#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 ZBDSQR */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download ZBDSQR + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE ZBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, */ /* LDU, C, LDC, RWORK, INFO ) */ /* CHARACTER UPLO */ /* INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU */ /* DOUBLE PRECISION D( * ), E( * ), RWORK( * ) */ /* COMPLEX*16 C( LDC, * ), U( LDU, * ), VT( LDVT, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > ZBDSQR computes the singular values and, optionally, the right and/or */ /* > left singular vectors from the singular value decomposition (SVD) of */ /* > a real N-by-N (upper or lower) bidiagonal matrix B using the implicit */ /* > zero-shift QR algorithm. The SVD of B has the form */ /* > */ /* > B = Q * S * P**H */ /* > */ /* > where S is the diagonal matrix of singular values, Q is an orthogonal */ /* > matrix of left singular vectors, and P is an orthogonal matrix of */ /* > right singular vectors. If left singular vectors are requested, this */ /* > subroutine actually returns U*Q instead of Q, and, if right singular */ /* > vectors are requested, this subroutine returns P**H*VT instead of */ /* > P**H, for given complex input matrices U and VT. When U and VT are */ /* > the unitary matrices that reduce a general matrix A to bidiagonal */ /* > form: A = U*B*VT, as computed by ZGEBRD, then */ /* > */ /* > A = (U*Q) * S * (P**H*VT) */ /* > */ /* > is the SVD of A. Optionally, the subroutine may also compute Q**H*C */ /* > for a given complex input matrix C. */ /* > */ /* > See "Computing Small Singular Values of Bidiagonal Matrices With */ /* > Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, */ /* > LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, */ /* > no. 5, pp. 873-912, Sept 1990) and */ /* > "Accurate singular values and differential qd algorithms," by */ /* > B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics */ /* > Department, University of California at Berkeley, July 1992 */ /* > for a detailed description of the algorithm. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] UPLO */ /* > \verbatim */ /* > UPLO is CHARACTER*1 */ /* > = 'U': B is upper bidiagonal; */ /* > = 'L': B is lower bidiagonal. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the matrix B. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] NCVT */ /* > \verbatim */ /* > NCVT is INTEGER */ /* > The number of columns of the matrix VT. NCVT >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] NRU */ /* > \verbatim */ /* > NRU is INTEGER */ /* > The number of rows of the matrix U. NRU >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] NCC */ /* > \verbatim */ /* > NCC is INTEGER */ /* > The number of columns of the matrix C. NCC >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] D */ /* > \verbatim */ /* > D is DOUBLE PRECISION array, dimension (N) */ /* > On entry, the n diagonal elements of the bidiagonal matrix B. */ /* > On exit, if INFO=0, the singular values of B in decreasing */ /* > order. */ /* > \endverbatim */ /* > */ /* > \param[in,out] E */ /* > \verbatim */ /* > E is DOUBLE PRECISION array, dimension (N-1) */ /* > On