DGEBRD(3)

🌸 NAME 🌸

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

okay so that's the NAME line, bestie — it's just the source file path, and it's giving “I live in a build directory” energy. periodt. 💅

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

• subroutine DGEBRD (m, n, a, lda, d, e, tauq, taup, work, lwork, info) — the call signature, no cap.
• DGEBRD — the routine itself, main character energy.

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💖 Function/Subroutine Documentation 💖

🌟 subroutine DGEBRD 🌟

subroutine DGEBRD (integer m, integer n, 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( * ) work, integer lwork, integer info)

DGEBRD — yes bestie, that's the routine name, and we're about to get into it. 🫶

🎀 Purpose: 🎀

DGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. slay. ✨

🌸 Parameters 🌸

• M — M is INTEGER. The number of rows in the matrix A. M >= 0.
• N — N is INTEGER. The number of columns in the matrix A. N >= 0.
• A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the M-by-N general matrix to be reduced. On exit:
  • if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors;
  • if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, 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 (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
• E — E is DOUBLE PRECISION array, dimension (min(M,N)-1). The off-diagonal elements of the bidiagonal matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
• TAUQ — TAUQ is DOUBLE PRECISION array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
• TAUP — TAUP is DOUBLE PRECISION array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
• WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= max(1,M,N). For optimum performance LWORK >= (M+N)*NB, where NB is the optimal blocksize.
  • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
• INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. 💀

👑 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:

If m >= n:

Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

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

If m < n:

Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

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; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 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).

The contents of A on exit are illustrated by the following examples:

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

  (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
  (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
  (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
  (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
  (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
  (  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). 🦋

📍 Definition at line 📍

Definition at line 203 of file dgebrd.f.

🧚 Author 🧚

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okay that's the whole page, bestie: the footer sign-off is LAPACK Version 3.12.0 and the path /home/abui...RC/dgebrd.f(3). Mwah 💋✨