SRC/dgghrd.f(3)

💅 SRC/dgghrd.f — from the LAPACK Library Functions Manual, bestie

okay so picture this: it's a C/Fortran library manual page, it's called SRC/dgghrd.f(3), and honestly? it's kind of iconic. let's get into it 🫶

✨ NAME ✨

SRC/dgghrd.f — that's the whole name, that's the vibe, periodt 💅

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🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine dgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, info)

DGGHRD 👑

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🦋 Function/Subroutine Documentation 🦋

subroutine dgghrd (character compq, character compz, integer n, integer ilo,
    integer ihi, double precision, dimension( lda, * ) a, integer lda,
    double precision, dimension( ldb, * ) b, integer ldb, double precision,
    dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, *
    ) z, integer ldz, integer info)

DGGHRD — say it with your chest 📣

💖 Purpose 💖

no cap, here's the whole deal:

!> DGGHRD reduces a pair of real matrices (A,B) to generalized upper !> Hessenberg form using orthogonal transformations, where A is a !> general matrix and B is upper triangular. The form of the !> generalized eigenvalue problem is

!>    A*x = lambda*B*x,

!> and B is typically made upper triangular by computing its QR !> factorization and moving the orthogonal matrix Q to the left side !> of the equation.

!> This subroutine simultaneously reduces A to a Hessenberg matrix H:

!>    Q**T*A*Z = H

!> and transforms B to another upper triangular matrix T:

!>    Q**T*B*Z = T

!> in order to reduce the problem to its standard form

!>    H*y = lambda*T*y

!> where y = Z**T*x.

!> The orthogonal matrices Q and Z are determined as products of Givens !> rotations. They may either be formed explicitly, or they may be !> postmultiplied into input matrices Q1 and Z1, so that

!>      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

!>      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

!> If Q1 is the orthogonal matrix from the QR factorization of B in the !> original equation A*x = lambda*B*x, then DGGHRD reduces the original !> problem to generalized Hessenberg form.

like... she really does it all 🤯

🎀 Parameters 🎀

COMPQ

!>          COMPQ is CHARACTER*1
!>          = 'N': do not compute Q;
!>          = 'I': Q is initialized to the unit matrix, and the
!>                 orthogonal matrix Q is returned;
!>          = 'V': Q must contain an orthogonal matrix Q1 on entry,
!>                 and the product Q1*Q is returned.

COMPZ

!>          COMPZ is CHARACTER*1
!>          = 'N': do not compute Z;
!>          = 'I': Z is initialized to the unit matrix, and the
!>                 orthogonal matrix Z is returned;
!>          = 'V': Z must contain an orthogonal matrix Z1 on entry,
!>                 and the product Z1*Z is returned.

N

!>          N is INTEGER
!>          The order of the matrices A and B.  N >= 0.

ILO

!>          ILO is INTEGER

IHI

!>          IHI is INTEGER

!>          ILO and IHI mark the rows and columns of A which are to be
!>          reduced.  It is assumed that A is already upper triangular
!>          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
!>          normally set by a previous call to DGGBAL; otherwise they
!>          should be set to 1 and N respectively.
!>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

ok bestie quick vibe check on ILO/IHI: those bounds are not suggestions, they're the rules. like, 1 <= ILO <= IHI <= N if N > 0, and ILO=1, IHI=0 if N=0. also DGGBAL is your girl for setting them, or just go 1 and N. ✨

A

!>          A is DOUBLE PRECISION array, dimension (LDA, N)
!>          On entry, the N-by-N general matrix to be reduced.
!>          On exit, the upper triangle and the first subdiagonal of A
!>          are overwritten with the upper Hessenberg matrix H, and the
!>          rest is set to zero.

she gets overwritten, so don't come crying to me about your original A 😭

LDA

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

B

!>          B is DOUBLE PRECISION array, dimension (LDB, N)
!>          On entry, the N-by-N upper triangular matrix B.
!>          On exit, the upper triangular matrix T = Q**T B Z.  The
!>          elements below the diagonal are set to zero.

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).

Q

!>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
!>          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
!>          typically from the QR factorization of B.
!>          On exit, if COMPQ='I', the orthogonal matrix Q, and if
!>          COMPQ = 'V', the product Q1*Q.
!>          Not referenced if COMPQ='N'.

LDQ

!>          LDQ is INTEGER
!>          The leading dimension of the array Q.
!>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

Z

!>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
!>          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
!>          On exit, if COMPZ='I', the orthogonal matrix Z, and if
!>          COMPZ = 'V', the product Z1*Z.
!>          Not referenced if COMPZ='N'.

LDZ

!>          LDZ is INTEGER
!>          The leading dimension of the array Z.
!>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

INFO

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

INFO = 0? slay, we move. INFO = -i? that means the i-th argument had an illegal value and honestly that's on you, not her 🙈

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👑 Author 👑

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

the whole squad showed up, we love a collab 💜

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🌈 Further Details: 🌈

!> This routine reduces A to Hessenberg and B to triangular form by !> an unblocked reduction, as described in _Matrix_Computations_, !> by Golub and Van Loan (Johns Hopkins Press.)

iykyk 📚 — unblocked reduction, Golub and Van Loan, Matrix Computations, Johns Hopkins Press. that's the citation, that's the lore.

Definition at line 205 of file dgghrd.f. 📍

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⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code. beep boop, the robots made this one 🤖✨

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and that's the whole page, bestie! 🌸💖

🎀 cute little sign-off: this page is brought to you by LAPACK, Version 3.12.0, living at SRC/dgghrd.f(3). go reduce something to generalized upper Hessenberg form, you absolute main character 🦋💅