork[*n + 1], &work[* n + 1], &work[(*n << 1) + 1], &i__1, &ierr); /* * CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1), */ /* * $ LWORK-2*N, IERR ) */ if (l2pert) { xsc = sqrt(small); i__1 = nr; for (p = 2; p <= i__1; ++p) { i__2 = p - 1; for (q = 1; q <= i__2; ++q) { /* Computing MIN */ r__3 = (r__1 = v[p + p * v_dim1], abs(r__1)), r__4 = (r__2 = v[q + q * v_dim1], abs( r__2)); temp1 = xsc * f2cmin(r__3,r__4); if ((r__1 = v[q + p * v_dim1], abs(r__1)) <= temp1) { v[q + p * v_dim1] = r_sign(&temp1, &v[q + p * v_dim1]); } /* L3968: */ } /* L3969: */ } } slacpy_("A", n, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1], n); if (l2pert) { xsc = sqrt(small); i__1 = nr; for (p = 2; p <= i__1; ++p) { i__2 = p - 1; for (q = 1; q <= i__2; ++q) { /* Computing MIN */ r__3 = (r__1 = v[p + p * v_dim1], abs(r__1)), r__4 = (r__2 = v[q + q * v_dim1], abs( r__2)); temp1 = xsc * f2cmin(r__3,r__4); v[p + q * v_dim1] = -r_sign(&temp1, &v[q + p * v_dim1]); /* L8971: */ } /* L8970: */ } } else { i__1 = nr - 1; i__2 = nr - 1; slaset_("L", &i__1, &i__2, &c_b34, &c_b34, &v[v_dim1 + 2], ldv); } /* Now, compute R2 = L3 * Q3, the LQ factorization. */ i__1 = *lwork - (*n << 1) - *n * nr - nr; sgelqf_(&nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + *n * nr + 1], &work[(*n << 1) + *n * nr + nr + 1], & i__1, &ierr); slacpy_("L", &nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + *n * nr + nr + 1], &nr); i__1 = nr; for (p = 1; p <= i__1; ++p) { temp1 = snrm2_(&p, &work[(*n << 1) + *n * nr + nr + p] , &nr); r__1 = 1.f / temp1; sscal_(&p, &r__1, &work[(*n << 1) + *n * nr + nr + p], &nr); /* L4950: */ } spocon_("L", &nr, &work[(*n << 1) + *n * nr + nr + 1], & nr, &c_b35, &temp1, &work[(*n << 1) + *n * nr + nr + nr * nr + 1], &iwork[*m + (*n << 1) + 1], & ierr); condr2 = 1.f / sqrt(temp1); if (condr2 >= cond_ok__) { /* (this overwrites the copy of R2, as it will not be */ /* needed in this branch, but it does not overwritte the */ /* Huseholder vectors of Q2.). */ slacpy_("U", &nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1], n); /* WORK(2*N+N*NR+1:2*N+N*NR+N) */ } } if (l2pert) { xsc = sqrt(small); i__1 = nr; for (q = 2; q <= i__1; ++q) { temp1 = xsc * v[q + q * v_dim1]; i__2 = q - 1; for (p = 1; p <= i__2; ++p) { /* V(p,q) = - SIGN( TEMP1, V(q,p) ) */ v[p + q * v_dim1] = -r_sign(&temp1, &v[p + q * v_dim1]); /* L4969: */ } /* L4968: */ } } else { i__1 = nr - 1; i__2 = nr - 1; slaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 1], ldv); } /* Second preconditioning finished; continue with Jacobi SVD */ /* The input matrix is lower trinagular. */ /* Recover the right singular vectors as solution of a well */ /* conditioned triangular matrix equation. */ if (condr1 < cond_ok__) { i__1 = *lwork - (*n << 1) - *n * nr - nr; sgesvj_("L", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[ 1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n * nr + nr + 1], &i__1, info); scalem = work[(*n << 1) + *n * nr + nr + 1]; numrank = i_nint(&work[(*n << 1) + *n * nr + nr + 2]); i__1 = nr; for (p = 1; p <= i__1; ++p) { scopy_(&nr, &v[p * v_dim1 + 1], &c__1, &u[p * u_dim1 + 1], &c__1); sscal_(&nr, &sva[p], &v[p * v_dim1 + 1], &c__1); /* L3970: */ } if (nr == *n) { /* :)) .. best case, R1 is inverted. The solution of this matrix */ /* equation is Q2*V2 = the product of the Jacobi rotations */ /* used in