entry, the N-1 offdiagonal elements of the bidiagonal */ /* > matrix B. */ /* > On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E */ /* > will contain the diagonal and superdiagonal elements of a */ /* > bidiagonal matrix orthogonally equivalent to the one given */ /* > as input. */ /* > \endverbatim */ /* > */ /* > \param[in,out] VT */ /* > \verbatim */ /* > VT is COMPLEX*16 array, dimension (LDVT, NCVT) */ /* > On entry, an N-by-NCVT matrix VT. */ /* > On exit, VT is overwritten by P**H * VT. */ /* > Not referenced if NCVT = 0. */ /* > \endverbatim */ /* > */ /* > \param[in] LDVT */ /* > \verbatim */ /* > LDVT is INTEGER */ /* > The leading dimension of the array VT. */ /* > LDVT >= f2cmax(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] U */ /* > \verbatim */ /* > U is COMPLEX*16 array, dimension (LDU, N) */ /* > On entry, an NRU-by-N matrix U. */ /* > On exit, U is overwritten by U * Q. */ /* > Not referenced if NRU = 0. */ /* > \endverbatim */ /* > */ /* > \param[in] LDU */ /* > \verbatim */ /* > LDU is INTEGER */ /* > The leading dimension of the array U. LDU >= f2cmax(1,NRU). */ /* > \endverbatim */ /* > */ /* > \param[in,out] C */ /* > \verbatim */ /* > C is COMPLEX*16 array, dimension (LDC, NCC) */ /* > On entry, an N-by-NCC matrix C. */ /* > On exit, C is overwritten by Q**H * C. */ /* > Not referenced if NCC = 0. */ /* > \endverbatim */ /* > */ /* > \param[in] LDC */ /* > \verbatim */ /* > LDC is INTEGER */ /* > The leading dimension of the array C. */ /* > LDC >= f2cmax(1,N) if NCC > 0; LDC >=1 if NCC = 0. */ /* > \endverbatim */ /* > */ /* > \param[out] RWORK */ /* > \verbatim */ /* > RWORK is DOUBLE PRECISION array, dimension (4*N) */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > < 0: If INFO = -i, the i-th argument had an illegal value */ /* > > 0: the algorithm did not converge; D and E contain the */ /* > elements of a bidiagonal matrix which is orthogonally */ /* > similar to the input matrix B; if INFO = i, i */ /* > elements of E have not converged to zero. */ /* > \endverbatim */ /* > \par Internal Parameters: */ /* ========================= */ /* > */ /* > \verbatim */ /* > TOLMUL DOUBLE PRECISION, default = f2cmax(10,f2cmin(100,EPS**(-1/8))) */ /* > TOLMUL controls the convergence criterion of the QR loop. */ /* > If it is positive, TOLMUL*EPS is the desired relative */ /* > precision in the computed singular values. */ /* > If it is negative, abs(TOLMUL*EPS*sigma_max) is the */ /* > desired absolute accuracy in the computed singular */ /* > values (corresponds to relative accuracy */ /* > abs(TOLMUL*EPS) in the largest singular value. */ /* > abs(TOLMUL) should be between 1 and 1/EPS, and preferably */ /* > between 10 (for fast convergence) and .1/EPS */ /* > (for there to be some accuracy in the results). */ /* > Default is to lose at either one eighth or 2 of the */ /* > available decimal digits in each computed singular value */ /* > (whichever is smaller). */ /* > */ /* > MAXITR INTEGER, default = 6 */ /* > MAXITR controls the maximum number of passes of the */ /* > algorithm through its inner loop. The algorithms stops */ /* > (and so fails to converge) if the number of passes */ /* > through the inner loop exceeds MAXITR*N**2. