SRC/zgebrd.f(3)

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💅 SRC/zgebrd.f(3) — the LAPACK manual page, but make it make sense

okay so bestie, you've stumbled into SRC/zgebrd.f(3) from the LAPACK Library Functions Manual and honestly? you're in for a treat. this is the page for one very specific, very iconic Fortran subroutine. let's get into it ✨

🎀 NAME 🎀

SRC/zgebrd.f — literally the file. that's the name. slay.

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine zgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)

ZGEBRD — that's her. that's the whole vibe.

🌸 Function/Subroutine Documentation 🌸

👑 subroutine zgebrd

here's the full signature, character for character, no cap:

subroutine zgebrd (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 lwork,
    integer info)

ZGEBRD 💖

⭐ Purpose:

!>
!> ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
!> 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.
!>

so like — she takes your general complex M-by-N matrix A and reduces it to upper or lower bidiagonal form B, using a unitary transformation, and the whole thing is this equation: Q**H * A * P = B. the way that works?? obsessed.

and the direction depends on the shape, bestie:

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.
!>

it's an INTEGER. it's the number of rows in the matrix A. and M >= 0, periodt.

`N`

!>          N is INTEGER
!>          The number of columns in the matrix A.  N >= 0.
!>

also an INTEGER. number of columns in A. N >= 0. that's her.

`A`

!>          A is 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.
!>

A is a COMPLEX*16 array, dimension (LDA,N). on entry: the M-by-N general matrix to be reduced. on exit it's giving completely transformed, so let's break it down:

if m >= n, the diagonal and the first superdiagonal get overwritten with the upper bidiagonal matrix B; the elements below the diagonal, together with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and the elements above the first superdiagonal, together with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
if m < n, the diagonal and the first subdiagonal get overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, together with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and the elements above the diagonal, together with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.

and yeah — See Further Details 💀 (we'll get there, don't worry)

`LDA`

!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,M).
!>

INTEGER. the leading dimension of the array A. rule: LDA >= max(1,M). respect the constraint or perish 🙈

`D`

!>          D is DOUBLE PRECISION array, dimension (min(M,N))
!>          The diagonal elements of the bidiagonal matrix B:
!>          D(i) = A(i,i).
!>

DOUBLE PRECISION array, dimension (min(M,N)). these are the diagonal elements of the bidiagonal matrix B, and it's literally just D(i) = A(i,i). main character energy on the diagonal 👑

`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.
!>

DOUBLE PRECISION array, dimension (min(M,N)-1). the off-diagonal besties 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*16 array, dimension (min(M,N))
!>          The scalar factors of the elementary reflectors which
!>          represent the unitary matrix Q. See Further Details.
!>

COMPLEX*16 array, dimension (min(M,N)). the scalar factors of the elementary reflectors that represent the unitary matrix Q. See Further Details 🫶

`TAUP`

!>          TAUP is COMPLEX*16 array, dimension (min(M,N))
!>          The scalar factors of the elementary reflectors which
!>          represent the unitary matrix P. See Further Details.
!>

COMPLEX*16 array, dimension (min(M,N)). the scalar factors of the elementary reflectors that represent the unitary matrix P. again: See Further Details ✨

`WORK`

!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

COMPLEX*16 array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, then WORK(1) returns the optimal LWORK. she's helpful like that 💜

`LWORK`

!>          LWORK is INTEGER
!>          The length of the array WORK.
!>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
!>          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.
!>

INTEGER. it's the length of the array WORK. the rules, fr fr:

LWORK >= 1, if MIN(M,N) = 0
LWORK >= MAX(M,N), otherwise
for optimum performance, LWORK >= (M+N)*NB, where NB is the optimal blocksize 🔥

aaand the special case: 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. no drama, no yelling 💅

`INFO`

!>          INFO is INTEGER
!>          = 0:  successful exit.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!>

INTEGER. the vibes report:

= 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.

the whole squad. legends only 🫶

🌈 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, and 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).
!>

okay so this is the part everyone skips and honestly? don't. it's cute. here's the deal: 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, and 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).

and here's what A looks like on exit — the contents are illustrated by these 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). look at that little diagram go 😭 it's literally art.

📌 Definition at line 204 of file zgebrd.f.

that's where she lives. line 204. icon behavior.

⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code.

(not a human, just a very diligent robot 🫡)

lowkey that's the whole page bestie — every parameter, every constraint, every !> comment preserved with love. go forth and bidiagonalize 💅✨

LAPACK — Version 3.12.0 🦋 and that's the SRC/zgebrd.f(3) page, signed, sealed, delivered, mwah 💖

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