SLABRD(3)

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/slabrd.f — that’s the file, bestie, and yes the NAME section is literally just the path. Iconic. 🫶

Hey bestie!! Welcome to SLABRD(3), straight out of the Library Functions Manual. Buckle up, we’re doing Fortran linear algebra and it’s gonna slay. 💅✨

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✨ SYNOPSIS ✨

🫶 Functions/Subroutines

subroutine SLABRD (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)

• 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. 👑

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💅 Function/Subroutine Documentation 💅

🎀 subroutine SLABRD 🎀

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 vibe: 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 it 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. 👑

🦋 Parameters

• M — M is INTEGER. It’s the number of rows in the matrix A. 🌈
• N — N is INTEGER. It’s the number of columns in the matrix A. 🌸
• NB — NB is INTEGER. It’s 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. 📌

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🤖 Author

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💖 that’s a wrap, bestie — brought to you by LAPACK, Version 3.12.0, from /home/abui...RC/slabrd.f(3) 🎀