#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 SGESVJ */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SGESVJ + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, */ /* LDV, WORK, LWORK, INFO ) */ /* INTEGER INFO, LDA, LDV, LWORK, M, MV, N */ /* CHARACTER*1 JOBA, JOBU, JOBV */ /* REAL A( LDA, * ), SVA( N ), V( LDV, * ), */ /* $ WORK( LWORK ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > SGESVJ computes the singular value decomposition (SVD) of a real */ /* > M-by-N matrix A, where M >= N. The SVD of A is written as */ /* > [++] [xx] [x0] [xx] */ /* > A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx] */ /* > [++] [xx] */ /* > where SIGMA is an N-by-N diagonal matrix, U is an M-by-N 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. */ /* > SGESVJ 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 structure of A. */ /* > = 'L': The input matrix A is lower triangular; */ /* > = 'U': The input matrix A is upper triangular; */ /* > = 'G': The input matrix A is general M-by-N matrix, M >= N. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBU */ /* > \verbatim */ /* > JOBU is CHARACTER*1 */ /* > Specifies whether to compute the left singular vectors */ /* > (columns of U): */ /* > = 'U': The left singular vectors corresponding to the nonzero */ /* > singular values are computed and returned in the leading */ /* > columns of A. See more details in the description of A. */ /* > The default numerical orthogonality threshold is set to */ /* > approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E'). */ /* > = 'C': Analogous to JOBU='U', except that user can control the */ /* > level of numerical orthogonality of the computed left */ /* > singular vectors. TOL can be set to TOL = CTOL*EPS, where */ /* > CTOL is given on input in the array WORK. */ /* > No CTOL smaller than ONE is allowed. CTOL greater */ /* > than 1 / EPS is meaningless. The option 'C' */ /* > can be used if M*EPS is satisfactory orthogonality */ /* > of the computed left singular vectors, so CTOL=M could */ /* > save few sweeps of Jacobi rotations. */ /* > See the descriptions of A and WORK(1). */ /* > = 'N': The matrix U is not computed. However, see the */ /* > description of A. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBV */ /* > \verbatim */ /* > JOBV is CHARACTER*1 */ /* > Specifies whether to compute the right singular vectors, that */ /* > is, the matrix V: */ /* > = 'V': the matrix V is computed and returned in the array V */ /* > = 'A': the Jacobi rotations are applied to the MV-by-N */ /* > array V. In other words, the right singular vector */ /* > matrix V is not computed explicitly; instead it is */ /* > applied to an MV-by-N matrix initially stored in the */ /* > first MV rows of V. */ /* > = 'N': the matrix V is not computed and the array V is not */ /* > referenced */ /* > \endverbatim */ /* > */ /* > \param[in] M */ /* > \verbatim */ /* > M is INTEGER */ /* > The number of rows of the input matrix A. 1/SLAMCH('E') > 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. */ /* > On exit, */ /* > If JOBU = 'U' .OR. JOBU = 'C': */ /* > If INFO = 0: */ /* > RANKA orthonormal columns of U are returned in the */ /* > leading RANKA columns of the array A. Here RANKA <= N */ /* > is the number of computed singular values of A that are */ /* > above the underflow threshold SLAMCH('S'). The singular */ /* > vectors corresponding to underflowed or zero singular */ /* > values are not computed. The value of RANKA is returned */ /* > in the array WORK as RANKA=NINT(WORK(2)). Also see the */ /* > descriptions of SVA and WORK. The computed columns of U */ /* > are mutually numerically orthogonal up to approximately */ /* > TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'), */ /* > see the description of JOBU. */ /* > If INFO > 0, */ /* > the procedure SGESVJ did not converge in the given number */ /* > of iterations (sweeps). In that case, the computed */ /* > columns of U may not be orthogonal up to TOL. The output */ /* > U (stored in A), SIGMA (given by the computed singular */ /* > values in SVA(1:N)) and V is still a decomposition of the */ /* > input matrix A in the sense that the residual */ /* > ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small. */ /* > If JOBU = 'N': */ /* > If INFO = 0: */ /* > Note that the left singular vectors are 'for free' in the */ /* > one-sided Jacobi SVD algorithm. However, if only the */ /* > singular values are needed, the level of numerical */ /* > orthogonality of U is not an issue and iterations are */ /* > stopped when the columns of the iterated matrix are */ /* > numerically orthogonal up to approximately M*EPS. Thus, */ /* > on exit, A contains the columns of U scaled with the */ /* > corresponding singular values. */ /* > If INFO > 0: */ /* > the procedure SGESVJ did not converge in the given number */ /* > of iterations (sweeps). */ /* > \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, */ /* > If INFO = 0 : */ /* > depending on the value SCALE = WORK(1), we have: */ /* > If SCALE = ONE: */ /* > SVA(1:N) contains the computed singular values of A. */ /* > During the computation SVA contains the Euclidean column */ /* > norms of the iterated matrices in the array A. */ /* > If SCALE .NE. ONE: */ /* > The singular values of A are SCALE*SVA(1:N), and this */ /* > factored representation is due to the fact that some of the */ /* > singular values of A