/* cbdsqr.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static doublereal c_b15 = -.125; static integer c__1 = 1; static real c_b49 = 1.f; static real c_b72 = -1.f; /* Subroutine */ int cbdsqr_(char *uplo, integer *n, integer *ncvt, integer * nru, integer *ncc, real *d__, real *e, complex *vt, integer *ldvt, complex *u, integer *ldu, complex *c__, integer *ldc, real *rwork, integer *info) { /* System generated locals */ integer c_dim1, c_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2; real r__1, r__2, r__3, r__4; doublereal d__1; /* Builtin functions */ double pow_dd(doublereal *, doublereal *), sqrt(doublereal), r_sign(real * , real *); /* Local variables */ real f, g, h__; integer i__, j, m; real r__, cs; integer ll; real sn, mu; integer nm1, nm12, nm13, lll; real eps, sll, tol, abse; integer idir; real abss; integer oldm; real cosl; integer isub, iter; real unfl, sinl, cosr, smin, smax, sinr; extern /* Subroutine */ int slas2_(real *, real *, real *, real *, real *) ; extern logical lsame_(char *, char *); real oldcs; extern /* Subroutine */ int clasr_(char *, char *, char *, integer *, integer *, real *, real *, complex *, integer *); integer oldll; real shift, sigmn, oldsn; extern /* Subroutine */ int cswap_(integer *, complex *, integer *, complex *, integer *); integer maxit; real sminl, sigmx; logical lower; extern /* Subroutine */ int csrot_(integer *, complex *, integer *, complex *, integer *, real *, real *), slasq1_(integer *, real *, real *, real *, integer *), slasv2_(real *, real *, real *, real * , real *, real *, real *, real *, real *); extern doublereal slamch_(char *); extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer *), xerbla_(char *, integer *); real sminoa; extern /* Subroutine */ int slartg_(real *, real *, real *, real *, real * ); real thresh; logical rotate; real tolmul; /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* CBDSQR 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 CGEBRD, 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. */ /* Arguments */ /* ========= */ /* UPLO (input) CHARACTER*1 */ /* = 'U': B is upper bidiagonal; */ /* = 'L': B is lower bidiagonal. */ /* N (input) INTEGER */ /* The order of the matrix B. N >= 0. */ /* NCVT (input) INTEGER */ /* The number of columns of the matrix VT. NCVT >= 0. */ /* NRU (input) INTEGER */ /* The number of rows of the matrix U. NRU >= 0. */ /* NCC (input) INTEGER */ /* The number of columns of the matrix C. NCC >= 0. */ /* D (input/output) REAL array, dimension (N) */ /* On entry, the n diagonal elements of the bidiagonal matrix B. */ /* On exit, if INFO=0, the singular values of B in decreasing */ /* order. */ /* E (input/output) REAL 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. */ /* VT (input/output) COMPLEX 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. */ /* LDVT (input) INTEGER */ /* The leading dimension of the array VT. */ /* LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. */ /* U (input/output) COMPLEX 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. */ /* LDU (input) INTEGER */ /* The leading dimension of the array U. LDU >= max(1,NRU). */ /* C (input/output) COMPLEX 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. */ /* LDC (input) INTEGER */ /* The leading dimension of the array C. */ /* LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. */ /* RWORK (workspace) REAL array, dimension (2*N) */ /* if NCVT = NRU = NCC = 0, (max(1, 4*N-4)) otherwise */ /* INFO (output) 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. */ /* Internal Parameters */ /* =================== */ /* TOLMUL REAL, default = max(10,min(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. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Test the input parameters. */ /* Parameter adjustments */ --d__; --e; vt_dim1 = *ldvt; vt_offset = 1 + vt_dim1; vt -= vt_offset; u_dim1 = *ldu; u_offset = 1 + u_dim1; u -= u_offset; c_dim1 = *ldc; c_offset = 1 + c_dim1; 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 < max(1,*n)) { *info = -9; } else if (*ldu < max(1,*nru)) { *info = -11; } else if (*ncc == 0 && *ldc < 1 || *ncc > 0 && *ldc < max(1,*n)) { *info = -13; } if (*info != 0) { i__1 = -(*info); xerbla_("CBDSQR", &i__1); 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) { slasq1_(n, &d__[1], &e[1], &rwork[1], info); return 0; } nm1 = *n - 1; nm12 = nm1 + nm1; nm13 = nm12 + nm1; idir = 0; /* Get machine constants */ eps = slamch_("Epsilon"); unfl = slamch_("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__) { slartg_(&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) { clasr_("R", "V", "F", nru, n, &rwork[1], &rwork[*n], &u[u_offset], ldu); } if (*ncc > 0) { clasr_("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__1 = (doublereal) eps; r__3 = 100.f, r__4 = pow_dd(&d__1, &c_b15); r__1 = 10.f, r__2 = dmin(r__3,r__4); tolmul = dmax(r__1,r__2); tol = tolmul * eps; /* Compute approximate maximum, minimum singular values */ smax = 0.f; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { /* Computing MAX */ r__2 = smax, r__3 = (r__1 = d__[i__], dabs(r__1)); smax = dmax(r__2,r__3); /* L20: */ } i__1 = *n - 1; for (i__ = 1; i__ <= i__1; ++i__) { /* Computing MAX */ r__2 = smax, r__3 = (r__1 = e[i__], dabs(r__1)); smax = dmax(r__2,r__3); /* L30: */ } sminl = 0.f; if (tol >= 0.f) { /* Relative accuracy desired */ sminoa = dabs(d__[1]); if (sminoa == 0.f) { goto L50; } mu = sminoa; i__1 = *n; for (i__ = 2; i__ <= i__1; ++i__) { mu = (r__2 = d__[i__], dabs(r__2)) * (mu / (mu + (r__1 = e[i__ - 1], dabs(r__1)))); sminoa = dmin(sminoa,mu); if (sminoa == 0.f) { goto L50; } /* L40: */ } L50: sminoa /= sqrt((real) (*n)); /* Computing MAX */ r__1 = tol * sminoa, r__2 = *n * 6 * *n * unfl; thresh = dmax(r__1,r__2); } else { /* Absolute accuracy desired */ /* Computing MAX */ r__1 = dabs(tol) * smax, r__2 = *n * 6 * *n * unfl; thresh = dmax(r__1,r__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.f && (r__1 = d__[m], dabs(r__1)) <= thresh) { d__[m] = 0.f; } smax = (r__1 = d__[m], dabs(r__1)); smin = smax; i__1 = m - 1; for (lll = 1; lll <= i__1; ++lll) { ll = m - lll; abss = (r__1 = d__[ll], dabs(r__1)); abse = (r__1 = e[ll], dabs(r__1)); if (tol < 0.f && abss <= thresh) { d__[ll] = 0.f; } if (abse <= thresh) { goto L80; } smin = dmin(smin,abss); /* Computing MAX */ r__1 = max(smax,abss); smax = dmax(r__1,abse); /* L70: */ } ll = 0; goto L90; L80: e[ll] = 0.f; /* 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 */ slasv2_(&d__[m - 1], &e[m - 1], &d__[m], &sigmn, &sigmx, &sinr, &cosr, &sinl, &cosl); d__[m - 1] = sigmx; e[m - 1] = 0.f; d__[m] = sigmn; /* Compute singular vectors, if desired */ if (*ncvt > 0) { csrot_(ncvt, &vt[m - 1 + vt_dim1], ldvt, &vt[m + vt_dim1], ldvt, & cosr, &sinr); } if (*nru > 0) { csrot_(nru, &u[(m - 1) * u_dim1 + 1], &c__1, &u[m * u_dim1 + 1], & c__1, &cosl, &sinl); } if (*ncc > 0) { csrot_(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 ((r__1 = d__[ll], dabs(r__1)) >= (r__2 = d__[m], dabs(r__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 ((r__2 = e[m - 1], dabs(r__2)) <= dabs(tol) * (r__1 = d__[m], dabs( r__1)) || tol < 0.f && (r__3 = e[m - 1], dabs(r__3)) <= thresh) { e[m - 1] = 0.f; goto L60; } if (tol >= 0.f) { /* If relative accuracy desired, */ /* apply convergence criterion forward */ mu = (r__1 = d__[ll], dabs(r__1)); sminl = mu; i__1 = m - 1; for (lll = ll; lll <= i__1; ++lll) { if ((r__1 = e[lll], dabs(r__1)) <= tol * mu) { e[lll] = 0.f; goto L60; } mu = (r__2 = d__[lll + 1], dabs(r__2)) * (mu / (mu + (r__1 = e[lll], dabs(r__1)))); sminl = dmin(sminl,mu); /* L100: */ } } } else { /* Run convergence test in backward direction */ /* First apply standard test to top of matrix */ if ((r__2 = e[ll], dabs(r__2)) <= dabs(tol) * (r__1 = d__[ll], dabs( r__1)) || tol < 0.f && (r__3 = e[ll], dabs(r__3)) <= thresh) { e[ll] = 0.f; goto L60; } if (tol >= 0.f) { /* If relative accuracy desired, */ /* apply convergence criterion backward */ mu = (r__1 = d__[m], dabs(r__1)); sminl = mu; i__1 = ll; for (lll = m - 1; lll >= i__1; --lll) { if ((r__1 = e[lll], dabs(r__1)) <= tol * mu) { e[lll] = 0.f; goto L60; } mu = (r__2 = d__[lll], dabs(r__2)) * (mu / (mu + (r__1 = e[ lll], dabs(r__1)))); sminl = dmin(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 */ r__1 = eps, r__2 = tol * .01f; if (tol >= 0.f && *n * tol * (sminl / smax) <= dmax(r__1,r__2)) { /* Use a zero shift to avoid loss of