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup complex16OTHERcomputational */ /* ===================================================================== */ /* Subroutine */ int zbdsqr_(char *uplo, integer *n, integer *ncvt, integer * nru, integer *ncc, doublereal *d__, doublereal *e, doublecomplex *vt, integer *ldvt, doublecomplex *u, integer *ldu, doublecomplex *c__, integer *ldc, doublereal *rwork, integer *info) { /* System generated locals */ integer c_dim1, c_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2; doublereal d__1, d__2, d__3, d__4; /* Local variables */ doublereal abse; integer idir; doublereal abss; integer oldm; doublereal cosl; integer isub, iter; doublereal unfl, sinl, cosr, smin, smax, sinr; extern /* Subroutine */ int dlas2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); doublereal f, g, h__; integer i__, j, m; doublereal r__; extern logical lsame_(char *, char *); doublereal oldcs; integer oldll; doublereal shift, sigmn, oldsn; integer maxit; doublereal sminl, sigmx; logical lower; extern /* Subroutine */ int zlasr_(char *, char *, char *, integer *, integer *, doublereal *, doublereal *, doublecomplex *, integer *), zdrot_(integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublereal *) , zswap_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), dlasq1_(integer *, doublereal *, doublereal *, doublereal *, integer *), dlasv2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); doublereal cs; integer ll; extern doublereal dlamch_(char *); doublereal sn, mu; extern /* Subroutine */ int dlartg_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *), xerbla_(char *, integer *, ftnlen), zdscal_(integer *, doublereal *, doublecomplex *, integer *); doublereal sminoa, thresh; logical rotate; integer nm1; doublereal tolmul; integer nm12, nm13, lll; doublereal eps, sll, tol; /* -- LAPACK computational routine (version 3.7.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* December 2016 */ /* ===================================================================== */ /* Test the input parameters. */ /* Parameter adjustments */ --d__; --e; vt_dim1 = *ldvt; vt_offset = 1 + vt_dim1 * 1; vt -= vt_offset; u_dim1 = *ldu; u_offset = 1 + u_dim1 * 1; u -= u_offset; c_dim1 = *ldc; c_offset = 1 + c_dim1 * 1; c__ -= c_offset; --rwork; /* Function Body */ *info = 0; lower = lsame_(uplo, "L"); if (! lsame_(uplo, "U") && ! lower) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*ncvt < 0) { *info = -3; } else if (*nru < 0) { *info = -4; } else if (*ncc < 0) { *info = -5; } else if (*ncvt == 0 && *ldvt < 1 || *ncvt > 0 && *ldvt < f2cmax(1,*n)) { *info = -9; } else if (*ldu < f2cmax(1,*nru)) { *info = -11; } else if (*ncc == 0 && *ldc < 1 || *ncc > 0 && *ldc < f2cmax(1,*n)) { *info = -13; } if (*info != 0) { i__1 = -(*info); xerbla_("ZBDSQR", &i__1, (ftnlen)6); return 0; } if (*n == 0) { return 0; } if (*n == 1) { goto L160; } /* ROTATE is true if any singular vectors desired, false otherwise */ rotate = *ncvt > 0 || *nru > 0 || *ncc > 0; /* If no singular vectors desired, use qd algorithm */ if (! rotate) { dlasq1_(n, &d__[1], &e[1], &rwork[1], info); /* If INFO equals 2, dqds didn't finish, try to finish */ if (*info != 2) { return 0; } *info = 0; } nm1 = *n - 1; nm12 = nm1 + nm1; nm13 = nm12 + nm1; idir = 0; /* Get machine constants */ eps = dlamch_("Epsilon"); unfl = dlamch_("Safe minimum"); /* If matrix lower bidiagonal, rotate to be upper bidiagonal */ /* by applying Givens rotations on the left */ if (lower) { i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { dlartg_(&d__[i__], &e[i__], &cs, &sn, &r__); d__[i__] = r__; e[i__] = sn * d__[i__ + 1]; d__[i__ + 1] = cs * d__[i__ + 1]; rwork[i__] = cs; rwork[nm1 + i__] = sn; /* L10: */ } /* Update singular vectors if desired */ if (*nru > 0) { zlasr_("R", "V", "F", nru, n, &rwork[1], &rwork[*n], &u[u_offset], ldu); } if (*ncc > 0) { zlasr_("L", "V", "F", n, ncc, &rwork[1], &rwork[*n], &c__[ c_offset], ldc); } } /* Compute singular values to relative accuracy TOL */ /* (By setting TOL to be negative, algorithm will compute */ /* singular values to absolute accuracy ABS(TOL)*norm(input matrix)) */ /* Computing MAX */ /* Computing MIN */ d__3 = 100., d__4 = pow_dd(&eps, &c_b15); d__1 = 10., d__2 = f2cmin(d__3,d__4); tolmul = f2cmax(d__1,d__2); tol = tolmul * eps; /* Compute approximate maximum, minimum singular values */ smax = 0.; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { /* Computing MAX */ d__2 = smax, d__3 = (d__1 = d__[i__], abs(d__1)); smax = f2cmax(d__2,d__3); /* L20: */ } i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { /* Computing MAX */ d__2 = smax, d__3 = (d__1 = e[i__], abs(d__1)); smax = f2cmax(d__2,d__3); /* L30: */ } sminl = 0.; if (tol >= 0.) { /* Relative accuracy desired */ sminoa = abs(d__[1]); if (sminoa == 0.) { goto L50; } mu = sminoa; i__1 = *n; for (i__ = 2; i__ <= i__1; ++i__) { mu = (d__2 = d__[i__], abs(d__2)) * (mu / (mu + (d__1 = e[i__ - 1] , abs(d__1)))); sminoa = f2cmin(sminoa,mu); if (sminoa == 0.) { goto L50; } /* L40: */ } L50: sminoa /= sqrt((doublereal) (*n)); /* Computing MAX */ d__1 = tol * sminoa, d__2 = *n * 6 * *n * unfl; thresh = f2cmax(d__1,d__2); } else { /* Absolute accuracy desired */ /* Computing MAX */ d__1 = abs(tol) * smax, d__2 = *n * 6 * *n * unfl; thresh = f2cmax(d__1,d__2); } /* Prepare for main iteration loop for the singular values */ /* (MAXIT is the maximum number of passes through the inner */ /* loop permitted before nonconvergence signalled.) */ maxit = *n * 6 * *n; iter = 0; oldll = -1; oldm = -1; /* M points to last element of unconverged part of matrix */ m = *n; /* Begin main iteration loop */ L60: /* Check for convergence or exceeding iteration count */ if (m <= 1) { goto L160; } if (iter > maxit) { goto L200; } /* Find diagonal block of matrix to work on */ if (tol < 0. && (d__1 = d__[m], abs(d__1)) <= thresh) { d__[m] = 0.; } smax = (d__1 = d__[m], abs(d__1)); smin = smax; i__1 = m - 1; for (lll = 1; lll <= i__1; ++lll) { ll = m - lll; abss = (d__1 = d__[ll], abs(d__1)); abse = (d__1 = e[ll], abs(d__1)); if (tol < 0. && abss <= thresh) { d__[ll] = 0.; } if (abse <= thresh) { goto L80; } smin = f2cmin(smin,abss); /* Computing MAX */ d__1 = f2cmax(smax,abss); smax = f2cmax(d__1,abse); /* L70: */ } ll = 0; goto L90; L80: e[ll] = 0.; /* Matrix splits since E(LL) = 0 */ if (ll == m - 1) { /* Convergence of bottom singular value, return