might underflow or overflow. */ /* > */ /* > If INFO > 0 : */ /* > the procedure SGESVJ did not converge in the given number of */ /* > iterations (sweeps) and SCALE*SVA(1:N) may not be accurate. */ /* > \endverbatim */ /* > */ /* > \param[in] MV */ /* > \verbatim */ /* > MV is INTEGER */ /* > If JOBV = 'A', then the product of Jacobi rotations in SGESVJ */ /* > is applied to the first MV rows of V. See the description of JOBV. */ /* > \endverbatim */ /* > */ /* > \param[in,out] V */ /* > \verbatim */ /* > V is REAL array, dimension (LDV,N) */ /* > If JOBV = 'V', then V contains on exit the N-by-N matrix of */ /* > the right singular vectors; */ /* > If JOBV = 'A', then V contains the product of the computed right */ /* > singular vector matrix and the initial matrix in */ /* > the array V. */ /* > If JOBV = 'N', then V is not referenced. */ /* > \endverbatim */ /* > */ /* > \param[in] LDV */ /* > \verbatim */ /* > LDV is INTEGER */ /* > The leading dimension of the array V, LDV >= 1. */ /* > If JOBV = 'V', then LDV >= f2cmax(1,N). */ /* > If JOBV = 'A', then LDV >= f2cmax(1,MV) . */ /* > \endverbatim */ /* > */ /* > \param[in,out] WORK */ /* > \verbatim */ /* > WORK is REAL array, dimension (LWORK) */ /* > On entry, */ /* > If JOBU = 'C' : */ /* > WORK(1) = CTOL, where CTOL defines the threshold for convergence. */ /* > The process stops if all columns of A are mutually */ /* > orthogonal up to CTOL*EPS, EPS=SLAMCH('E'). */ /* > It is required that CTOL >= ONE, i.e. it is not */ /* > allowed to force the routine to obtain orthogonality */ /* > below EPSILON. */ /* > On exit, */ /* > WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N) */ /* > are the computed singular vcalues of A. */ /* > (See description of SVA().) */ /* > WORK(2) = NINT(WORK(2)) is the number of the computed nonzero */ /* > singular values. */ /* > WORK(3) = NINT(WORK(3)) is the number of the computed singular */ /* > values that are larger than the underflow threshold. */ /* > WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi */ /* > rotations needed for numerical convergence. */ /* > WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep. */ /* > This is useful information in cases when SGESVJ did */ /* > not converge, as it can be used to estimate whether */ /* > the output is still useful and for post festum analysis. */ /* > WORK(6) = the largest absolute value over all sines of the */ /* > Jacobi rotation angles in the last sweep. It can be */ /* > useful for a post festum analysis. */ /* > \endverbatim */ /* > */ /* > \param[in] LWORK */ /* > \verbatim */ /* > LWORK is INTEGER */ /* > length of WORK, WORK >= MAX(6,M+N) */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit. */ /* > < 0: if INFO = -i, then the i-th argument had an illegal value */ /* > > 0: SGESVJ did not converge in the maximal allowed number (30) */ /* > of sweeps. The output may still be useful. See the */ /* > description of WORK. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date June 2017 */ /* > \ingroup realGEcomputational */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane */ /* > rotations. The rotations are implemented as fast scaled rotations of */ /* > Anda and Park [1]. In the case of underflow of the Jacobi angle, a */ /* > modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses */ /* > column interchanges of de Rijk [2]. The relative accuracy of the computed */ /* > singular values and the accuracy of the computed singular vectors (in */ /* > angle metric) is as guaranteed by the theory of Demmel and Veselic [3]. */ /* > The condition number that determines the accuracy in the full rank case */ /* > is essentially min_{D=diag} kappa(A*D), where kappa(.) is the */ /* > spectral condition number. The best performance of this Jacobi SVD */ /* > procedure is achieved if used in an accelerated version of Drmac and */ /* > Veselic [5,6], and it is the kernel routine in the SIGMA library [7]. */ /* > Some tunning parameters (marked with [TP]) are available for the */ /* > implementer. \n */ /* > The computational range for the nonzero singular values is the machine */ /* > number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even */ /* > denormalized singular values can be computed with the corresponding */ /* > gradual loss of accurate digits. */ /* > */ /* > \par Contributors: */ /* ================== */ /* > */ /* > Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) */ /* > */ /* > \par References: */ /* ================ */ /* > */ /* > [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling. \n */ /* > SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174. \n\n */ /* > [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the */ /* > singular value decomposition on a vector computer. \n */ /* > SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371. \n\n */ /* > [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR. \n */ /* > [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular */ /* > value computation in floating point arithmetic. \n */ /* > SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222. \n\n */ /* > [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. \n */ /* > SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. \n */ /* > LAPACK Working note 169. \n\n */ /* > [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. \n */ /* > SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. \n */ /* > LAPACK Working note 170. \n\n */ /* > [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, */ /* > QSVD, (H,K)-SVD computations.