relative accuracy */ shift = 0.f; } else { /* Compute the shift from 2-by-2 block at end of matrix */ if (idir == 1) { sll = (r__1 = d__[ll], dabs(r__1)); slas2_(&d__[m - 1], &e[m - 1], &d__[m], &shift, &r__); } else { sll = (r__1 = d__[m], dabs(r__1)); slas2_(&d__[ll], &e[ll], &d__[ll + 1], &shift, &r__); } /* Test if shift negligible, and if so set to zero */ if (sll > 0.f) { /* Computing 2nd power */ r__1 = shift / sll; if (r__1 * r__1 < eps) { shift = 0.f; } } } /* Increment iteration count */ iter = iter + m - ll; /* If SHIFT = 0, do simplified QR iteration */ if (shift == 0.f) { if (idir == 1) { /* Chase bulge from top to bottom */ /* Save cosines and sines for later singular vector updates */ cs = 1.f; oldcs = 1.f; i__1 = m - 1; for (i__ = ll; i__ <= i__1; ++i__) { r__1 = d__[i__] * cs; slartg_(&r__1, &e[i__], &cs, &sn, &r__); if (i__ > ll) { e[i__ - 1] = oldsn * r__; } r__1 = oldcs * r__; r__2 = d__[i__ + 1] * sn; slartg_(&r__1, &r__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; clasr_("L", "V", "F", &i__1, ncvt, &rwork[1], &rwork[*n], &vt[ ll + vt_dim1], ldvt); } if (*nru > 0) { i__1 = m - ll + 1; clasr_("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; clasr_("L", "V", "F", &i__1, ncc, &rwork[nm12 + 1], &rwork[ nm13 + 1], &c__[ll + c_dim1], ldc); } /* Test convergence */ if ((r__1 = e[m - 1], dabs(r__1)) <= thresh) { e[m - 1] = 0.f; } } else { /* Chase bulge from bottom to top */ /* Save cosines and sines for later singular vector updates */ cs = 1.f; oldcs = 1.f; i__1 = ll + 1; for (i__ = m; i__ >= i__1; --i__) { r__1 = d__[i__] * cs; slartg_(&r__1, &e[i__ - 1], &cs, &sn, &r__); if (i__ < m) { e[i__] = oldsn * r__; } r__1 = oldcs * r__; r__2 = d__[i__ - 1] * sn; slartg_(&r__1, &r__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; clasr_("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; clasr_("R", "V", "B", nru, &i__1, &rwork[1], &rwork[*n], &u[ ll * u_dim1 + 1], ldu); } if (*ncc > 0) { i__1 = m - ll + 1; clasr_("L", "V", "B", &i__1, ncc, &rwork[1], &rwork[*n], &c__[ ll + c_dim1], ldc); } /* Test convergence */ if ((r__1 = e[ll], dabs(r__1)) <= thresh) { e[ll] = 0.f; } } } else { /* Use nonzero shift */ if (idir == 1) { /* Chase bulge from top to bottom */ /* Save cosines and sines for later singular vector updates */ f = ((r__1 = d__[ll], dabs(r__1)) - shift) * (r_sign(&c_b49, &d__[ ll]) + shift / d__[ll]); g = e[ll]; i__1 = m - 1; for (i__ = ll; i__ <= i__1; ++i__) { slartg_(&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]; slartg_(&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; clasr_("L", "V", "F", &i__1, ncvt, &rwork[1], &rwork[*n], &vt[ ll + vt_dim1], ldvt); } if (*nru > 0) { i__1 = m - ll + 1; clasr_("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; clasr_("L", "V", "F", &i__1, ncc, &rwork[nm12 + 1], &rwork[ nm13 + 1], &c__[ll + c_dim1], ldc); } /* Test convergence */ if ((r__1 = e[m - 1], dabs(r__1)) <= thresh) { e[m - 1] = 0.f; } } else { /* Chase bulge from bottom to top */ /* Save cosines and sines for later singular vector updates */ f = ((r__1 = d__[m], dabs(r__1)) - shift) * (r_sign(&c_b49, &d__[ m]) + shift / d__[m]); g = e[m - 1]; i__1 = ll + 1; for (i__ = m; i__ >= i__1; --i__) { slartg_(&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]; slartg_(&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 ((r__1 = e[ll], dabs(r__1)) <= thresh) { e[ll] = 0.f; } /* Update singular vectors if desired */ if (*ncvt > 0) { i__1 = m - ll + 1; clasr_("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; clasr_("R", "V", "B", nru, &i__1, &rwork[1], &rwork[*n], &u[ ll * u_dim1 + 1], ldu); } if (*ncc > 0) { i__1 = m - ll + 1; clasr_("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.f) { d__[i__] = -d__[i__]; /* Change sign of singular vectors, if desired */ if (*ncvt > 0) { csscal_(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) { cswap_(ncvt, &vt[isub + vt_dim1], ldvt, &vt[*n + 1 - i__ + vt_dim1], ldvt); } if (*nru > 0) { cswap_(nru, &u[isub * u_dim1 + 1], &c__1, &u[(*n + 1 - i__) * u_dim1 + 1], &c__1); } if (*ncc > 0) { cswap_(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.f) { ++(*info); } /* L210: */ } L220: return 0; /* End of CBDSQR */ } /* cbdsqr_ */