to top of loop */ --m; goto L60; } L90: ++ll; /* E(LL) through E(M-1) are nonzero, E(LL-1) is zero */ if (ll == m - 1) { /* 2 by 2 block, handle separately */ dlasv2_(&d__[m - 1], &e[m - 1], &d__[m], &sigmn, &sigmx, &sinr, &cosr, &sinl, &cosl); d__[m - 1] = sigmx; e[m - 1] = 0.; d__[m] = sigmn; /* Compute singular vectors, if desired */ if (*ncvt > 0) { zdrot_(ncvt, &vt[m - 1 + vt_dim1], ldvt, &vt[m + vt_dim1], ldvt, & cosr, &sinr); } if (*nru > 0) { zdrot_(nru, &u[(m - 1) * u_dim1 + 1], &c__1, &u[m * u_dim1 + 1], & c__1, &cosl, &sinl); } if (*ncc > 0) { zdrot_(ncc, &c__[m - 1 + c_dim1], ldc, &c__[m + c_dim1], ldc, & cosl, &sinl); } m += -2; goto L60; } /* If working on new submatrix, choose shift direction */ /* (from larger end diagonal element towards smaller) */ if (ll > oldm || m < oldll) { if ((d__1 = d__[ll], abs(d__1)) >= (d__2 = d__[m], abs(d__2))) { /* Chase bulge from top (big end) to bottom (small end) */ idir = 1; } else { /* Chase bulge from bottom (big end) to top (small end) */ idir = 2; } } /* Apply convergence tests */ if (idir == 1) { /* Run convergence test in forward direction */ /* First apply standard test to bottom of matrix */ if ((d__2 = e[m - 1], abs(d__2)) <= abs(tol) * (d__1 = d__[m], abs( d__1)) || tol < 0. && (d__3 = e[m - 1], abs(d__3)) <= thresh) { e[m - 1] = 0.; goto L60; } if (tol >= 0.) { /* If relative accuracy desired, */ /* apply convergence criterion forward */ mu = (d__1 = d__[ll], abs(d__1)); sminl = mu; i__1 = m - 1; for (lll = ll; lll <= i__1; ++lll) { if ((d__1 = e[lll], abs(d__1)) <= tol * mu) { e[lll] = 0.; goto L60; } mu = (d__2 = d__[lll + 1], abs(d__2)) * (mu / (mu + (d__1 = e[ lll], abs(d__1)))); sminl = f2cmin(sminl,mu); /* L100: */ } } } else { /* Run convergence test in backward direction */ /* First apply standard test to top of matrix */ if ((d__2 = e[ll], abs(d__2)) <= abs(tol) * (d__1 = d__[ll], abs(d__1) ) || tol < 0. && (d__3 = e[ll], abs(d__3)) <= thresh) { e[ll] = 0.; goto L60; } if (tol >= 0.) { /* If relative accuracy desired, */ /* apply convergence criterion backward */ mu = (d__1 = d__[m], abs(d__1)); sminl = mu; i__1 = ll; for (lll = m - 1; lll >= i__1; --lll) { if ((d__1 = e[lll], abs(d__1)) <= tol * mu) { e[lll] = 0.; goto L60; } mu = (d__2 = d__[lll], abs(d__2)) * (mu / (mu + (d__1 = e[lll] , abs(d__1)))); sminl = f2cmin(sminl,mu); /* L110: */ } } } oldll = ll; oldm = m; /* Compute shift. First, test if shifting would ruin relative */ /* accuracy, and if so set the shift to zero. */ /* Computing MAX */ d__1 = eps, d__2 = tol * .01; if (tol >= 0. && *n * tol * (sminl / smax) <= f2cmax(d__1,d__2)) { /* Use a zero shift to avoid loss of relative accuracy */ shift = 0.; } else { /* Compute the shift from 2-by-2 block at end of matrix */ if (idir == 1) { sll = (d__1 = d__[ll], abs(d__1)); dlas2_(&d__[m - 1], &e[m - 1], &d__[m], &shift, &r__); } else { sll = (d__1 = d__[m], abs(d__1)); dlas2_(&d__[ll], &e[ll], &d__[ll + 1], &shift, &r__); } /* Test if shift negligible, and if so set to zero */ if (sll > 0.) { /* Computing 2nd power */ d__1 = shift / sll; if (d__1 * d__1 < eps) { shift = 0.; } } } /* Increment iteration count */ iter = iter + m - ll; /* If SHIFT = 0, do simplified QR iteration */ if (shift == 0.) { if (idir == 1) { /* Chase bulge from top to bottom */ /* Save cosines and sines for later singular vector updates */ cs = 1.; oldcs = 1.; i__1 = m - 1; for (i__ = ll; i__ <= i__1; ++i__) { d__1 = d__[i__] * cs; dlartg_(&d__1, &e[i__], &cs, &sn, &r__); if (i__ > ll) { e[i__ - 1] = oldsn * r__; } d__1 = oldcs * r__; d__2 = d__[i__ + 1] * sn; dlartg_(&d__1, &d__2, &oldcs, &oldsn, &d__[i__]); rwork[i__ - ll + 1] = cs; rwork[i__ - ll + 1 + nm1] = sn; rwork[i__ - ll + 1 + nm12] = oldcs; rwork[i__ - ll + 1 + nm13] = oldsn; /* L120: */ } h__ = d__[m] * cs; d__[m] = h__ * oldcs; e[m - 1] = h__ * oldsn; /* Update singular vectors */ if (*ncvt > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "F", &i__1, ncvt, &rwork[1], &rwork[*n], &vt[ ll + vt_dim1], ldvt); } if (*nru > 0) { i__1 = m - ll + 1; zlasr_("R", "V", "F", nru, &i__1, &rwork[nm12 + 1], &rwork[ nm13 + 1], &u[ll * u_dim1 + 1], ldu); } if (*ncc > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "F", &i__1, ncc, &rwork[nm12 + 1], &rwork[ nm13 + 1], &c__[ll + c_dim1], ldc); } /* Test convergence */ if ((d__1 = e[m - 1], abs(d__1)) <= thresh) { e[m - 1] = 0.; } } else { /* Chase bulge from bottom to top */ /* Save cosines and sines for later singular vector updates */ cs = 1.; oldcs = 1.; i__1 = ll + 1; for (i__ = m; i__ >= i__1; --i__) { d__1 = d__[i__] * cs; dlartg_(&d__1, &e[i__ - 1], &cs, &sn, &r__); if (i__ < m) { e[i__] = oldsn * r__; } d__1 = oldcs * r__; d__2 = d__[i__ - 1] * sn; dlartg_(&d__1, &d__2, &oldcs, &oldsn, &d__[i__]); rwork[i__ - ll] = cs; rwork[i__ - ll + nm1] = -sn; rwork[i__ - ll + nm12] = oldcs; rwork[i__ - ll + nm13] = -oldsn; /* L130: */ } h__ = d__[ll] * cs; d__[ll] = h__ * oldcs; e[ll] = h__ * oldsn; /* Update singular vectors */ if (*ncvt > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "B", &i__1, ncvt, &rwork[nm12 + 1], &rwork[ nm13 + 1], &vt[ll + vt_dim1], ldvt); } if (*nru > 0) { i__1 = m - ll + 1; zlasr_("R", "V", "B", nru, &i__1, &rwork[1], &rwork[*n], &u[ ll * u_dim1 + 1], ldu); } if (*ncc > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "B", &i__1, ncc, &rwork[1], &rwork[*n], &c__[ ll + c_dim1], ldc); } /* Test convergence */ if ((d__1 = e[ll], abs(d__1)) <= thresh) { e[ll] = 0.; } } } else { /* Use nonzero shift */ if (idir == 1) { /* Chase bulge from top to bottom */ /* Save cosines and sines for later singular vector updates */ f = ((d__1 = d__[ll], abs(d__1)) - shift) * (d_sign(&c_b49, &d__[ ll]) + shift / d__[ll]); g = e[ll]; i__1 = m - 1; for (i__ = ll; i__ <= i__1; ++i__) { dlartg_(&f, &g, &cosr, &sinr, &r__); if (i__ > ll) { e[i__ - 1] = r__; } f = cosr * d__[i__] + sinr * e[i__]; e[i__] = cosr * e[i__] - sinr * d__[i__]; g = sinr * d__[i__ + 1]; d__[i__ + 1] = cosr * d__[i__ + 1]; dlartg_(&f, &g, &cosl, &sinl, &r__); d__[i__] = r__; f = cosl * e[i__] + sinl * d__[i__ + 1]; d__[i__ + 1] = cosl * d__[i__ + 1] - sinl * e[i__]; if (i__ < m - 1) { g = sinl * e[i__ + 1]; e[i__ + 1] = cosl * e[i__ + 1]; } rwork[i__ - ll + 1] = cosr; rwork[i__ - ll + 1 + nm1] = sinr; rwork[i__ - ll + 1 + nm12] = cosl; rwork[i__ - ll + 1 + nm13] = sinl; /* L140: */ } e[m - 1] = f; /* Update singular vectors */ if (*ncvt > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "F", &i__1, ncvt, &rwork[1], &rwork[*n], &vt[ ll + vt_dim1], ldvt); } if (*nru > 0) { i__1 = m - ll + 1; zlasr_("R", "V", "F", nru, &i__1, &rwork[nm12 + 1], &rwork[ nm13 + 1], &u[ll * u_dim1 + 1], ldu); } if (*ncc > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "F", &i__1, ncc, &rwork[nm12 + 1], &rwork[ nm13 + 1], &c__[ll + c_dim1], ldc); } /* Test convergence */ if ((d__1 = e[m - 1], abs(d__1)) <= thresh) { e[m - 1] = 0.; } } else { /* Chase bulge from bottom to top */ /* Save cosines and sines for later singular vector updates */ f = ((d__1 = d__[m], abs(d__1)) - shift) * (d_sign(&c_b49, &d__[m] ) + shift / d__[m]); g = e[m - 1]; i__1 = ll + 1; for (i__ = m; i__ >= i__1; --i__) { dlartg_(&f, &g, &cosr, &sinr, &r__); if (i__ < m) { e[i__] = r__; } f = cosr * d__[i__] + sinr * e[i__ - 1]; e[i__ - 1] = cosr * e[i__ - 1] - sinr * d__[i__]; g = sinr * d__[i__ - 1]; d__[i__ - 1] = cosr * d__[i__ - 1]; dlartg_(&f, &g, &cosl, &sinl, &r__); d__[i__] = r__; f = cosl * e[i__ - 1] + sinl * d__[i__ - 1]; d__[i__ - 1] = cosl * d__[i__ - 1] - sinl * e[i__ - 1]; if (i__ > ll + 1) { g = sinl * e[i__ - 2]; e[i__ - 2] = cosl * e[i__ - 2]; } rwork[i__ - ll] = cosr; rwork[i__ - ll + nm1] = -sinr; rwork[i__ - ll + nm12] = cosl; rwork[i__ - ll + nm13] = -sinl; /* L150: */ } e[ll] = f; /* Test convergence */ if ((d__1 = e[ll], abs(d__1)) <= thresh) { e[ll] = 0.; } /* Update singular vectors if desired */ if (*ncvt > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "B", &i__1, ncvt, &rwork[nm12 + 1], &rwork[ nm13 + 1], &vt[ll + vt_dim1], ldvt); } if (*nru > 0) { i__1 = m - ll + 1; zlasr_("R", "V", "B", nru, &i__1, &rwork[1], &rwork[*n], &u[ ll * u_dim1 + 1], ldu); } if (*ncc > 0) { i__1 = m - ll + 1; zlasr_("L", "V", "B", &i__1, ncc, &rwork[1], &rwork[*n], &c__[ ll + c_dim1], ldc); } } } /* QR iteration finished, go back and check convergence */ goto L60; /* All singular values converged, so make them positive */ L160: i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if (d__[i__] < 0.) { d__[i__] = -d__[i__]; /* Change sign of singular vectors, if desired */ if (*ncvt > 0) { zdscal_(ncvt, &c_b72, &vt[i__ + vt_dim1], ldvt); } } /* L170: */ } /* Sort the singular values into decreasing order (insertion sort on */ /* singular values, but only one transposition per singular vector) */ i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { /* Scan for smallest D(I) */ isub = 1; smin = d__[1]; i__2 = *n + 1 - i__; for (j = 2; j <= i__2; ++j) { if (d__[j] <= smin) { isub = j; smin = d__[j]; } /* L180: */ } if (isub != *n + 1 - i__) { /* Swap singular values and vectors */ d__[isub] = d__[*n + 1 - i__]; d__[*n + 1 - i__] = smin; if (*ncvt > 0) { zswap_(ncvt, &vt[isub + vt_dim1], ldvt, &vt[*n + 1 - i__ + vt_dim1], ldvt); } if (*nru > 0) { zswap_(nru, &u[isub * u_dim1 + 1], &c__1, &u[(*n + 1 - i__) * u_dim1 + 1], &c__1); } if (*ncc > 0) { zswap_(ncc, &c__[isub + c_dim1], ldc, &c__[*n + 1 - i__ + c_dim1], ldc); } } /* L190: */ } goto L220; /* Maximum number of iterations exceeded, failure to converge */ L200: *info = 0; i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { if (e[i__] != 0.) { ++(*info); } /* L210: */ } L220: return 0; /* End of ZBDSQR */ } /* zbdsqr_ */