\n */ /* > Department of Mathematics, University of Zagreb, 2008. */ /* > */ /* > \par Bugs, Examples and Comments: */ /* ================================= */ /* > */ /* > Please report all bugs and send interesting test examples and comments to */ /* > drmac@math.hr. Thank you. */ /* ===================================================================== */ /* Subroutine */ int sgesvj_(char *joba, char *jobu, char *jobv, integer *m, integer *n, real *a, integer *lda, real *sva, integer *mv, real *v, integer *ldv, real *work, integer *lwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, v_dim1, v_offset, i__1, i__2, i__3, i__4, i__5; real r__1, r__2; /* Local variables */ real aapp, aapq, aaqq, ctol; integer ierr; real bigtheta; extern real sdot_(integer *, real *, integer *, real *, integer *); integer pskipped; real aapp0, temp1; extern real snrm2_(integer *, real *, integer *); integer i__, p, q; real t, large, apoaq, aqoap; extern logical lsame_(char *, char *); real theta; extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); real small, sfmin; logical lsvec; real fastr[5], epsln; logical applv, rsvec, uctol, lower, upper; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); logical rotok; integer n2; extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *, integer *); integer n4; extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *, real *, integer *), srotm_(integer *, real *, integer *, real *, integer *, real *); real rootsfmin; extern /* Subroutine */ int sgsvj0_(char *, integer *, integer *, real *, integer *, real *, real *, integer *, real *, integer *, real *, real *, real *, integer *, real *, integer *, integer *), sgsvj1_(char *, integer *, integer *, integer *, real *, integer * , real *, real *, integer *, real *, integer *, real *, real *, real *, integer *, real *, integer *, integer *); integer n34; real cs, sn; extern real slamch_(char *); extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen); integer ijblsk, swband; extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, real *, integer *, integer *, real *, integer *, integer *); extern integer isamax_(integer *, real *, integer *); integer blskip; real mxaapq; extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *, real *, real *, integer *); real thsign; extern /* Subroutine */ int slassq_(integer *, real *, integer *, real *, real *); real mxsinj; integer ir1, emptsw, notrot, iswrot, jbc; real big; integer kbl, lkahead, igl, ibr, jgl, nbl; real skl; logical goscale; real tol; integer mvl; logical noscale; real rootbig, rooteps; integer rowskip; real roottol; /* -- 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 2017 */ /* ===================================================================== */ /* from BLAS */ /* from LAPACK */ /* from BLAS */ /* from LAPACK */ /* Test the input arguments */ /* Parameter adjustments */ --sva; a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; v_dim1 = *ldv; v_offset = 1 + v_dim1 * 1; v -= v_offset; --work; /* Function Body */ lsvec = lsame_(jobu, "U"); uctol = lsame_(jobu, "C"); rsvec = lsame_(jobv, "V"); applv = lsame_(jobv, "A"); upper = lsame_(joba, "U"); lower = lsame_(joba, "L"); if (! (upper || lower || lsame_(joba, "G"))) { *info = -1; } else if (! (lsvec || uctol || lsame_(jobu, "N"))) { *info = -2; } else if (! (rsvec || applv || lsame_(jobv, "N"))) { *info = -3; } else if (*m < 0) { *info = -4; } else if (*n < 0 || *n > *m) { *info = -5; } else if (*lda < *m) { *info = -7; } else if (*mv < 0) { *info = -9; } else if (rsvec && *ldv < *n || applv && *ldv < *mv) { *info = -11; } else if (uctol && work[1] <= 1.f) { *info = -12; } else /* if(complicated condition) */ { /* Computing MAX */ i__1 = *m + *n; if (*lwork < f2cmax(i__1,6)) { *info = -13; } else { *info = 0; } } /* #:( */ if (*info != 0) { i__1 = -(*info); xerbla_("SGESVJ", &i__1, (ftnlen)6); return 0; } /* #:) Quick return for void matrix */ if (*m == 0 || *n == 0) { return 0; } /* Set numerical parameters */ /* The stopping criterion for Jacobi rotations is */ /* max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS */ /* where EPS is the round-off and CTOL is defined as follows: */ if (uctol) { /* ... user controlled */ ctol = work[1]; } else { /* ... default */ if (lsvec || rsvec || applv) { ctol = sqrt((real) (*m)); } else { ctol = (real) (*m); } } /* ... and the machine dependent parameters are */ /* [!] (Make sure that SLAMCH() works properly on the target machine.) */ epsln = slamch_("Epsilon"); rooteps = sqrt(epsln); sfmin = slamch_("SafeMinimum"); rootsfmin = sqrt(sfmin); small = sfmin / epsln; big = slamch_("Overflow"); /* BIG = ONE / SFMIN */ rootbig = 1.f / rootsfmin; large = big / sqrt((real) (*m * *n)); bigtheta = 1.f / rooteps; tol = ctol * epsln; roottol = sqrt(tol); if ((real) (*m) * epsln >= 1.f) { *info = -4; i__1 = -(*info); xerbla_("SGESVJ", &i__1, (ftnlen)6); return 0; } /* Initialize the right singular vector matrix. */ if (rsvec) { mvl = *n; slaset_("A", &mvl, n, &c_b17, &c_b18, &v[v_offset], ldv); } else if (applv) { mvl = *mv; } rsvec = rsvec || applv; /* Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N ) */ /* (!) If necessary, scale A to protect the largest singular value */ /* from overflow. It is possible that saving the largest singular */ /* value