SRC/dlabrd.f(3)

💖 NAME 💖

SRC/dlabrd.f — yep, that's the page, bestie. It lives in the Library Functions Manual, and it's the whole vibe for the dlabrd subroutine. Slay. ✨

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✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

• subroutine dlabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) — DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. Periodt.

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🦋 Function/Subroutine Documentation 🦋

subroutine dlabrd (integer m, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tauq, double precision, dimension( * ) taup, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( ldy, * ) y, integer ldy)

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. It's giving very important auxiliary routine energy. 💅

🎀 Purpose 🎀

Okay so, DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q**T * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form. No cap.

This is an auxiliary routine called by DGEBRD. Iconic behavior. 👑

🌸 Parameters 🌸

• M — M is INTEGER. The number of rows in the matrix A.
• N — N is INTEGER. The number of columns in the matrix A.
• NB — NB is INTEGER. The number of leading rows and columns of A to be reduced.
• A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged.
  • If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.
  • If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.
  • See Further Details.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• D — D is DOUBLE PRECISION array, dimension (NB). The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).
• E — E is DOUBLE PRECISION array, dimension (NB). The off-diagonal elements of the first NB rows and columns of the reduced matrix.
• TAUQ — TAUQ is DOUBLE PRECISION array, dimension (NB). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
• TAUP — TAUP is DOUBLE PRECISION array, dimension (NB). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
• X — X is DOUBLE PRECISION array, dimension (LDX,NB). The m-by-nb matrix X required to update the unreduced part of A.
• LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,M).
• Y — Y is DOUBLE PRECISION array, dimension (LDY,NB). The n-by-nb matrix Y required to update the unreduced part of A.
• LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,N).

👑 Author 👑

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

🧚 Further Details 🧚

The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**T which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y**T - X*U**T.

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

  (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
  (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
  (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
  (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
  (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
  (  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

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Definition at line 208 of file dlabrd.f.

💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code.

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Okay bestie, that's the whole page. Final sign-off vibe: LAPACK Version 3.12.0, source path SRC/dlabrd.f(3). (The original footer has no date line, so I'm not inventing one — we keep the facts cute and correct. 💅💖)