#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 SBDSVDX */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SBDSVDX + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SBDSVDX( UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU, */ /* $ NS, S, Z, LDZ, WORK, IWORK, INFO ) */ /* CHARACTER JOBZ, RANGE, UPLO */ /* INTEGER IL, INFO, IU, LDZ, N, NS */ /* REAL VL, VU */ /* INTEGER IWORK( * ) */ /* REAL D( * ), E( * ), S( * ), WORK( * ), */ /* Z( LDZ, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > SBDSVDX computes the singular value decomposition (SVD) of a real */ /* > N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, */ /* > where S is a diagonal matrix with non-negative diagonal elements */ /* > (the singular values of B), and U and VT are orthogonal matrices */ /* > of left and right singular vectors, respectively. */ /* > */ /* > Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] */ /* > and superdiagonal E = [ e_1 e_2 ... e_N-1 ], SBDSVDX computes the */ /* > singular value decompositon of B through the eigenvalues and */ /* > eigenvectors of the N*2-by-N*2 tridiagonal matrix */ /* > */ /* > | 0 d_1 | */ /* > | d_1 0 e_1 | */ /* > TGK = | e_1 0 d_2 | */ /* > | d_2 . . | */ /* > | . . . | */ /* > */ /* > If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then */ /* > (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / */ /* > sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and */ /* > P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ]. */ /* > */ /* > Given a TGK matrix, one can either a) compute -s,-v and change signs */ /* > so that the singular values (and corresponding vectors) are already in */ /* > descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder */ /* > the values (and corresponding vectors). SBDSVDX implements a) by */ /* > calling SSTEVX (bisection plus inverse iteration, to be replaced */ /* > with a version of the Multiple Relative Robust Representation */ /* > algorithm. (See P. Willems and B. Lang, A framework for the MR^3 */ /* > algorithm: theory and implementation, SIAM J. Sci. Comput., */ /* > 35:740-766, 2013.) */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] UPLO */ /* > \verbatim */ /* > UPLO is CHARACTER*1 */ /* > = 'U': B is upper bidiagonal; */ /* > = 'L': B is lower bidiagonal. */ /* > \endverbatim */ /* > */ /* > \param[in] JOBZ */ /* > \verbatim */ /* > JOBZ is CHARACTER*1 */ /* > = 'N': Compute singular values only; */ /* > = 'V': Compute singular values and singular vectors. */ /* > \endverbatim */ /* > */ /* > \param[in] RANGE */ /* > \verbatim */ /* > RANGE is CHARACTER*1 */ /* > = 'A': all singular values will be found. */ /* > = 'V': all singular values in the half-open interval [VL,VU) */ /* > will be found. */ /* > = 'I': the IL-th through IU-th singular values will be found. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the bidiagonal matrix. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] D */ /* > \verbatim */ /* > D is REAL array, dimension (N) */ /* > The n diagonal elements of the bidiagonal matrix B. */ /* > \endverbatim */ /* > */ /* > \param[in] E */ /* > \verbatim */ /* > E is REAL array, dimension (f2cmax(1,N-1)) */ /* > The (n-1) superdiagonal elements of the bidiagonal matrix */ /* > B in elements 1 to N-1. */ /* > \endverbatim */ /* > */ /* > \param[in] VL */ /* > \verbatim */ /* > VL is REAL */ /* > If RANGE='V', the lower bound of the interval to */ /* > be searched for singular values. VU > VL. */ /* > Not referenced if RANGE = 'A' or 'I'. */ /* > \endverbatim */ /* > */ /* > \param[in] VU */ /* > \verbatim */ /* > VU is REAL */ /* > If RANGE='V', the upper bound of the interval to */ /* > be searched for singular values. VU > VL. */ /* > Not referenced if RANGE = 'A' or 'I'. */ /* > \endverbatim */ /* > */ /* > \param[in] IL */ /* > \verbatim */ /* > IL is INTEGER */ /* > If RANGE='I', the index of the */ /* > smallest singular value to be returned. */ /* > 1 <= IL <= IU <= f2cmin(M,N), if f2cmin(M,N) > 0. */ /* > Not referenced if RANGE = 'A' or 'V'. */ /* > \endverbatim */ /* > */ /* > \param[in] IU */ /* > \verbatim */ /* > IU is INTEGER */ /* > If RANGE='I', the index of the */ /* > largest singular value to be returned. */ /* > 1 <= IL <= IU <= f2cmin(M,N), if f2cmin(M,N) > 0. */ /* > Not referenced if RANGE = 'A' or 'V'. */ /* > \endverbatim */ /* > */ /* > \param[out] NS */ /* > \verbatim */ /* > NS is INTEGER */ /* > The total number of singular values found. 0 <= NS <= N. */ /* > If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1. */ /* > \endverbatim */ /* > */ /* > \param[out] S */ /* > \verbatim */ /* > S is REAL array, dimension (N) */ /* > The first NS elements contain the selected singular values in */ /* > ascending order. */ /* > \endverbatim */ /* > */ /* > \param[out] Z */ /* > \verbatim */ /* > Z is REAL array, dimension (2*N,K) */ /* > If JOBZ = 'V', then if INFO = 0 the first NS columns of Z */ /* > contain the singular vectors of the matrix B corresponding to */ /* > the selected singular values, with U in rows 1 to N and V */ /* > in rows N+1 to N*2, i.e. */ /* > Z = [ U ] */ /* > [ V ] */ /* > If JOBZ = 'N', then Z is not referenced. */ /* > Note: The user must ensure that at least K = NS+1 columns are */ /* > supplied in the array Z; if RANGE = 'V', the exact value of */ /* > NS is not known in advance and an upper bound must be used. */ /* > \endverbatim */ /* > */ /* > \param[in] LDZ */ /* > \verbatim */ /* > LDZ is INTEGER */ /* > The leading dimension of the array Z. LDZ >= 1, and if */ /* > JOBZ = 'V', LDZ >= f2cmax(2,N*2). */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is REAL array, dimension (14*N) */ /* > \endverbatim */ /* > */ /* > \param[out] IWORK */ /* > \verbatim */ /* > IWORK is INTEGER array, dimension (12*N) */ /* > If JOBZ = 'V', then if INFO = 0, the first NS elements of */ /* > IWORK are zero. If INFO > 0, then IWORK contains the indices */ /* > of the eigenvectors that failed to converge in DSTEVX. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > < 0: if INFO = -i, the i-th argument had an illegal value */ /* > > 0: if INFO = i, then i eigenvectors failed to converge */ /* > in SSTEVX. The indices of the eigenvectors */ /* > (as returned by SSTEVX) are stored in the */ /* > array IWORK. */ /* > if INFO = N*2 + 1, an internal error occurred. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date June 2016 */ /* > \ingroup realOTHEReigen */ /* ===================================================================== */ /* Subroutine */ int sbdsvdx_(char *uplo, char *jobz, char *range, integer *n, real *d__, real *e, real *vl, real *vu, integer *il, integer *iu, integer *ns, real *s, real *z__, integer *ldz, real *work, integer * iwork, integer *info) { /* System generated locals */ integer z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5; real r__1, r__2, r__3, r__4; doublereal d__1; /* Local variables */ real emin; integer ntgk; real smin, smax; extern real sdot_(integer *, real *, integer *, real *, integer *); real nrmu, nrmv; logical sveq0; extern real snrm2_(integer *, real *, integer *); integer i__, idbeg, j, k; real sqrt2; integer idend, isbeg; extern logical lsame_(char *, char *); integer idtgk, ietgk; extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); integer iltgk, itemp, icolz; logical allsv; integer idptr; logical indsv; integer ieptr, iutgk; real vltgk; logical lower; real zjtji; logical split, valsv; integer isplt; real ortol, vutgk; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *), sswap_(integer *, real *, integer *, real *, integer * ); logical wantz; char rngvx[1]; integer irowu, irowv; extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *, real *, integer *); integer irowz, iifail; real mu; extern real slamch_(char *); extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen); extern integer isamax_(integer *, real *, integer *); real abstol; extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *, real *, real *, integer *); real thresh; integer iiwork; extern /* Subroutine */ int mecago_(), sstevx_(char *, char *, integer *, real *, real *, real *, real *, integer *, integer *, real *, integer *, real *, real *, integer *, real *, integer *, integer *, integer *); real eps; integer nsl; real tol, ulp; integer nru, nrv; /* -- LAPACK driver routine (version 3.8.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* November 2017 */ /* ===================================================================== */ /* Test the input parameters. */ /* Parameter adjustments */ --d__; --e; --s; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; --work; --iwork; /* Function Body */ allsv = lsame_(range, "A"); valsv = lsame_(range, "V"); indsv = lsame_(range, "I"); wantz = lsame_(jobz, "V"); lower = lsame_(uplo, "L"); *info = 0; if (! lsame_(uplo, "U") && ! lower) { *info = -1; } else if (! (wantz || lsame_(jobz, "N"))) { *info = -2; } else if (! (allsv || valsv || indsv)) { *info = -3; } else if (*n < 0) { *info = -4; } else if (*n > 0) { if (valsv) { if (*vl < 0.f) { *info = -7; } else if (*vu <= *vl) { *info = -8; } } else if (indsv) { if (*il < 1 || *il > f2cmax(1,*n)) { *info = -9; } else if (*iu < f2cmin(*n,*il) || *iu > *n) { *info = -10; } } } if (*info == 0) { if (*ldz < 1 || wantz && *ldz < *n << 1) { *info = -14; } } if (*info != 0) { i__1 = -(*info); xerbla_("SBDSVDX", &i__1, (ftnlen)7); return 0; } /* Quick return if possible (N.LE.1) */ *ns = 0; if (*n == 0) { return 0; } if (*n == 1) { if (allsv || indsv) { *ns = 1; s[1] = abs(d__[1]); } else { if (*vl < abs(d__[1]) && *vu >= abs(d__[1])) { *ns = 1; s[1] = abs(d__[1]); } } if (wantz) { z__[z_dim1 + 1] = r_sign(&c_b10, &d__[1]); z__[z_dim1 + 2] = 1.f; } return 0; } abstol = slamch_("Safe Minimum") * 2; ulp = slamch_("Precision"); eps = slamch_("Epsilon"); sqrt2 = sqrt(2.f); ortol = sqrt(ulp); /* Criterion for splitting is taken from SBDSQR when singular */ /* values are computed to relative accuracy TOL. (See J. Demmel and */ /* W. Kahan, Accurate singular values of bidiagonal matrices, SIAM */ /* J. Sci. and Stat. Comput., 11:873–912, 1990.) */ /* Computing MAX */ /* Computing