destroys the information about the small ones. */ /* This initial scaling is almost minimal in the sense that the */ /* goal is to make sure that no column norm overflows, and that */ /* SQRT(N)*max_i SVA(i) does not overflow. If INFinite entries */ /* in A are detected, the procedure returns with INFO=-6. */ skl = 1.f / sqrt((real) (*m) * (real) (*n)); noscale = TRUE_; goscale = TRUE_; if (lower) { /* the input matrix is M-by-N lower triangular (trapezoidal) */ i__1 = *n; for (p = 1; p <= i__1; ++p) { aapp = 0.f; aaqq = 1.f; i__2 = *m - p + 1; slassq_(&i__2, &a[p + p * a_dim1], &c__1, &aapp, &aaqq); if (aapp > big) { *info = -6; i__2 = -(*info); xerbla_("SGESVJ", &i__2, (ftnlen)6); return 0; } aaqq = sqrt(aaqq); if (aapp < big / aaqq && noscale) { sva[p] = aapp * aaqq; } else { noscale = FALSE_; sva[p] = aapp * (aaqq * skl); if (goscale) { goscale = FALSE_; i__2 = p - 1; for (q = 1; q <= i__2; ++q) { sva[q] *= skl; /* L1873: */ } } } /* L1874: */ } } else if (upper) { /* the input matrix is M-by-N upper triangular (trapezoidal) */ i__1 = *n; for (p = 1; p <= i__1; ++p) { aapp = 0.f; aaqq = 1.f; slassq_(&p, &a[p * a_dim1 + 1], &c__1, &aapp, &aaqq); if (aapp > big) { *info = -6; i__2 = -(*info); xerbla_("SGESVJ", &i__2, (ftnlen)6); return 0; } aaqq = sqrt(aaqq); if (aapp < big / aaqq && noscale) { sva[p] = aapp * aaqq; } else { noscale = FALSE_; sva[p] = aapp * (aaqq * skl); if (goscale) { goscale = FALSE_; i__2 = p - 1; for (q = 1; q <= i__2; ++q) { sva[q] *= skl; /* L2873: */ } } } /* L2874: */ } } else { /* the input matrix is M-by-N general dense */ 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 = -6; i__2 = -(*info); xerbla_("SGESVJ", &i__2, (ftnlen)6); return 0; } aaqq = sqrt(aaqq); if (aapp < big / aaqq && noscale) { sva[p] = aapp * aaqq; } else { noscale = FALSE_; sva[p] = aapp * (aaqq * skl); if (goscale) { goscale = FALSE_; i__2 = p - 1; for (q = 1; q <= i__2; ++q) { sva[q] *= skl; /* L3873: */ } } } /* L3874: */ } } if (noscale) { skl = 1.f; } /* Move the smaller part of the spectrum from the underflow threshold */ /* (!) Start by determining the position of the nonzero entries of the */ /* array SVA() relative to ( SFMIN, BIG ). */ aapp = 0.f; aaqq = big; i__1 = *n; for (p = 1; p <= i__1; ++p) { if (sva[p] != 0.f) { /* Computing MIN */ r__1 = aaqq, r__2 = sva[p]; aaqq = f2cmin(r__1,r__2); } /* Computing MAX */ r__1 = aapp, r__2 = sva[p]; aapp = f2cmax(r__1,r__2); /* L4781: */ } /* #:) Quick return for zero matrix */ if (aapp == 0.f) { if (lsvec) { slaset_("G", m, n, &c_b17, &c_b18, &a[a_offset], lda); } work[1] = 1.f; work[2] = 0.f; work[3] = 0.f; work[4] = 0.f; work[5] = 0.f; work[6] = 0.f; return 0; } /* #:) Quick return for one-column matrix */ if (*n == 1) { if (lsvec) { slascl_("G", &c__0, &c__0, &sva[1], &skl, m, &c__1, &a[a_dim1 + 1] , lda, &ierr); } work[1] = 1.f / skl; if (sva[1] >= sfmin) { work[2] = 1.f; } else { work[2] = 0.f; } work[3] = 0.f; work[4] = 0.f; work[5] = 0.f; work[6] = 0.f; return 0; } /* Protect small singular values from underflow, and try to */ /* avoid underflows/overflows in computing Jacobi rotations. */ sn = sqrt(sfmin / epsln); temp1 = sqrt(big / (real) (*n)); if (aapp <= sn || aaqq >= temp1 || sn <= aaqq && aapp <= temp1) { /* Computing MIN */ r__1 = big, r__2 = temp1 / aapp; temp1 = f2cmin(r__1,r__2); /* AAQQ = AAQQ*TEMP1 */ /* AAPP = AAPP*TEMP1 */ } else if (aaqq <= sn && aapp <= temp1) { /* Computing MIN */ r__1 = sn / aaqq, r__2 = big / (aapp * sqrt((real) (*n))); temp1 = f2cmin(r__1,r__2); /* AAQQ = AAQQ*TEMP1 */ /* AAPP = AAPP*TEMP1 */ } else if (aaqq >= sn && aapp >= temp1) { /* Computing MAX */ r__1 = sn / aaqq, r__2 = temp1 / aapp; temp1 = f2cmax(r__1,r__2); /* AAQQ = AAQQ*TEMP1 */ /* AAPP = AAPP*TEMP1 */ } else if (aaqq <= sn && aapp >= temp1) { /* Computing MIN */ r__1 = sn / aaqq, r__2 = big / (sqrt((real) (*n)) * aapp); temp1 = f2cmin(r__1,r__2); /* AAQQ = AAQQ*TEMP1 */ /* AAPP = AAPP*TEMP1 */ } else { temp1 = 1.f; } /* Scale, if necessary */ if (temp1 != 1.f) { slascl_("G", &c__0, &c__0, &c_b18, &temp1, n, &c__1, &sva[1], n, & ierr); } skl = temp1 * skl; if (skl != 1.f) { slascl_(joba, &c__0, &c__0, &c_b18, &skl, m, n, &a[a_offset], lda, & ierr); skl = 1.f / skl; } /* Row-cyclic Jacobi SVD algorithm with column pivoting */ emptsw = *n * (*n - 1) / 2; notrot = 0; fastr[0] = 0.f; /* A is represented in factored form A = A * diag(WORK), where diag(WORK) */ /* is initialized to identity. WORK is updated during fast scaled */ /* rotations. */ i__1 = *n; for (q = 1; q <= i__1; ++q) { work[q] = 1.f; /* L1868: */ } swband = 3; /* [TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective */ /* if SGESVJ is used as a computational routine in the preconditioned */ /* Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure */ /* works on pivots inside a band-like region around the diagonal. */ /* The boundaries are determined dynamically, based on the number of */ /* pivots above a threshold. */ kbl = f2cmin(8,*n); /* [TP] KBL is a tuning parameter that defines the tile size in the */ /* tiling of the p-q loops of pivot pairs. In general, an optimal */ /* value of KBL depends on the matrix dimensions and on the */ /* parameters of the computer's memory. */ nbl = *n / kbl; if (nbl * kbl != *n) { ++nbl; } /* Computing 2nd power */ i__1 = kbl; blskip = i__1 * i__1; /* [TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. */ rowskip = f2cmin(5,kbl); /* [TP] ROWSKIP is a tuning