SGESVJ, premultiplied with the orthogonal matrix */ /* from the second QR factorization. */ strsm_("L", "U", "N", "N", &nr, &nr, &c_b35, &a[ a_offset], lda, &v[v_offset], ldv); } else { /* is inverted to get the product of the Jacobi rotations */ /* used in SGESVJ. The Q-factor from the second QR */ /* factorization is then built in explicitly. */ strsm_("L", "U", "T", "N", &nr, &nr, &c_b35, &work[(* n << 1) + 1], n, &v[v_offset], ldv); if (nr < *n) { i__1 = *n - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1], ldv); i__1 = *n - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1 + 1], ldv); i__1 = *n - nr; i__2 = *n - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr + 1) * v_dim1], ldv); } i__1 = *lwork - (*n << 1) - *n * nr - nr; sormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &work[*n + 1], &v[v_offset], ldv, &work[(*n << 1) + *n * nr + nr + 1], &i__1, &ierr); } } else if (condr2 < cond_ok__) { /* :) .. the input matrix A is very likely a relative of */ /* the Kahan matrix :) */ /* The matrix R2 is inverted. The solution of the matrix equation */ /* is Q3^T*V3 = the product of the Jacobi rotations (appplied to */ /* the lower triangular L3 from the LQ factorization of */ /* R2=L3*Q3), pre-multiplied with the transposed Q3. */ i__1 = *lwork - (*n << 1) - *n * nr - nr; sgesvj_("L", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[ 1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n * nr + nr + 1], &i__1, info); scalem = work[(*n << 1) + *n * nr + nr + 1]; numrank = i_nint(&work[(*n << 1) + *n * nr + nr + 2]); i__1 = nr; for (p = 1; p <= i__1; ++p) { scopy_(&nr, &v[p * v_dim1 + 1], &c__1, &u[p * u_dim1 + 1], &c__1); sscal_(&nr, &sva[p], &u[p * u_dim1 + 1], &c__1); /* L3870: */ } strsm_("L", "U", "N", "N", &nr, &nr, &c_b35, &work[(*n << 1) + 1], n, &u[u_offset], ldu); i__1 = nr; for (q = 1; q <= i__1; ++q) { i__2 = nr; for (p = 1; p <= i__2; ++p) { work[(*n << 1) + *n * nr + nr + iwork[*n + p]] = u[p + q * u_dim1]; /* L872: */ } i__2 = nr; for (p = 1; p <= i__2; ++p) { u[p + q * u_dim1] = work[(*n << 1) + *n * nr + nr + p]; /* L874: */ } /* L873: */ } if (nr < *n) { i__1 = *n - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1], ldv); i__1 = *n - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1 + 1], ldv); i__1 = *n - nr; i__2 = *n - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr + 1) * v_dim1], ldv); } i__1 = *lwork - (*n << 1) - *n * nr - nr; sormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, & work[*n + 1], &v[v_offset], ldv, &work[(*n << 1) + *n * nr + nr + 1], &i__1, &ierr); } else { /* Last line of defense. */ /* #:( This is a rather pathological case: no scaled condition */ /* improvement after two pivoted QR factorizations. Other */ /* possibility is that the rank revealing QR factorization */ /* or the condition estimator has failed, or the COND_OK */ /* is set very close to ONE (which is unnecessary). Normally, */ /* this branch should never be executed, but in rare cases of */ /* failure of the RRQR or condition estimator, the last line of */ /* defense ensures that SGEJSV completes the task. */ /* Compute the full SVD of L3 using SGESVJ with explicit */ /* accumulation of Jacobi rotations. */ i__1 = *lwork - (*n << 1) - *n * nr - nr; sgesvj_("L", "U", "V", &nr, &nr, &v[v_offset], ldv, &sva[ 1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n * nr + nr + 1], &i__1, info); scalem = work[(*n << 1) + *n * nr + nr + 1]; numrank = i_nint(&work[(*n << 1) + *n * nr + nr + 2]); if (nr < *n) { i__1 = *n - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1], ldv); i__1 = *n - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1 + 1], ldv); i__1 = *n - nr; i__2 = *n - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr + 1) * v_dim1], ldv); } i__1 = *lwork - (*n << 1) - *n * nr - nr; sormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, & work[*n + 1], &v[v_offset], ldv, &work[(*n << 1) + *n * nr + nr + 1], &i__1, &ierr); i__1 = *lwork - (*n << 1) - *n * nr - nr; sormlq_("L", "T", &nr, &nr, &nr, &work[(*n << 1) + 1], n, &work[(*n << 1) + *n * nr + 1], &u[u_offset], ldu, &work[(*n << 1) + *n * nr + nr + 1], &i__1, & ierr); i__1 = nr; for (q = 1; q <= i__1; ++q) { i__2 = nr; for (p = 1; p <= i__2; ++p) { work[(*n << 1) + *n * nr + nr + iwork[*n + p]] = u[p + q * u_dim1]; /* L772: */ } i__2 = nr; for (p = 1; p <= i__2; ++p) { u[p + q * u_dim1] = work[(*n << 1) + *n * nr + nr + p]; /* L774: */ } /* L773: */ } } /* Permute the rows of V using the (column) permutation from the */ /* first QRF. Also, scale the columns to make them unit in */ /* Euclidean norm. This applies to all cases. */ temp1 = sqrt((real) (*n)) * epsln; i__1 = *n; for (q = 1; q <= i__1; ++q) { i__2 = *n; for (p = 1; p <= i__2; ++p) { work[(*n << 1) + *n * nr + nr + iwork[p]] = v[p + q * v_dim1]; /* L972: */ } i__2 = *n; for (p = 1; p <= i__2; ++p) { v[p + q * v_dim1] = work[(*n << 1) + *n * nr + nr + p] ; /* L973: */ } xsc = 1.f / snrm2_(n, &v[q * v_dim1 + 1], &c__1); if (xsc < 1.f - temp1 || xsc > temp1 + 1.f) { sscal_(n, &xsc, &v[q * v_dim1 + 1], &c__1); } /* L1972: */ } /* At this moment, V contains the right singular vectors of A. */ /* Next, assemble the left singular vector matrix U (M x N). */ if (nr < *m) { i__1 = *m - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 + u_dim1], ldu); if (nr < n1) { i__1 = n1 - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) * u_dim1 + 1], ldu); i__1 = *m - nr; i__2 = n1 - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 + (nr + 1) * u_dim1], ldu); } } /* The Q matrix from the first QRF is built into the left singular */ /* matrix U. This applies to all cases. */ i__1 = *lwork - *n; sormqr_("Left", "No_Tr", m, &n1, n, &a[a_offset], lda, &work[ 1], &u[u_offset], ldu, &work[*n + 1], &i__1, &ierr); /* The columns of U are normalized. The cost is O(M*N) flops. */ temp1 = sqrt((real) (*m)) * epsln; i__1 = nr; for (p = 1; p <= i__1; ++p) { xsc = 1.f / snrm2_(m, &u[p * u_dim1 + 1], &c__1); if (xsc < 1.f - temp1 || xsc > temp1 + 1.f) { sscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1); } /* L1973: */ } /* If the initial QRF is computed with row pivoting, the left */ /* singular vectors must be adjusted. */ if (rowpiv) { i__1 = *m - 1; slaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1) + 1], &c_n1); } } else { /* the second QRF is not needed */ slacpy_("Upper", n, n, &a[a_offset], lda, &work[*n + 1], n); if (l2pert) { xsc = sqrt(small); i__1 = *n; for (p = 2; p <= i__1; ++p) { temp1 = xsc * work[*n + (p - 1) * *n + p]; i__2 = p - 1; for (q = 1; q <= i__2; ++q) { work[*n + (q - 1) * *n + p] = -r_sign(&temp1, & work[*n + (p - 1) * *n + q]); /* L5971: */ } /* L5970: */ } } else { i__1 = *n - 1; i__2 = *n - 