gebd2(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 NAME 💖

okay bestie, this is gebd2(3) from the Library Functions Manual, and the function is gebd2 — gebd2: reduction to bidiagonal, level 2 ✨

💅 SYNOPSIS 💅

📚 Functions 📚

  • subroutine cgebd2 (m, n, a, lda, d, e, tauq, taup, work, info) — CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
  • subroutine dgebd2 (m, n, a, lda, d, e, tauq, taup, work, info) — DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
  • subroutine sgebd2 (m, n, a, lda, d, e, tauq, taup, work, info) — SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
  • subroutine zgebd2 (m, n, a, lda, d, e, tauq, taup, work, info) — ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

🌸 Detailed Description 🌸

the detailed function tea starts right here, bestie — keep scrolling 💅

🎀 Function Documentation 🎀

🌟 subroutine cgebd2 🌟

subroutine cgebd2 (integer m, integer n, complex, dimension( lda, * ) a,
    integer lda, real, dimension( * ) d, real, dimension( * ) e, complex,
    dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( *
    ) work, integer info)

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

💫 Purpose 💫

CGEBD2 reduces a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation: Q**H * A * P = B.

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

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 COMPLEX 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 unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
    • On exit, 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 unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, 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 REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
  • E — E is REAL 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 COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
  • TAUP — TAUP is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
  • WORK — WORK is COMPLEX array, dimension (max(M,N)).
  • 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**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex 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**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, v and u are complex 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 189 of file cgebd2.f.

🌟 subroutine dgebd2 🌟

subroutine dgebd2 (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 info)

DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

💫 Purpose 💫

DGEBD2 reduces a real general m by n matrix A to upper or lower real 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.

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.
    • On exit, 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(M,N)).
  • 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 🌈

v(1:i-1) = 0, v(i) = 1, and v(i+1:m) gets stored on exit in A(i+1:m,i)

u(1:i) = 0, u(i+1) = 1, and u(i+2:n) gets stored on exit in A(i,i+2:n)

okay and then — plot twist — tauq lives in TAUQ(i) and taup lives in TAUP(i). keeping the receipts 🧾

If m < n, the whole thing flips:

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) gets stored on exit in A(i+2:m,i)

u(1:i-1) = 0, u(i) = 1, and u(i+1:n) gets stored on exit in A(i,i+1:n)

tauq is stored in TAUQ(i) and taup in TAUP(i). same energy, different shape, it's giving symmetrical 💅

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). like, that's the whole map of the crime scene bestie, memorize it 🗺️✨

Definition at line 188 of file dgebd2.f.

🌈 subroutine sgebd2

subroutine sgebd2 (integer m, integer n, real, dimension( lda, * ) a,
    integer lda, real, dimension( * ) d, real, dimension( * ) e, real,
    dimension( * ) tauq, real, dimension( * ) taup, real, dimension( * )
    work, integer info)

SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. yes, unblocked — she's doing it the honest way, one step at a time, no shortcuts, no cap 🫶

💖 Purpose

!>
!> SGEBD2 reduces a real general 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.
!>

✨ Parameters

  • M — INTEGER. The number of rows in the matrix A. M >= 0. (zero rows is allowed, she's not gonna judge 🙈)
  • N — INTEGER. The number of columns in the matrix A. N >= 0.
  • A — REAL 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. (yes queen, scroll down, it's all there 👇)
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M). disrespect this and it's over for you 💀
  • D — REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). main character diagonal, literally 👑
  • E — REAL 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 — REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
  • TAUP — REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
  • WORK — REAL array, dimension (max(M,N)). (just vibes and scratch space ✨)
  • INFO — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. bestie you fed her something cursed, check your args 🚨

🎀 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

(an absolute supergroup, no notes 🫡)

🦋 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). same diagram, real numbers this time, we love consistency 🌸

Definition at line 188 of file sgebd2.f.

💜 subroutine zgebd2

subroutine zgebd2 (integer m, integer n, 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( * ) work, integer info)

ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. the complex one. the sophisticated one. she's got complex*16 everything and she knows it 💅✨

⭐ Purpose

!>
!> ZGEBD2 reduces a complex general m by n matrix A to upper or lower
!> real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!>
!> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
!>
ok important detail bestie: B comes out real bidiagonal even though A went in complex. she strips the imaginary drama right out. also note it's Q**H, conjugate transpose, not Q**T — complex rules, different game 🔥

🧚 Parameters

  • M — INTEGER. The number of rows in the matrix A. M >= 0.
  • N — INTEGER. The number of columns in the matrix A. N >= 0.
  • A — COMPLEX*16 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 unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary 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 unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
    • See Further Details.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • D — DOUBLE PRECISION array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). yes, DOUBLE PRECISION — real output, remember? 😌
  • E — 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 — COMPLEX*16 array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
  • TAUP — COMPLEX*16 array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
  • WORK —
  • WORK — a COMPLEX*16 array, dimension (max(M,N)). the workspace bestie, periodt 💖
  • INFO — an INTEGER status check, and it’s giving exit-code realness:
    • = 0: successful exit, we love to see it ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value. oop, illegal value energy 🫠

👑 Author

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

💅 Further Details: 💅

okay so 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**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex 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**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, v and u are complex 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 188 of file zgebd2.f.

🧚 Author

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