parameter. */ lkahead = 1; /* [TP] LKAHEAD is a tuning parameter. */ /* Quasi block transformations, using the lower (upper) triangular */ /* structure of the input matrix. The quasi-block-cycling usually */ /* invokes cubic convergence. Big part of this cycle is done inside */ /* canonical subspaces of dimensions less than M. */ /* Computing MAX */ i__1 = 64, i__2 = kbl << 2; if ((lower || upper) && *n > f2cmax(i__1,i__2)) { /* [TP] The number of partition levels and the actual partition are */ /* tuning parameters. */ n4 = *n / 4; n2 = *n / 2; n34 = n4 * 3; if (applv) { q = 0; } else { q = 1; } if (lower) { /* This works very well on lower triangular matrices, in particular */ /* in the framework of the preconditioned Jacobi SVD (xGEJSV). */ /* The idea is simple: */ /* [+ 0 0 0] Note that Jacobi transformations of [0 0] */ /* [+ + 0 0] [0 0] */ /* [+ + x 0] actually work on [x 0] [x 0] */ /* [+ + x x] [x x]. [x x] */ i__1 = *m - n34; i__2 = *n - n34; i__3 = *lwork - *n; sgsvj0_(jobv, &i__1, &i__2, &a[n34 + 1 + (n34 + 1) * a_dim1], lda, &work[n34 + 1], &sva[n34 + 1], &mvl, &v[n34 * q + 1 + ( n34 + 1) * v_dim1], ldv, &epsln, &sfmin, &tol, &c__2, & work[*n + 1], &i__3, &ierr); i__1 = *m - n2; i__2 = n34 - n2; i__3 = *lwork - *n; sgsvj0_(jobv, &i__1, &i__2, &a[n2 + 1 + (n2 + 1) * a_dim1], lda, & work[n2 + 1], &sva[n2 + 1], &mvl, &v[n2 * q + 1 + (n2 + 1) * v_dim1], ldv, &epsln, &sfmin, &tol, &c__2, &work[*n + 1], &i__3, &ierr); i__1 = *m - n2; i__2 = *n - n2; i__3 = *lwork - *n; sgsvj1_(jobv, &i__1, &i__2, &n4, &a[n2 + 1 + (n2 + 1) * a_dim1], lda, &work[n2 + 1], &sva[n2 + 1], &mvl, &v[n2 * q + 1 + ( n2 + 1) * v_dim1], ldv, &epsln, &sfmin, &tol, &c__1, & work[*n + 1], &i__3, &ierr); i__1 = *m - n4; i__2 = n2 - n4; i__3 = *lwork - *n; sgsvj0_(jobv, &i__1, &i__2, &a[n4 + 1 + (n4 + 1) * a_dim1], lda, & work[n4 + 1], &sva[n4 + 1], &mvl, &v[n4 * q + 1 + (n4 + 1) * v_dim1], ldv, &epsln, &sfmin, &tol, &c__1, &work[*n + 1], &i__3, &ierr); i__1 = *lwork - *n; sgsvj0_(jobv, m, &n4, &a[a_offset], lda, &work[1], &sva[1], &mvl, &v[v_offset], ldv, &epsln, &sfmin, &tol, &c__1, &work[*n + 1], &i__1, &ierr); i__1 = *lwork - *n; sgsvj1_(jobv, m, &n2, &n4, &a[a_offset], lda, &work[1], &sva[1], & mvl, &v[v_offset], ldv, &epsln, &sfmin, &tol, &c__1, & work[*n + 1], &i__1, &ierr); } else if (upper) { i__1 = *lwork - *n; sgsvj0_(jobv, &n4, &n4, &a[a_offset], lda, &work[1], &sva[1], & mvl, &v[v_offset], ldv, &epsln, &sfmin, &tol, &c__2, & work[*n + 1], &i__1, &ierr); i__1 = *lwork - *n; sgsvj0_(jobv, &n2, &n4, &a[(n4 + 1) * a_dim1 + 1], lda, &work[n4 + 1], &sva[n4 + 1], &mvl, &v[n4 * q + 1 + (n4 + 1) * v_dim1], ldv, &epsln, &sfmin, &tol, &c__1, &work[*n + 1], &i__1, &ierr); i__1 = *lwork - *n; sgsvj1_(jobv, &n2, &n2, &n4, &a[a_offset], lda, &work[1], &sva[1], &mvl, &v[v_offset], ldv, &epsln, &sfmin, &tol, &c__1, & work[*n + 1], &i__1, &ierr); i__1 = n2 + n4; i__2 = *lwork - *n; sgsvj0_(jobv, &i__1, &n4, &a[(n2 + 1) * a_dim1 + 1], lda, &work[ n2 + 1], &sva[n2 + 1], &mvl, &v[n2 * q + 1 + (n2 + 1) * v_dim1], ldv, &epsln, &sfmin, &tol, &c__1, &work[*n + 1], &i__2, &ierr); } } for (i__ = 1; i__ <= 30; ++i__) { mxaapq = 0.f; mxsinj = 0.f; iswrot = 0; notrot = 0; pskipped = 0; /* Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs */ /* 1 <= p < q <= N. This is the first step toward a blocked implementation */ /* of the rotations. New implementation, based on block transformations, */ /* is under development. */ i__1 = nbl; for (ibr = 1; ibr <= i__1; ++ibr) { igl = (ibr - 1) * kbl + 1; /* Computing MIN */ i__3 = lkahead, i__4 = nbl - ibr; i__2 = f2cmin(i__3,i__4); for (ir1 = 0; ir1 <= i__2; ++ir1) { igl += ir1 * kbl; /* Computing MIN */ i__4 = igl + kbl - 1, i__5 = *n - 1; i__3 = f2cmin(i__4,i__5); for (p = igl; p <= i__3; ++p) { i__4 = *n - p + 1; q = isamax_(&i__4, &sva[p], &c__1) + p - 1; if (p != q) { sswap_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1); if (rsvec) { sswap_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], &c__1); } temp1 = sva[p]; sva[p] = sva[q]; sva[q] = temp1; temp1 = work[p]; work[p] = work[q]; work[q] = temp1; } if (ir1 == 0) { /* Column norms are periodically updated by explicit */ /* norm computation. */ /* Caveat: */ /* Unfortunately, some BLAS implementations compute SNRM2(M,A(1,p),1) */ /* as SQRT(SDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to */ /* overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and to */ /* underflow for ||A(:,p)||_2 < SQRT(underflow_threshold). */ /* Hence, SNRM2 cannot be trusted, not even in the case when */ /* the true norm is far from the under(over)flow boundaries. */ /* If properly implemented SNRM2 is available, the IF-THEN-ELSE */ /* below should read "AAPP = SNRM2( M, A(1,p), 1 ) * WORK(p)". */ if (sva[p] < rootbig && sva[p] > rootsfmin) { sva[p] = snrm2_(m, &a[p * a_dim1 + 1], &c__1) * work[p]; } else { temp1 = 0.f; aapp = 1.f; slassq_(m, &a[p * a_dim1 + 1], &c__1, &temp1, & aapp); sva[p] = temp1 * sqrt(aapp) * work[p]; } aapp = sva[p]; } else { aapp = sva[p]; } if (aapp > 0.f) { pskipped = 0; /* Computing MIN */ i__5 = igl + kbl - 1; i__4 = f2cmin(i__5,*n); for (q = p + 1; q <= i__4; ++q) { aaqq = sva[q]; if (aaqq > 0.f) { aapp0 = aapp; if (aaqq >= 1.f) { rotok = small * aapp <= aaqq; if (aapp < big / aaqq) { aapq = sdot_(m, &a[p * a_dim1 + 1], & c__1, &a[q * a_dim1 + 1], & c__1) * work[p] * work[q] / aaqq / aapp; } else { scopy_(m, &a[p * a_dim1 + 1], &c__1, & work[*n + 1], &c__1); slascl_("G", &c__0, &c__0, &aapp, & work[p], m, &c__1, &work[*n + 1], lda, &ierr); aapq = sdot_(m, &work[*n + 1], &c__1, &a[q * a_dim1 + 1], &c__1) * work[q] / aaqq; } } else { rotok = aapp <= aaqq / small; if (aapp > small / aaqq) { aapq = sdot_(m, &a[p * a_dim1 + 1], & c__1, &a[q * a_dim1 + 1], & c__1) * work[p] * work[q] / aaqq / aapp; } else { scopy_(m, &a[q * a_dim1 + 1], &c__1, & work[*n + 1], &c__1); slascl_("G", &c__0, &c__0, &aaqq, & work[q], m, &c__1, &work[*n + 1], lda, &ierr); aapq = sdot_(m, &work[*n + 1], &c__1, &a[p * a_dim1 + 1], &c__1) * work[p] / aapp; } } /* Computing MAX */ r__1 = mxaapq, r__2 = abs(aapq); mxaapq = f2cmax(r__1,r__2); /* TO rotate or NOT to rotate, THAT is the question ... */ if (abs(aapq) > tol) { /* [RTD] ROTATED = ROTATED + ONE */ if (ir1 == 0) { notrot = 0; pskipped = 0; ++iswrot; } if (rotok) { aqoap = aaqq / aapp; apoaq = aapp / aaqq; theta = (r__1 = aqoap - apoaq, abs( r__1)) * -.5f / aapq; if (abs(theta) > bigtheta) { t = .5f / theta; fastr[2] = t * work[p] / work[q]; fastr[3] = -t * work[q] / work[p]; srotm_(m, &a[p * a_dim1 + 1], & c__1, &a[q * a_dim1 + 1], &c__1, fastr); if (rsvec) { srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], &c__1, fastr); } /* Computing MAX */ r__1 = 0.f, r__2 = t * apoaq * aapq + 1.f; sva[q] = aaqq * sqrt((f2cmax(r__1, r__2))); /* Computing MAX */ r__1 = 0.f, r__2 = 1.f - t * aqoap * aapq; aapp *= sqrt((f2cmax(r__1,r__2))); /* Computing MAX */ r__1 = mxsinj, r__2 = abs(t); mxsinj = f2cmax(r__1,r__2); } else { thsign = -r_sign(&c_b18, &aapq); t = 1.f / (theta + thsign * sqrt( theta * theta + 1.f)); cs = sqrt(1.f / (t * t + 1.f)); sn = t * cs; /* Computing MAX */ r__1 = mxsinj, r__2 = abs(sn); mxsinj = f2cmax(r__1,r__2); /* Computing MAX */ r__1 = 0.f, r__2 = t * apoaq * aapq + 1.f; sva[q] = aaqq * sqrt((f2cmax(r__1, r__2))); /* Computing MAX */ r__1 = 0.f, r__2 = 1.f - t * aqoap * aapq; aapp *= sqrt((f2cmax(r__1,r__2))); apoaq = work[p] / work[q]; aqoap = work[q] / work[p]; if (work[p] >= 1.f) { if (work[q] >= 1.f) { fastr[2] = t * apoaq; fastr[3] = -t * aqoap; work[p] *= cs; work[q] *= cs; srotm_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1, fastr); if (rsvec) { srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[ q * v_dim1 + 1], &c__1, fastr); } } else { r__1 = -t * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[ p * a_dim1 + 1], &c__1); r__1 = cs * sn * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[ q * a_dim1 + 1], &c__1); work[p] *= cs; work[q] /= cs; if (rsvec) { r__1 = -t * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], & c__1, &v[p * v_dim1 + 1], &c__1); r__1 = cs * sn * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], & c__1, &v[q * v_dim1 + 1], &c__1); } } } else { if (work[q] >= 1.f) { r__1 = t * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[ q * a_dim1 + 1], &c__1); r__1 = -cs * sn * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[ p * a_dim1 + 1], &c__1); work[p] /= cs; work[q] *= cs; if (rsvec) { r__1 = t * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], & c__1, &v[q * v_dim1 + 1], &c__1); r__1 = -cs * sn * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], & c__1, &v[p * v_dim1 + 1], &c__1); } } else { if (work[p] >= work[q]) { r__1 = -t * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[p * a_dim1 + 1], &c__1); r__1 = cs * sn * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1); work[p] *= cs; work[q] /= cs; if (rsvec) { r__1 = -t * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &c__1, &v[p * v_dim1 + 1], & c__1); r__1 = cs * sn * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], & c__1); } } else { r__1 = t * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1); r__1 = -cs * sn * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[p * a_dim1 + 1], &c__1); work[p] /= cs; work[q] *= cs; if (rsvec) { r__1 = t * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], & c__1); r__1 = -cs * sn * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &c__1, &v[p * v_dim1 + 1], & c__1); } } } } } } else { scopy_(m, &a[p * a_dim1 + 1], &c__1, & work[*n + 1], &c__1); slascl_("G", &c__0, &c__0, &aapp, & c_b18, m, &c__1, &work[*n + 1] , lda, &ierr); slascl_("G", &c__0, &c__0, &aaqq, & c_b18, m, &c__1, &a[q * a_dim1 + 1], lda, &ierr); temp1 = -aapq * work[p] / work[q]; saxpy_(m, &temp1, &work[*n + 1], & c__1, &a[q * a_dim1 + 1], & c__1); slascl_("G", &c__0, &c__0, &c_b18, & aaqq, m, &c__1, &a[q * a_dim1 + 1], lda, &ierr); /* Computing MAX */ r__1 = 0.f, r__2 = 1.f - aapq * aapq; sva[q] = aaqq * sqrt((f2cmax(r__1,r__2))) ; mxsinj = f2cmax(mxsinj,sfmin); } /* END IF ROTOK THEN ... ELSE */ /* In the case of cancellation in updating SVA(q), SVA(p) */ /* recompute SVA(q), SVA(p). */ /* Computing 2nd power */ r__1 = sva[q] / aaqq; if (r__1 * r__1 <= rooteps) { if (aaqq < rootbig && aaqq > rootsfmin) { sva[q] = snrm2_(m, &a[q * a_dim1 + 1], &c__1) * work[q]; } else { t = 0.f; aaqq = 1.f; slassq_(m, &a[q * a_dim1 + 1], & c__1, &t, &aaqq); sva[q] = t * sqrt(aaqq) * work[q]; } } if (aapp / aapp0 <= rooteps) { if (aapp < rootbig && aapp > rootsfmin) { aapp = snrm2_(m, &a[p * a_dim1 + 1], &c__1) * work[p]; } else { t = 0.f; aapp = 1.f; slassq_(m, &a[p * a_dim1 + 1], & c__1, &t, &aapp); aapp = t * sqrt(aapp) * work[p]; } sva[p] = aapp; } } else { /* A(:,p) and A(:,q) already numerically orthogonal */ if (ir1 == 0) { ++notrot; } /* [RTD] SKIPPED = SKIPPED + 1 */ ++pskipped; } } else { /* A(:,q) is zero column */ if (ir1 == 0) { ++notrot; } ++pskipped; } if (i__ <= swband && pskipped > rowskip) { if (ir1 == 0) { aapp = -aapp; } notrot = 0; goto L2103; } /* L2002: */ } /* END q-LOOP */ L2103: /* bailed out of q-loop */ sva[p] = aapp; } else { sva[p] = aapp; if (ir1 == 0 && aapp == 0.f) { /* Computing MIN */ i__4 = igl + kbl - 1; notrot = notrot + f2cmin(i__4,*n) - p; } } /* L2001: */ } /* end of the p-loop */ /* end of doing the block ( ibr, ibr ) */ /* L1002: */ } /* end of ir1-loop */ /* ... go to the off diagonal blocks */ igl = (ibr - 1) * kbl + 1; i__2 = nbl; for (jbc = ibr + 1; jbc <= i__2; ++jbc) { jgl = (jbc - 1) * kbl + 1; /* doing the block at ( ibr, jbc ) */ ijblsk = 0; /* Computing MIN */ i__4 = igl + kbl - 1; i__3 = f2cmin(i__4,*n); for (p = igl; p <= i__3; ++p) { aapp = sva[p]; if (aapp > 0.f) { pskipped = 0; /* Computing MIN */ i__5 = jgl + kbl - 1; i__4 = f2cmin(i__5,*n); for (q = jgl; q <= i__4; ++q) { aaqq = sva[q]; if (aaqq > 0.f) { aapp0 = aapp; /* Safe Gram matrix computation */ if (aaqq >= 1.f) { if (aapp >= aaqq) { rotok = small * aapp <= aaqq; } else { rotok = small * aaqq <= aapp; } if (aapp < big / aaqq) { aapq = sdot_(m, &a[p * a_dim1 + 1], & c__1, &a[q * a_dim1 + 1], & c__1) * work[p] * work[q] / aaqq / aapp; } else { scopy_(m, &a[p * a_dim1 + 1], &c__1, & work[*n + 1], &c__1); slascl_("G", &c__0, &c__0, &aapp, & work[p], m, &c__1, &work[*n + 1], lda, &ierr); aapq = sdot_(m, &work[*n + 1], &c__1, &a[q * a_dim1 + 1], &c__1) * work[q] / aaqq; } } else { if (aapp >= aaqq) { rotok = aapp <= aaqq / small; } else { rotok = aaqq <= aapp / small; } if (aapp > small / aaqq) { aapq = sdot_(m, &a[p * a_dim1 + 1], & c__1, &a[q * a_dim1 + 1], & c__1) * work[p] * work[q] / aaqq / aapp; } else { scopy_(m, &a[q * a_dim1 + 1], &c__1, & work[*n + 1], &c__1); slascl_("G", &c__0, &c__0, &aaqq, & work[q], m, &c__1, &work[*n + 1], lda, &ierr); aapq = sdot_(m, &work[*n + 1], &c__1, &a[p * a_dim1 + 1], &c__1) * work[p] / aapp; } } /* Computing MAX */ r__1 = mxaapq, r__2 = abs(aapq); mxaapq = f2cmax(r__1,r__2); /* TO rotate or NOT to rotate, THAT is the question ... */ if (abs(aapq) > tol) { notrot = 0; /* [RTD] ROTATED = ROTATED + 1 */ pskipped = 0; ++iswrot; if (rotok) { aqoap = aaqq / aapp; apoaq = aapp / aaqq; theta = (r__1 = aqoap - apoaq, abs( r__1)) * -.5f / aapq; if (aaqq > aapp0) { theta = -theta; } if (abs(theta) > bigtheta) { t = .5f / theta; fastr[2] = t * work[p] / work[q]; fastr[3] = -t * work[q] / work[p]; srotm_(m, &a[p * a_dim1 + 1], & c__1, &a[q * a_dim1 + 1], &c__1, fastr); if (rsvec) { srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], &c__1, fastr); } /* Computing MAX */ r__1 = 0.f, r__2 = t * apoaq * aapq + 1.f; sva[q] = aaqq * sqrt((f2cmax(r__1, r__2))); /* Computing MAX */ r__1 = 0.f, r__2 = 1.f - t * aqoap * aapq; aapp *= sqrt((f2cmax(r__1,r__2))); /* Computing MAX */ r__1 = mxsinj, r__2 = abs(t); mxsinj = f2cmax(r__1,r__2); } else { thsign = -r_sign(&c_b18, &aapq); if (aaqq > aapp0) { thsign = -thsign; } t = 1.f / (theta + thsign * sqrt( theta * theta + 1.f)); cs = sqrt(1.f / (t * t + 1.f)); sn = t * cs; /* Computing MAX */ r__1 = mxsinj, r__2 = abs(sn); mxsinj = f2cmax(r__1,r__2); /* Computing MAX */ r__1 = 0.f, r__2 = t * apoaq * aapq + 1.f; sva[q] = aaqq * sqrt((f2cmax(r__1, r__2))); /* Computing MAX */ r__1 = 0.f, r__2 = 1.f - t * aqoap * aapq; aapp *= sqrt((f2cmax(r__1,r__2))); apoaq = work[p] / work[q]; aqoap = work[q] / work[p]; if (work[p] >= 1.f) { if (work[q] >= 1.f) { fastr[2] = t * apoaq; fastr[3] = -t * aqoap; work[p] *= cs; work[q] *= cs; srotm_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1, fastr); if (rsvec) { srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[ q * v_dim1 + 1], &c__1, fastr); } } else { r__1 = -t * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[ p * a_dim1 + 1], &c__1); r__1 = cs * sn * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[ q * a_dim1 + 1], &c__1); if (rsvec) { r__1 = -t * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], & c__1, &v[p * v_dim1 + 1], &c__1); r__1 = cs * sn * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], & c__1, &v[q * v_dim1 + 1], &c__1); } work[p] *= cs; work[q] /= cs; } } else { if (work[q] >= 1.f) { r__1 = t * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[ q * a_dim1 + 1], &c__1); r__1 = -cs * sn * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[ p * a_dim1 + 1], &c__1); if (rsvec) { r__1 = t * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], & c__1, &v[q * v_dim1 + 1], &c__1); r__1 = -cs * sn * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], & c__1, &v[p * v_dim1 + 1], &c__1); } work[p] /= cs; work[q] *= cs; } else { if (work[p] >= work[q]) { r__1 = -t * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[p * a_dim1 + 1], &c__1); r__1 = cs * sn * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1); work[p] *= cs; work[q] /= cs; if (rsvec) { r__1 = -t * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &c__1, &v[p * v_dim1 + 1], & c__1); r__1 = cs * sn * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], & c__1); } } else { r__1 = t * apoaq; saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1); r__1 = -cs * sn * aqoap; saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[p * a_dim1 + 1], &c__1); work[p] /= cs; work[q] *= cs; if (rsvec) { r__1 = t * apoaq; saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], & c__1); r__1 = -cs * sn * aqoap; saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &c__1, &v[p * v_dim1 + 1], & c__1); } } } } } } else { if (aapp > aaqq) { scopy_(m, &a[p * a_dim1 + 1], & c__1, &work[*n + 1], & c__1); slascl_("G", &c__0, &c__0, &aapp, &c_b18, m, &c__1, &work[* n + 1], lda, &ierr); slascl_("G", &c__0, &c__0, &aaqq, &c_b18, m, &c__1, &a[q * a_dim1 + 1], lda, &ierr); temp1 = -aapq * work[p] / work[q]; saxpy_(m, &temp1, &work[*n + 1], & c__1, &a[q * a_dim1 + 1], &c__1); slascl_("G", &c__0, &c__0, &c_b18, &aaqq, m, &c__1, &a[q * a_dim1 + 1], lda, &ierr); /* Computing MAX */ r__1 = 0.f, r__2 = 1.f - aapq * aapq; sva[q] = aaqq * sqrt((f2cmax(r__1, r__2))); mxsinj = f2cmax(mxsinj,sfmin); } else { scopy_(m, &a[q * a_dim1 + 1], & c__1, &work[*n + 1], & c__1); slascl_("G", &c__0, &c__0, &aaqq, &c_b18, m, &c__1, &work[* n + 1], lda, &ierr); slascl_("G", &c__0, &c__0, &aapp, &c_b18, m, &c__1, &a[p * a_dim1 + 1], lda, &ierr); temp1 = -aapq * work[q] / work[p]; saxpy_(m, &temp1, &work[*n + 1], & c__1, &a[p * a_dim1 + 1], &c__1); slascl_("G", &c__0, &c__0, &c_b18, &aapp, m, &c__1, &a[p * a_dim1 + 1], lda, &ierr); /* Computing MAX */ r__1 = 0.f, r__2 = 1.f - aapq * aapq; sva[p] = aapp * sqrt((f2cmax(r__1, r__2))); mxsinj = f2cmax(mxsinj,sfmin); } } /* END IF ROTOK THEN ... ELSE */ /* In the case of cancellation in updating SVA(q) */ /* Computing 2nd power */ r__1 = sva[q] / aaqq; if (r__1 * r__1 <= rooteps) { if (aaqq < rootbig && aaqq > rootsfmin) { sva[q] = snrm2_(m, &a[q * a_dim1 + 1], &c__1) * work[q]; } else { t = 0.f; aaqq = 1.f; slassq_(m, &a[q * a_dim1 + 1], & c__1, &t, &aaqq); sva[q] = t * sqrt(aaqq) * work[q]; } } /* Computing 2nd power */ r__1 = aapp / aapp0; if (r__1 * r__1 <= rooteps) { if (aapp < rootbig && aapp > rootsfmin) { aapp = snrm2_(m, &a[p * a_dim1 + 1], &c__1) * work[p]; } else { t = 0.f; aapp = 1.f; slassq_(m, &a[p * a_dim1 + 1], & c__1, &t, &aapp); aapp = t * sqrt(aapp) * work[p]; } sva[p] = aapp; } /* end of OK rotation */ } else { ++notrot; /* [RTD] SKIPPED = SKIPPED + 1 */ ++pskipped; ++ijblsk; } } else { ++notrot; ++pskipped; ++ijblsk; } if (i__ <= swband && ijblsk >= blskip) { sva[p] = aapp; notrot = 0; goto L2011; } if (i__ <= swband && pskipped > rowskip) { aapp = -aapp; notrot = 0; goto L2203; } /* L2200: */ } /* end of the q-loop */ L2203: sva[p] = aapp; } else { if (aapp == 0.f) { /* Computing MIN */ i__4 = jgl + kbl - 1; notrot = notrot + f2cmin(i__4,*n) - jgl + 1; } if (aapp < 0.f) { notrot = 0; } } /* L2100: */ } /* end of the p-loop */ /* L2010: */ } /* end of the jbc-loop */ L2011: /* 2011 bailed out of the jbc-loop */ /* Computing MIN */ i__3 = igl + kbl - 1; i__2 = f2cmin(i__3,*n); for (p = igl; p <= i__2; ++p) { sva[p] = (r__1 = sva[p], abs(r__1)); /* L2012: */ } /* ** */ /* L2000: */ } /* 2000 :: end of the ibr-loop */ if (sva[*n] < rootbig && sva[*n] > rootsfmin) { sva[*n] = snrm2_(m, &a[*n * a_dim1 + 1], &c__1) * work[*n]; } else { t = 0.f; aapp = 1.f; slassq_(m, &a[*n * a_dim1 + 1], &c__1, &t, &aapp); sva[*n] = t * sqrt(aapp) * work[*n]; } /* Additional steering devices */ if (i__ < swband && (mxaapq <= roottol || iswrot <= *n)) { swband = i__; } if (i__ > swband + 1 && mxaapq < sqrt((real) (*n)) * tol && (real) (* n) * mxaapq * mxsinj < tol) { goto L1994; } if (notrot >= emptsw) { goto L1994; } /* L1993: */ } /* end i=1:NSWEEP loop */ /* #:( Reaching this point means that the procedure has not converged. */ *info = 29; goto L1995; L1994: /* #:) Reaching this point means numerical convergence after the i-th */ /* sweep. */ *info = 0; /* #:) INFO = 0 confirms successful iterations. */ L1995: /* Sort the singular values and find how many are above */ /* the underflow threshold. */ n2 = 0; n4 = 0; i__1 = *n - 1; for (p = 1; p <= i__1; ++p) { i__2 = *n - p + 1; q = isamax_(&i__2, &sva[p], &c__1) + p - 1; if (p != q) { temp1 = sva[p]; sva[p] = sva[q]; sva[q] = temp1; temp1 = work[p]; work[p] = work[q]; work[q] = temp1; sswap_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1); if (rsvec) { sswap_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], & c__1); } } if (sva[p] != 0.f) { ++n4; if (sva[p] * skl > sfmin) { ++n2; } } /* L5991: */ } if (sva[*n] != 0.f) { ++n4; if (sva[*n] * skl > sfmin) { ++n2; } } /* Normalize the left singular vectors. */ if (lsvec || uctol) { i__1 = n2; for (p = 1; p <= i__1; ++p) { r__1 = work[p] / sva[p]; sscal_(m, &r__1, &a[p * a_dim1 + 1], &c__1); /* L1998: */ } } /* Scale the product of Jacobi rotations (assemble the fast rotations). */ if (rsvec) { if (applv) { i__1 = *n; for (p = 1; p <= i__1; ++p) { sscal_(&mvl, &work[p], &v[p * v_dim1 + 1], &c__1); /* L2398: */ } } else { i__1 = *n; for (p = 1; p <= i__1; ++p) { temp1 = 1.f / snrm2_(&mvl, &v[p * v_dim1 + 1], &c__1); sscal_(&mvl, &temp1, &v[p * v_dim1 + 1], &c__1); /* L2399: */ } } } /* Undo scaling, if necessary (and possible). */ if (skl > 1.f && sva[1] < big / skl || skl < 1.f && sva[f2cmax(n2,1)] > sfmin / skl) { i__1 = *n; for (p = 1; p <= i__1; ++p) { sva[p] = skl * sva[p]; /* L2400: */ } skl = 1.f; } work[1] = skl; /* The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE */ /* then some of the singular values may overflow or underflow and */ /* the spectrum is given in this factored representation. */ work[2] = (real) n4; /* N4 is the number of computed nonzero singular values of A. */ work[3] = (real) n2; /* N2 is the number of singular values of A greater than SFMIN. */ /* If N2