1; slaset_("Lower", &i__1, &i__2, &c_b34, &c_b34, &work[*n + 2], n); } i__1 = *lwork - *n - *n * *n; sgesvj_("Upper", "U", "N", n, n, &work[*n + 1], n, &sva[1], n, &u[u_offset], ldu, &work[*n + *n * *n + 1], &i__1, info); scalem = work[*n + *n * *n + 1]; numrank = i_nint(&work[*n + *n * *n + 2]); i__1 = *n; for (p = 1; p <= i__1; ++p) { scopy_(n, &work[*n + (p - 1) * *n + 1], &c__1, &u[p * u_dim1 + 1], &c__1); sscal_(n, &sva[p], &work[*n + (p - 1) * *n + 1], &c__1); /* L6970: */ } strsm_("Left", "Upper", "NoTrans", "No UD", n, n, &c_b35, &a[ a_offset], lda, &work[*n + 1], n); i__1 = *n; for (p = 1; p <= i__1; ++p) { scopy_(n, &work[*n + p], n, &v[iwork[p] + v_dim1], ldv); /* L6972: */ } temp1 = sqrt((real) (*n)) * epsln; i__1 = *n; for (p = 1; p <= i__1; ++p) { xsc = 1.f / snrm2_(n, &v[p * v_dim1 + 1], &c__1); if (xsc < 1.f - temp1 || xsc > temp1 + 1.f) { sscal_(n, &xsc, &v[p * v_dim1 + 1], &c__1); } /* L6971: */ } /* Assemble the left singular vector matrix U (M x N). */ if (*n < *m) { i__1 = *m - *n; slaset_("A", &i__1, n, &c_b34, &c_b34, &u[*n + 1 + u_dim1] , ldu); if (*n < n1) { i__1 = n1 - *n; slaset_("A", n, &i__1, &c_b34, &c_b34, &u[(*n + 1) * u_dim1 + 1], ldu); i__1 = *m - *n; i__2 = n1 - *n; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[*n + 1 + (*n + 1) * u_dim1], ldu); } } i__1 = *lwork - *n; sormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[ 1], &u[u_offset], ldu, &work[*n + 1], &i__1, &ierr); temp1 = sqrt((real) (*m)) * epsln; i__1 = n1; for (p = 1; p <= i__1; ++p) { xsc = 1.f / snrm2_(m, &u[p * u_dim1 + 1], &c__1); if (xsc < 1.f - temp1 || xsc > temp1 + 1.f) { sscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1); } /* L6973: */ } if (rowpiv) { i__1 = *m - 1; slaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1) + 1], &c_n1); } } /* end of the >> almost orthogonal case << in the full SVD */ } else { /* This branch deploys a preconditioned Jacobi SVD with explicitly */ /* accumulated rotations. It is included as optional, mainly for */ /* experimental purposes. It does perform well, and can also be used. */ /* In this implementation, this branch will be automatically activated */ /* if the condition number sigma_max(A) / sigma_min(A) is predicted */ /* to be greater than the overflow threshold. This is because the */ /* a posteriori computation of the singular vectors assumes robust */ /* implementation of BLAS and some LAPACK procedures, capable of working */ /* in presence of extreme values. Since that is not always the case, ... */ i__1 = nr; for (p = 1; p <= i__1; ++p) { i__2 = *n - p + 1; scopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], & c__1); /* L7968: */ } if (l2pert) { xsc = sqrt(small / epsln); i__1 = nr; for (q = 1; q <= i__1; ++q) { temp1 = xsc * (r__1 = v[q + q * v_dim1], abs(r__1)); i__2 = *n; for (p = 1; p <= i__2; ++p) { if (p > q && (r__1 = v[p + q * v_dim1], abs(r__1)) <= temp1 || p < q) { v[p + q * v_dim1] = r_sign(&temp1, &v[p + q * v_dim1]); } if (p < q) { v[p + q * v_dim1] = -v[p + q * v_dim1]; } /* L5968: */ } /* L5969: */ } } else { i__1 = nr - 1; i__2 = nr - 1; slaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 1], ldv); } i__1 = *lwork - (*n << 1); sgeqrf_(n, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*n << 1) + 1], &i__1, &ierr); slacpy_("L", n, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1], n); i__1 = nr; for (p = 1; p <= i__1; ++p) { i__2 = nr - p + 1; scopy_(&i__2, &v[p + p * v_dim1], ldv, &u[p + p * u_dim1], & c__1); /* L7969: */ } if (l2pert) { xsc = sqrt(small / epsln); i__1 = nr; for (q = 2; q <= i__1; ++q) { i__2 = q - 1; for (p = 1; p <= i__2; ++p) { /* Computing MIN */ r__3 = (r__1 = u[p + p * u_dim1], abs(r__1)), r__4 = ( r__2 = u[q + q * u_dim1], abs(r__2)); temp1 = xsc * f2cmin(r__3,r__4); u[p + q * u_dim1] = -r_sign(&temp1, &u[q + p * u_dim1] ); /* L9971: */ } /* L9970: */ } } else { i__1 = nr - 1; i__2 = nr - 1; slaset_("U", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 1], ldu); } i__1 = *lwork - (*n << 1) - *n * nr; sgesvj_("L", "U", "V", &nr, &nr, &u[u_offset], ldu, &sva[1], n, & v[v_offset], ldv, &work[(*n << 1) + *n * nr + 1], &i__1, info); scalem = work[(*n << 1) + *n * nr + 1]; numrank = i_nint(&work[(*n << 1) + *n * nr + 2]); if (nr < *n) { i__1 = *n - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1], ldv); i__1 = *n - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1 + 1], ldv); i__1 = *n - nr; i__2 = *n - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr + 1) * v_dim1], ldv); } i__1 = *lwork - (*n << 1) - *n * nr - nr; sormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &work[*n + 1], &v[v_offset], ldv, &work[(*n << 1) + *n * nr + nr + 1] , &i__1, &ierr); /* Permute the rows of V using the (column) permutation from the */ /* first QRF. Also, scale the columns to make them unit in */ /* Euclidean norm. This applies to all cases. */ temp1 = sqrt((real) (*n)) * epsln; i__1 = *n; for (q = 1; q <= i__1; ++q) { i__2 = *n; for (p = 1; p <= i__2; ++p) { work[(*n << 1) + *n * nr + nr + iwork[p]] = v[p + q * v_dim1]; /* L8972: */ } i__2 = *n; for (p = 1; p <= i__2; ++p) { v[p + q * v_dim1] = work[(*n << 1) + *n * nr + nr + p]; /* L8973: */ } xsc = 1.f / snrm2_(n, &v[q * v_dim1 + 1], &c__1); if (xsc < 1.f - temp1 || xsc > temp1 + 1.f) { sscal_(n, &xsc, &v[q * v_dim1 + 1], &c__1); } /* L7972: */ } /* At this moment, V contains the right singular vectors of A. */ /* Next, assemble the left singular vector matrix U (M x N). */ if (nr < *m) { i__1 = *m - nr; slaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 + u_dim1], ldu); if (nr < n1) { i__1 = n1 - nr; slaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) * u_dim1 + 1], ldu); i__1 = *m - nr; i__2 = n1 - nr; slaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 + ( nr + 1) * u_dim1], ldu); } } i__1 = *lwork - *n; sormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[1], & u[u_offset], ldu, &work[*n + 1], &i__1, &ierr); if (rowpiv) { i__1 = *m - 1; slaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1) + 1], &c_n1); } } if (transp) { i__1 = *n; for (p = 1; p <= i__1; ++p) { sswap_(n, &u[p * u_dim1 + 1], &c__1, &v[p * v_dim1 + 1], & c__1); /* L6974: */ } } } /* end of the full SVD */ /* Undo scaling, if necessary (and possible) */ if (uscal2 <= big / sva[1] * uscal1) { slascl_("G", &c__0, &c__0, &uscal1, &uscal2, &nr, &c__1, &sva[1], n, & ierr); uscal1 = 1.f; uscal2 = 1.f; } if (nr < *n) { i__1 = *n; for (p = nr + 1; p <= i__1; ++p) { sva[p] = 0.f; /* L3004: */ } } work[1] = uscal2 * scalem; work[2] = uscal1; if (errest) { work[3] = sconda; } if (lsvec && rsvec) { work[4] = condr1; work[5] = condr2; } if (l2tran) { work[6] = entra; work[7] = entrat; } iwork[1] = nr; iwork[2] = numrank; iwork[3] = warning; return 0; } /* sgejsv_ */