bdsvdx(3)

📦

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bdsvdx — bidiagonal SVD, bisection 💅

okay so this is the bdsvdx(3) page from the Library Functions Manual, and it's all about computing a bidiagonal SVD by bisection. not me being obsessed with tridiagonal matrices, but here we are 🦋

✨ NAME ✨

bdsvdx — bdsvdx: bidiagonal SVD, bisection 🎀

🌸 SYNOPSIS 🌸

Functions

the Fortran subroutines on deck are these two icons, no cap:

       subroutine dbdsvdx (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s,
           z, ldz, work, iwork, info)

that one is DBDSVDX 💖

       subroutine sbdsvdx (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s,
           z, ldz, work, iwork, info)

and that one is SBDSVDX ✨

🦋 Detailed Description

the detailed description is literally the function documentation right below — both dbdsvdx and sbdsvdx get their full moment 💅

📚 Function Documentation

`dbdsvdx` — `DBDSVDX` 💜

here's the exact signature, bestie:

       subroutine dbdsvdx (character uplo, character jobz, character range, integer
           n, double precision, dimension( * ) d, double precision, dimension( * )
           e, double precision vl, double precision vu, integer il, integer iu,
           integer ns, double precision, dimension( * ) s, double precision,
           dimension( ldz, * ) z, integer ldz, double precision, dimension( * )
           work, integer, dimension( * ) iwork, integer info)

DBDSVDX

🎯 Purpose

so DBDSVDX 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. periodt.

given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] and superdiagonal E = [ e_1 e_2 ... e_N-1 ], DBDSVDX computes the singular value decomposition 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 DGESVD/DGESDD) or b) compute s,v and reorder the values (and corresponding vectors). DBDSVDX implements a) by calling DSTEVX (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.) 🔥

🎀 Parameters

UPLO — CHARACTER*1. = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal. 👑
JOBZ — CHARACTER*1. = 'N': Compute singular values only; = 'V': Compute singular values and singular vectors. ✨
RANGE — 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. 💖
N — INTEGER. The order of the bidiagonal matrix. N >= 0. 🦋
D — DOUBLE PRECISION array, dimension (N). The n diagonal elements of the bidiagonal matrix B. 🍓
E — DOUBLE PRECISION array, dimension (max(1,N-1)). The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1. 🌸
VL — DOUBLE PRECISION. If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'. 💜
VU — DOUBLE PRECISION. If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'. 🌈
IL — INTEGER. If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'. 🎀
IU — INTEGER. If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'. ⭐
NS — INTEGER. The total number of singular values found. 0 <= NS <= N. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1. 💅
S — DOUBLE PRECISION array, dimension (N). The first NS elements contain the selected singular values in ascending order. 🫶
Z — DOUBLE PRECISION 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. 🧚

LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2). 💀
WORK — DOUBLE PRECISION array, dimension (14*N). 🔥
IWORK — 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. 😭
INFO — 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 DSTEVX. The indices of the eigenvectors (as returned by DSTEVX) are stored in the array IWORK. If INFO = N*2 + 1, an internal error occurred. 🙈

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

📌 Definition

Definition at line 224 of file dbdsvdx.f. 🎀

`sbdsvdx` — `SBDSVDX` 💙

here's the exact signature for this one too:

       subroutine sbdsvdx (character uplo, character jobz, character range, integer
           n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu,
           integer il, integer iu, integer ns, real, dimension( * ) s, real,
           dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer,
           dimension( * ) iwork, integer info)

SBDSVDX

🎯 Purpose

so 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. periodt.

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 decomposition 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  .   .     |
       |              .   .   . |

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.) 🔥

🎀 Parameters

UPLO — CHARACTER*1. = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal. 👑
JOBZ — CHARACTER*1. = 'N': Compute singular values only; = 'V': Compute singular values and singular vectors. ✨
RANGE — 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. 💖
N — INTEGER. The order of the bidiagonal matrix. N >= 0. 🦋
D — REAL array, dimension (N). The n diagonal elements of the bidiagonal matrix B. 🍓
E — REAL array, dimension (max(1,N-1)). The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1. 🌸
VL — 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'. 💜
VU —
VU — VU is REAL. Bestie, if RANGE='V', this is the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'. 🦋
IL — IL is INTEGER. If RANGE='I', this is the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'. ✨
IU — IU is INTEGER. If RANGE='I', this is the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'. 💖
NS — 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. 🎀
S — S is REAL array, dimension (N). The first NS elements contain the selected singular values in ascending order. 🌸
Z — 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 ]

LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2). 🌈
WORK — WORK is REAL array, dimension (14*N). ⭐
IWORK — 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. 🧚
INFO — 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.

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 224 of file sbdsvdx.f. 👑

Author

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That's the footer, bestie: LAPACK, version 3.12.0, page bdsvdx(3) — bye! ✨🦋

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bdsvdx(3)

bdsvdx — bidiagonal SVD, bisection 💅

✨ NAME ✨

🌸 SYNOPSIS 🌸

Functions

🦋 Detailed Description

📚 Function Documentation

dbdsvdx — DBDSVDX 💜

🎯 Purpose

🎀 Parameters

👑 Author

📌 Definition

sbdsvdx — SBDSVDX 💙

🎯 Purpose

🎀 Parameters

Author

Author

`dbdsvdx` — `DBDSVDX` 💜

`sbdsvdx` — `SBDSVDX` 💙