SRC/zlabrd.f(3)

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Welcome to the Library Functions Manual page for SRC/zlabrd.f, bestie. This is the ZLABRD tea: bidiagonal reductions, unitary transformations, and all the arrays you need to keep track of. Let’s get into it 💅✨

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💖 NAME 💖

SRC/zlabrd.f

✨ SYNOPSIS ✨

Functions/Subroutines

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

🫶 Function/Subroutine Documentation 🫶

   subroutine zlabrd (integer m, integer n, integer nb, complex*16, dimension(
       lda, * ) a, integer lda, double precision, dimension( * ) d, double
       precision, dimension( * ) e, complex*16, dimension( * ) tauq,
       complex*16, dimension( * ) taup, complex*16, dimension( ldx, * ) x,
       integer ldx, complex*16, dimension( ldy, * ) y, integer ldy)

ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form. It does this by a unitary transformation Q**H * A * P, and it returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. Iconic behavior, no cap 🔥

🎯 Purpose

Okay so here’s the main character summary: ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q**H * 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.

This is an auxiliary routine called by ZGEBRD 👑

🧾 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 COMPLEX*16 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 unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary 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 unitary 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 unitary 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 COMPLEX*16 array, dimension (NB); the scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
• TAUP — TAUP is COMPLEX*16 array, dimension (NB); the scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
• X — X is COMPLEX*16 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 COMPLEX*16 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**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex 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**H 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**H - X*U**H.

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).

Definition at line 210 of file zlabrd.f.

🤖 Author

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That’s the whole tea ☕ — this page is from LAPACK Version 3.12.0, file SRC/zlabrd.f(3). Stay iconic, bestie! 💅✨