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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_slasq4 (i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1, dn2, tau, ttype, g) |
| | SLASQ4: computes an approximation TAU to the smallest eigenvalue using values of d from the previous transform.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_dlasq4 (i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1, dn2, tau, ttype, g) |
| | DLASQ4: computes an approximation TAU to the smallest eigenvalue using values of d from the previous transform.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_qlasq4 (i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1, dn2, tau, ttype, g) |
| | QLASQ4: computes an approximation TAU to the smallest eigenvalue using values of d from the previous transform.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_slasq5 (i0, n0, z, pp, tau, sigma, dmin, dmin1, dmin2, dn, dnm1, dnm2, ieee, eps) |
| | SLASQ5: computes one dqds transform in ping-pong form, one version for IEEE machines another for non IEEE machines.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_dlasq5 (i0, n0, z, pp, tau, sigma, dmin, dmin1, dmin2, dn, dnm1, dnm2, ieee, eps) |
| | DLASQ5: computes one dqds transform in ping-pong form, one version for IEEE machines another for non IEEE machines.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_qlasq5 (i0, n0, z, pp, tau, sigma, dmin, dmin1, dmin2, dn, dnm1, dnm2, ieee, eps) |
| | QLASQ5: computes one dqds transform in ping-pong form, one version for IEEE machines another for non IEEE machines.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_slasq6 (i0, n0, z, pp, dmin, dmin1, dmin2, dn, dnm1, dnm2) |
| | SLASQ6: computes one dqd (shift equal to zero) transform in ping-pong form, with protection against underflow and overflow.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_dlasq6 (i0, n0, z, pp, dmin, dmin1, dmin2, dn, dnm1, dnm2) |
| | DLASQ6: computes one dqd (shift equal to zero) transform in ping-pong form, with protection against underflow and overflow.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_qlasq6 (i0, n0, z, pp, dmin, dmin1, dmin2, dn, dnm1, dnm2) |
| | QLASQ6: computes one dqd (shift equal to zero) transform in ping-pong form, with protection against underflow and overflow.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_slasq3 (i0, n0, z, pp, dmin, sigma, desig, qmax, nfail, iter, ndiv, ieee, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau) |
| | SLASQ3: checks for deflation, computes a shift (TAU) and calls dqds. In case of failure it changes shifts, and tries again until output is positive.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_dlasq3 (i0, n0, z, pp, dmin, sigma, desig, qmax, nfail, iter, ndiv, ieee, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau) |
| | DLASQ3: checks for deflation, computes a shift (TAU) and calls dqds. In case of failure it changes shifts, and tries again until output is positive.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_qlasq3 (i0, n0, z, pp, dmin, sigma, desig, qmax, nfail, iter, ndiv, ieee, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau) |
| | QLASQ3: checks for deflation, computes a shift (TAU) and calls dqds. In case of failure it changes shifts, and tries again until output is positive.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_sbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info) |
| | SBDSQR: 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**T 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**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by SGEBRD, then A = (U*Q) * S * (P**T*VT) is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real 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.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_dbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info) |
| | DBDSQR: 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**T 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**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by DGEBRD, then A = (U*Q) * S * (P**T*VT) is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real 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.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_qbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info) |
| | QBDSQR: 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**T 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**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by QGEBRD, then A = (U*Q) * S * (P**T*VT) is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real 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.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_slasq1 (n, d, e, work, info) |
| | SLASQ1: computes the singular values of a real N-by-N bidiagonal matrix with diagonal D and off-diagonal E. The singular values are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. The algorithm was first presented in "Accurate singular values and differential qd algorithms" by K. V. Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, 1994, and the present implementation is described in "An implementation of
the dqds Algorithm (Positive Case)", LAPACK Working Note.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_dlasq1 (n, d, e, work, info) |
| | DLASQ1: computes the singular values of a real N-by-N bidiagonal matrix with diagonal D and off-diagonal E. The singular values are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. The algorithm was first presented in "Accurate singular values and differential qd algorithms" by K. V. Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, 1994, and the present implementation is described in "An implementation of
the dqds Algorithm (Positive Case)", LAPACK Working Note.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_qlasq1 (n, d, e, work, info) |
| | QLASQ1: computes the singular values of a real N-by-N bidiagonal matrix with diagonal D and off-diagonal E. The singular values are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. The algorithm was first presented in "Accurate singular values and differential qd algorithms" by K. V. Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, 1994, and the present implementation is described in "An implementation of
the dqds Algorithm (Positive Case)", LAPACK Working Note.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_slasq2 (n, z, info) |
| | SLASQ2: computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. To see the relation of Z to the tridiagonal matrix, let L be a unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and let U be an upper bidiagonal matrix with 1's above and diagonal Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the symmetric tridiagonal to which it is similar. Note : SLASQ2 defines a logical variable, IEEE, which is true on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs, and false otherwise. This variable is passed to SLASQ3.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_dlasq2 (n, z, info) |
| | DLASQ2: computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. To see the relation of Z to the tridiagonal matrix, let L be a unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and let U be an upper bidiagonal matrix with 1's above and diagonal Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the symmetric tridiagonal to which it is similar. Note : DLASQ2 defines a logical variable, IEEE, which is true on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs, and false otherwise. This variable is passed to DLASQ3.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_qlasq2 (n, z, info) |
| | QLASQ2: computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. To see the relation of Z to the tridiagonal matrix, let L be a unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and let U be an upper bidiagonal matrix with 1's above and diagonal Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the symmetric tridiagonal to which it is similar. Note : QLASQ2 defines a logical variable, IEEE, which is true on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs, and false otherwise. This variable is passed to QLASQ3.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_cbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info) |
| | 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.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_zbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info) |
| | 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.
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| pure subroutine, public | la_lapack_svd_bidiag_qr::la_wbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info) |
| | WBDSQR: 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 WGEBRD, 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.
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1.13.2