SRC/slabrd.f(3)

💖 NAME 💖

okay so this is the SRC/slabrd.f(3) page from the Library Functions Manual, and the name is literally SRC/slabrd.f — that's it, bestie. It's all about one Fortran subroutine, slabrd, and she's iconic. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

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

🌸 Function/Subroutine Documentation 🌸

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

SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

🎀 Purpose 🎀

okay so here's the tea: SLABRD 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. 🦋

This is an auxiliary routine called by SGEBRD. no cap.

🍓 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 REAL 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 REAL 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 REAL array, dimension (NB). The off-diagonal elements of the first NB rows and columns of the reduced matrix.
• TAUQ — TAUQ is REAL array, dimension (NB). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
• TAUP — TAUP is REAL array, dimension (NB). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
• X — X is REAL 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 REAL 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).

Definition at line 208 of file slabrd.f. 📌

👑 Author 👑

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That’s the whole page, bestie! Source/version: LAPACK Version 3.12.0, page SRC/slabrd.f(3) — no date printed on this one. 🫶💖💅