SBDSQR(3)

💖 SBDSQR(3) — Library Functions Manual 💖

okay so bestie, welcome to the Library Functions Manual page for SBDSQR(3). The header/footer path tag is /home/abui...RC/sbdsqr.f(3), and we are about to keep every single argument, every status code, and every internal parameter, because we don't do missing details here. periodt. 💅

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sbdsqr.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

   subroutine SBDSQR (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
       ldc, work, info)
       SBDSQR

🛠️ Function/Subroutine Documentation 🛠️

   subroutine SBDSQR (character uplo, integer n, integer ncvt, integer nru,
       integer ncc, real, dimension( * ) d, real, dimension( * ) e, real,
       dimension( ldvt, * ) vt, integer ldvt, real, dimension( ldu, * ) u,
       integer ldu, real, dimension( ldc, * ) c, integer ldc, real, dimension(
       * ) work, integer info)
       SBDSQR

🎯 Purpose 🎯

okay so 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.

For the algorithm deets, the page points you to 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 to B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992. 💖

🫶 Parameters 🫶

• UPLO — CHARACTER*1. = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.
• N — INTEGER. The order of the matrix B. N >= 0.
• NCVT — INTEGER. The number of columns of the matrix VT. NCVT >= 0.
• NRU — INTEGER. The number of rows of the matrix U. NRU >= 0.
• NCC — INTEGER. The number of columns of the matrix C. NCC >= 0.
• D — 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 — 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 — REAL array, dimension (LDVT, NCVT). On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**T * VT. Not referenced if NCVT = 0.
• LDVT — INTEGER. The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
• U — REAL 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 — INTEGER. The leading dimension of the array U. LDU >= max(1,NRU).
• C — REAL array, dimension (LDC, NCC). On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**T * C. Not referenced if NCC = 0.
• LDC — INTEGER. The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
• WORK — REAL array, dimension (4*N).
• INFO — INTEGER.
  • = 0: successful exit.
  • < 0: If INFO = -i, the i-th argument had an illegal value.
  • > 0:
    • if NCVT = NRU = NCC = 0,
      • = 1, a split was marked by a positive value in E
      • = 2, current block of Z not diagonalized after 30*N iterations (in inner while loop)
      • = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
    • else NCVT = NRU = NCC = 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 algorithm stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.

📝 Note 📝

Bug report from Cezary Dendek. On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is removed since it can overflow pretty easily (for N larger or equal than 18,919). We instead use MAXITDIVN = MAXITR*N.

👑 Author 👑

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

Definition at line 238 of file sbdsqr.f.

🫶 Author (auto-generated) 🫶

Generated automatically by Doxygen for LAPACK from the source code.

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