MIN */ d__1 = (doublereal) eps; r__3 = 100.f, r__4 = pow_dd(&d__1, &c_b14); r__1 = 10.f, r__2 = f2cmin(r__3,r__4); tol = f2cmax(r__1,r__2) * eps; /* Compute approximate maximum, minimum singular values. */ i__ = isamax_(n, &d__[1], &c__1); smax = (r__1 = d__[i__], abs(r__1)); i__1 = *n - 1; i__ = isamax_(&i__1, &e[1], &c__1); /* Computing MAX */ r__2 = smax, r__3 = (r__1 = e[i__], abs(r__1)); smax = f2cmax(r__2,r__3); /* Compute threshold for neglecting D's and E's. */ smin = abs(d__[1]); if (smin != 0.f) { mu = smin; i__1 = *n; for (i__ = 2; i__ <= i__1; ++i__) { mu = (r__2 = d__[i__], abs(r__2)) * (mu / (mu + (r__1 = e[i__ - 1] , abs(r__1)))); smin = f2cmin(smin,mu); if (smin == 0.f) { myexit_(); } } } smin /= sqrt((real) (*n)); thresh = tol * smin; /* Check for zeros in D and E (splits), i.e. submatrices. */ i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { if ((r__1 = d__[i__], abs(r__1)) <= thresh) { d__[i__] = 0.f; } if ((r__1 = e[i__], abs(r__1)) <= thresh) { e[i__] = 0.f; } } if ((r__1 = d__[*n], abs(r__1)) <= thresh) { d__[*n] = 0.f; } /* Pointers for arrays used by SSTEVX. */ idtgk = 1; ietgk = idtgk + (*n << 1); itemp = ietgk + (*n << 1); iifail = 1; iiwork = iifail + (*n << 1); /* Set RNGVX, which corresponds to RANGE for SSTEVX in TGK mode. */ /* VL,VU or IL,IU are redefined to conform to implementation a) */ /* described in the leading comments. */ iltgk = 0; iutgk = 0; vltgk = 0.f; vutgk = 0.f; if (allsv) { /* All singular values will be found. We aim at -s (see */ /* leading comments) with RNGVX = 'I'. IL and IU are set */ /* later (as ILTGK and IUTGK) according to the dimension */ /* of the active submatrix. */ *(unsigned char *)rngvx = 'I'; if (wantz) { i__1 = *n << 1; i__2 = *n + 1; slaset_("F", &i__1, &i__2, &c_b19, &c_b19, &z__[z_offset], ldz); } } else if (valsv) { /* Find singular values in a half-open interval. We aim */ /* at -s (see leading comments) and we swap VL and VU */ /* (as VUTGK and VLTGK), changing their signs. */ *(unsigned char *)rngvx = 'V'; vltgk = -(*vu); vutgk = -(*vl); i__1 = idtgk + (*n << 1) - 1; for (i__ = idtgk; i__ <= i__1; ++i__) { work[i__] = 0.f; } /* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */ scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2); i__1 = *n - 1; scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2); i__1 = *n << 1; sstevx_("N", "V", &i__1, &work[idtgk], &work[ietgk], &vltgk, &vutgk, & iltgk, &iltgk, &abstol, ns, &s[1], &z__[z_offset], ldz, &work[ itemp], &iwork[iiwork], &iwork[iifail], info); if (*ns == 0) { return 0; } else { if (wantz) { i__1 = *n << 1; slaset_("F", &i__1, ns, &c_b19, &c_b19, &z__[z_offset], ldz); } } } else if (indsv) { /* Find the IL-th through the IU-th singular values. We aim */ /* at -s (see leading comments) and indices are mapped into */ /* values, therefore mimicking SSTEBZ, where */ /* GL = GL - FUDGE*TNORM*ULP*N - FUDGE*TWO*PIVMIN */ /* GU = GU + FUDGE*TNORM*ULP*N + FUDGE*PIVMIN */ iltgk = *il; iutgk = *iu; *(unsigned char *)rngvx = 'V'; i__1 = idtgk + (*n << 1) - 1; for (i__ = idtgk; i__ <= i__1; ++i__) { work[i__] = 0.f; } /* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */ scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2); i__1 = *n - 1; scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2); i__1 = *n << 1; sstevx_("N", "I", &i__1, &work[idtgk], &work[ietgk], &vltgk, &vltgk, & iltgk, &iltgk, &abstol, ns, &s[1], &z__[z_offset], ldz, &work[ itemp], &iwork[iiwork], &iwork[iifail], info); vltgk = s[1] - smax * 2.f * ulp * *n; i__1 = idtgk + (*n << 1) - 1; for (i__ = idtgk; i__ <= i__1; ++i__) { work[i__] = 0.f; } /* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */ scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2); i__1 = *n - 1; scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2); i__1 = *n << 1; sstevx_("N", "I", &i__1, &work[idtgk], &work[ietgk], &vutgk, &vutgk, & iutgk, &iutgk, &abstol, ns, &s[1], &z__[z_offset], ldz, &work[ itemp], &iwork[iiwork], &iwork[iifail], info); vutgk = s[1] + smax * 2.f * ulp * *n; vutgk = f2cmin(vutgk,0.f); /* If VLTGK=VUTGK, SSTEVX returns an error message, */ /* so if needed we change VUTGK slightly. */ if (vltgk == vutgk) { vltgk -= tol; } if (wantz) { i__1 = *n << 1; i__2 = *iu - *il + 1; slaset_("F", &i__1, &i__2, &c_b19, &c_b19, &z__[z_offset], ldz); } } /* Initialize variables and pointers for S, Z, and WORK. */ /* NRU, NRV: number of rows in U and V for the active submatrix */ /* IDBEG, ISBEG: offsets for the entries of D and S */ /* IROWZ, ICOLZ: offsets for the rows and columns of Z */ /* IROWU, IROWV: offsets for the rows of U and V */ *ns = 0; nru = 0; nrv = 0; idbeg = 1; isbeg = 1; irowz = 1; icolz = 1; irowu = 2; irowv = 1; split = FALSE_; sveq0 = FALSE_; /* Form the tridiagonal TGK matrix. */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { s[i__] = 0.f; } /* S( 1:N ) = ZERO */ work[ietgk + (*n << 1) - 1] = 0.f; i__1 = idtgk + (*n << 1) - 1; for (i__ = idtgk; i__ <= i__1; ++i__) { work[i__] = 0.f; } /* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */ scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2); i__1 = *n - 1; scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2); /* Check for splits in two levels, outer level */ /* in E and inner level in D. */ i__1 = *n << 1; for (ieptr = 2; ieptr <= i__1; ieptr += 2) { if (work[ietgk + ieptr - 1] == 0.f) { /* Split in E (this piece of B is square) or bottom */ /* of the (input bidiagonal) matrix. */ isplt = idbeg; idend = ieptr - 1; i__2 = idend; for (idptr = idbeg; idptr <= i__2; idptr += 2) { if (work[ietgk + idptr - 1] == 0.f) { /* Split in D (rectangular submatrix). Set the number */ /* of rows in U and V (NRU and NRV) accordingly. */ if (idptr == idbeg) { /* D=0 at the top. */ sveq0 = TRUE_; if (idbeg == idend) { nru = 1; nrv = 1; } } else if (idptr == idend) { /* D=0 at the bottom. */ sveq0 = TRUE_; nru = (idend - isplt) / 2 + 1; nrv = nru; if (isplt != idbeg) { ++nru; } } else { if (isplt == idbeg) { /* Split: top rectangular submatrix. */ nru = (idptr - idbeg) / 2; nrv = nru + 1; } else { /* Split: middle square submatrix. */ nru = (idptr - isplt) / 2 + 1; nrv = nru; } } } else if (idptr == idend) { /* Last entry of D in the active submatrix. */ if (isplt == idbeg) { /* No split (trivial case). */ nru = (idend - idbeg) / 2 + 1; nrv = nru; } else { /* Split: bottom rectangular submatrix. */ nrv = (idend - isplt) / 2 + 1; nru = nrv + 1; } } ntgk = nru + nrv; if (ntgk > 0) { /* Compute eigenvalues/vectors of the active */ /* submatrix according to RANGE: */ /* if RANGE='A' (ALLSV) then RNGVX = 'I' */ /* if RANGE='V' (VALSV) then RNGVX = 'V' */ /* if RANGE='I' (INDSV) then RNGVX = 'V' */ iltgk = 1; iutgk = ntgk / 2; if (allsv || vutgk == 0.f) { if (sveq0 || smin < eps || ntgk % 2 > 0) { /* Special case: eigenvalue equal to zero or very */ /* small, additional eigenvector is needed. */ ++iutgk; } } /* Workspace needed by SSTEVX: */ /* WORK( ITEMP: ): 2*5*NTGK */ /* IWORK( 1: ): 2*6*NTGK */ sstevx_(jobz, rngvx, &ntgk, &work[idtgk + isplt - 1], & work[ietgk + isplt - 1], &vltgk, &vutgk, &iltgk, & iutgk, &abstol, &nsl, &s[isbeg], &z__[irowz + icolz * z_dim1], ldz, &work[itemp], &iwork[iiwork] , &iwork[iifail], info); if (*info != 0) { /* Exit with the error code from SSTEVX. */ return 0; } emin = (r__1 = s[isbeg], abs(r__1)); i__3 = isbeg + nsl - 1; for (i__ = isbeg; i__ <= i__3; ++i__) { if ((r__1 = s[i__], abs(r__1)) > emin) { emin = s[i__]; } } /* EMIN = ABS( MAXVAL( S( ISBEG:ISBEG+NSL-1 ) ) ) */ if (nsl > 0 && wantz) { /* Normalize u=Z([2,4,...],:) and v=Z([1,3,...],:), */ /* changing the sign of v as discussed in the leading */ /* comments. The norms of u and v may be (slightly) */ /* different from 1/sqrt(2) if the corresponding */ /* eigenvalues are very small or too close. We check */ /* those norms and, if needed, reorthogonalize the */ /* vectors. */ if (nsl > 1 && vutgk == 0.f && ntgk % 2 == 0 && emin == 0.f && ! split) { /* D=0 at the top or bottom of the active submatrix: */ /* one eigenvalue is equal to zero; concatenate the */ /* eigenvectors corresponding to the two smallest */ /* eigenvalues. */ i__3 = irowz + ntgk - 1; for (i__ = irowz; i__ <= i__3; ++i__) { z__[i__ + (icolz + nsl - 2) * z_dim1] += z__[ i__ + (icolz + nsl - 1) * z_dim1]; z__[i__ + (icolz + nsl - 1) * z_dim1] = 0.f; } /* Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-2 ) = */ /* $ Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-2 ) + */ /* $ Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-1 ) */ /* Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-1 ) = */ /* $ ZERO */ /* IF( IUTGK*2.GT.NTGK ) THEN */ /* Eigenvalue equal to zero or very small. */ /* NSL = NSL - 1 */ /* END IF */ } /* Computing MIN */ i__4 = nsl - 1, i__5 = nru - 1; i__3 = f2cmin(i__4,i__5); for (i__ = 0; i__ <= i__3; ++i__) { nrmu = snrm2_(&nru, &z__[irowu + (icolz + i__) * z_dim1], &c__2); if (nrmu == 0.f) { *info = (*n << 1) + 1; return 0; } r__1 = 1.f / nrmu; sscal_(&nru, &r__1, &z__[irowu + (icolz + i__) * z_dim1], &c__2); if (nrmu != 1.f && (r__1 = nrmu - ortol, abs(r__1) ) * sqrt2 > 1.f) { i__4 = i__ - 1; for (j = 0; j <= i__4; ++j) { zjtji = -sdot_(&nru, &z__[irowu + (icolz + j) * z_dim1], &c__2, &z__[irowu + (icolz + i__) * z_dim1], &c__2); saxpy_(&nru, &zjtji, &z__[irowu + (icolz + j) * z_dim1], &c__2, &z__[irowu + (icolz + i__) * z_dim1], &c__2); } nrmu = snrm2_(&nru, &z__[irowu + (icolz + i__) * z_dim1], &c__2); r__1 = 1.f / nrmu; sscal_(&nru, &r__1, &z__[irowu + (icolz + i__) * z_dim1], &c__2); } } /* Computing MIN */ i__4 = nsl - 1, i__5 = nrv - 1; i__3 = f2cmin(i__4,i__5); for (i__ = 0; i__ <= i__3; ++i__) { nrmv = snrm2_(&nrv, &z__[irowv + (icolz + i__) * z_dim1], &c__2); if (nrmv == 0.f) { *info = (*n << 1) + 1; return 0; } r__1 = -1.f / nrmv; sscal_(&nrv, &r__1, &z__[irowv + (icolz + i__) * z_dim1], &c__2); if (nrmv != 1.f && (r__1 = nrmv - ortol, abs(r__1) ) * sqrt2 > 1.f) { i__4 = i__ - 1; for (j = 0; j <= i__4; ++j) { zjtji = -sdot_(&nrv, &z__[irowv + (icolz + j) * z_dim1], &c__2, &z__[irowv + (icolz + i__) * z_dim1], &c__2); saxpy_(&nru, &zjtji, &z__[irowv + (icolz + j) * z_dim1], &c__2, &z__[irowv + (icolz + i__) * z_dim1], &c__2); } nrmv = snrm2_(&nrv, &z__[irowv + (icolz + i__) * z_dim1], &c__2); r__1 = 1.f / nrmv; sscal_(&nrv, &r__1, &z__[irowv + (icolz + i__) * z_dim1], &c__2); } } if (vutgk == 0.f && idptr < idend && ntgk % 2 > 0) { /* D=0 in the middle of the active submatrix (one */ /* eigenvalue is equal to zero): save the corresponding */ /* eigenvector for later use (when bottom of the */ /* active submatrix is reached). */ split = TRUE_; i__3 = irowz + ntgk - 1; for (i__ = irowz; i__ <= i__3; ++i__) { z__[i__ + (*n + 1) * z_dim1] = z__[i__ + (*ns + nsl) * z_dim1]; z__[i__ + (*ns + nsl) * z_dim1] = 0.f; } /* Z( IROWZ:IROWZ+NTGK-1,N+1 ) = */ /* $ Z( IROWZ:IROWZ+NTGK-1,NS+NSL ) */ /* Z( IROWZ:IROWZ+NTGK-1,NS+NSL ) = */ /* $ ZERO */ } } /* ** WANTZ **! */ nsl = f2cmin(nsl,nru); sveq0 = FALSE_; /* Absolute values of the eigenvalues of TGK. */ i__3 = nsl - 1; for (i__ = 0; i__ <= i__3; ++i__) { s[isbeg + i__] = (r__1 = s[isbeg + i__], abs(r__1)); } /* Update pointers for TGK, S and Z. */ isbeg += nsl; irowz += ntgk; icolz += nsl; irowu = irowz; irowv = irowz + 1; isplt = idptr + 1; *ns += nsl; nru = 0; nrv = 0; } /* ** NTGK.GT.0 **! */ if (irowz < *n << 1 && wantz) { i__3 = irowz - 1; for (i__ = 1; i__ <= i__3; ++i__) { z__[i__ + icolz * z_dim1] = 0.f; } /* Z( 1:IROWZ-1, ICOLZ ) = ZERO */ } } /* ** IDPTR loop **! */ if (split && wantz) { /* Bring back eigenvector corresponding */ /* to eigenvalue equal to zero. */ i__2 = idend - ntgk + 1; for (i__ = idbeg; i__ <= i__2; ++i__) { z__[i__ + (isbeg - 1) * z_dim1] += z__[i__ + (*n + 1) * z_dim1]; z__[i__ + (*n + 1) * z_dim1] = 0.f; } /* Z( IDBEG:IDEND-NTGK+1,ISBEG-1 ) = */ /* $ Z( IDBEG:IDEND-NTGK+1,ISBEG-1 ) + */ /* $ Z( IDBEG:IDEND-NTGK+1,N+1 ) */ /* Z( IDBEG:IDEND-NTGK+1,N+1 ) = 0 */ } --irowv; ++irowu; idbeg = ieptr + 1; sveq0 = FALSE_; split = FALSE_; } /* ** Check for split in E **! */ } /* Sort the singular values into decreasing order (insertion sort on */ /* singular values, but only one transposition per singular vector) */ /* ** IEPTR loop **! */ i__1 = *ns - 1; for (i__ = 1; i__ <= i__1; ++i__) { k = 1; smin = s[1]; i__2 = *ns + 1 - i__; for (j = 2; j <= i__2; ++j) { if (s[j] <= smin) { k = j; smin = s[j]; } } if (k != *ns + 1 - i__) { s[k] = s[*ns + 1 - i__]; s[*ns + 1 - i__] = smin; if (wantz) { i__2 = *n << 1; sswap_(&i__2, &z__[k * z_dim1 + 1], &c__1, &z__[(*ns + 1 - i__) * z_dim1 + 1], &c__1); } } } /* If RANGE=I, check for singular values/vectors to be discarded. */ if (indsv) { k = *iu - *il + 1; if (k < *ns) { i__1 = *ns; for (i__ = k + 1; i__ <= i__1; ++i__) { s[i__] = 0.f; } /* S( K+1:NS ) = ZERO */ if (wantz) { i__1 = *n << 1; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *ns; for (j = k + 1; j <= i__2; ++j) { z__[i__ + j * z_dim1] = 0.f; } } /* Z( 1:N*2,K+1:NS ) = ZERO */ } *ns = k; } } /* Reorder Z: U = Z( 1:N,1:NS ), V = Z( N+1:N*2,1:NS ). */ /* If B is a lower diagonal, swap U and V. */ if (wantz) { i__1 = *ns; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *n << 1; scopy_(&i__2, &z__[i__ * z_dim1 + 1], &c__1, &work[1], &c__1); if (lower) { scopy_(n, &work[2], &c__2, &z__[*n + 1 + i__ * z_dim1], &c__1) ; scopy_(n, &work[1], &c__2, &z__[i__ * z_dim1 + 1], &c__1); } else { scopy_(n, &work[2], &c__2, &z__[i__ * z_dim1 + 1], &c__1); scopy_(n, &work[1], &c__2, &z__[*n + 1 + i__ * z_dim1], &c__1) ; } } } return 0; /* End of SBDSVDX */ } /* sbdsvdx_ */