CLAHQR(3)

🌸 NAME 🌸

okay so the name is literally this entire path, no cap:

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

and that's the manual name bestie, we love a specific queen who knows exactly where she lives. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine CLAHQR (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info)

CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. It's giving math. It's giving linear algebra. It's giving slay. 🦋

🎀 Function/Subroutine Documentation 🎀

subroutine CLAHQR (logical wantt, logical wantz, integer n, integer ilo,
    integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex,
    dimension( * ) w, integer iloz, integer ihiz, complex, dimension( ldz, *
    ) z, integer ldz, integer info)

CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. Periodt. 💖

Purpose

okay so this is the whole vibe: CLAHQR is an auxiliary routine that gets called by CHSEQR and it's basically updating the eigenvalues and Schur decomposition that CHSEQR already computed, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. Here's the exact word, bestie:

!>
!>    CLAHQR is an auxiliary routine called by CHSEQR to update the
!>    eigenvalues and Schur decomposition already computed by CHSEQR, by
!>    dealing with the Hessenberg submatrix in rows and columns ILO to
!>    IHI.
!>

Parameters

WANTT

WANTT is a LOGICAL and it's like, do you want the full Schur form T or are you just here for the eigenvalues? Choose your fighter. ⭐

!>          WANTT is LOGICAL
!>          = .TRUE. : the full Schur form T is required;
!>          = .FALSE.: only eigenvalues are required.
!>

WANTZ

WANTZ is also a LOGICAL, and this one asks: do you want the matrix of Schur vectors Z, or nah? 👑

!>          WANTZ is LOGICAL
!>          = .TRUE. : the matrix of Schur vectors Z is required;
!>          = .FALSE.: Schur vectors are not required.
!>

N

N is an INTEGER — the order of the matrix H. And it's like N >= 0, no cap. ✨

!>          N is INTEGER
!>          The order of the matrix H.  N >= 0.
!>

ILO

ILO is an INTEGER:

!>          ILO is INTEGER
!>

IHI

IHI is an INTEGER and this is where the deets get real. The assumption is that H is already upper triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 unless ILO = 1. CLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but it applies transformations to all of H if WANTT is .TRUE.. Also, for the record: 1 <= ILO <= max(1,IHI); IHI <= N. 💅

!>          IHI is INTEGER
!>          It is assumed that H is already upper triangular in rows and
!>          columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
!>          CLAHQR works primarily with the Hessenberg submatrix in rows
!>          and columns ILO to IHI, but applies transformations to all of
!>          H if WANTT is .TRUE..
!>          1 <= ILO <= max(1,IHI); IHI <= N.
!>

H

H is a COMPLEX array, dimension (LDH,N). On entry, it's the upper Hessenberg matrix H. On exit, if INFO is zero and WANTT is .TRUE., then H is upper triangular in rows and columns ILO:IHI. If INFO is zero and WANTT is .FALSE., then the contents of H are unspecified on exit (iykyk). The output state of H in case INF is positive is below under the description of INFO. 🫶

⚠️ heads up: the original says INF here — not INFO — and I'm keeping it exactly as written because facts are facts. Don't come for me. 💀

!>          H is COMPLEX array, dimension (LDH,N)
!>          On entry, the upper Hessenberg matrix H.
!>          On exit, if INFO is zero and if WANTT is .TRUE., then H
!>          is upper triangular in rows and columns ILO:IHI.  If INFO
!>          is zero and if WANTT is .FALSE., then the contents of H
!>          are unspecified on exit.  The output state of H in case
!>          INF is positive is below under the description of INFO.
!>

LDH

LDH is an INTEGER — the leading dimension of the array H, and it's gotta be LDH >= max(1,N). 🌈

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

W

W is a COMPLEX array, dimension (N). The computed eigenvalues ILO to IHI are stored in the corresponding elements of W. If WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i). Obsessed with a consistent ordering. 💜

!>          W is COMPLEX array, dimension (N)
!>          The computed eigenvalues ILO to IHI are stored in the
!>          corresponding elements of W. If WANTT is .TRUE., the
!>          eigenvalues are stored in the same order as on the diagonal
!>          of the Schur form returned in H, with W(i) = H(i,i).
!>

ILOZ

ILOZ is an INTEGER:

!>          ILOZ is INTEGER
!>

IHIZ

IHIZ is an INTEGER. It specifies the rows of Z to which transformations must be applied if WANTZ is .TRUE.. And: 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. 🎀

!>          IHIZ is INTEGER
!>          Specify the rows of Z to which transformations must be
!>          applied if WANTZ is .TRUE..
!>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
!>

Z

Z is a COMPLEX array, dimension (LDZ,N). If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by CHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced. So like, mind ya business bestie. 🙈

!>          Z is COMPLEX array, dimension (LDZ,N)
!>          If WANTZ is .TRUE., on entry Z must contain the current
!>          matrix Z of transformations accumulated by CHSEQR, and on
!>          exit Z has been updated; transformations are applied only to
!>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
!>          If WANTZ is .FALSE., Z is not referenced.
!>

LDZ

LDZ is an INTEGER — the leading dimension of the array Z, and it's LDZ >= max(1,N). ⭐

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

INFO

INFO is an INTEGER and it's the one that tells you if everything went slay or if it flopped. Here's the whole breakdown, bestie:

• = 0: successful exit — slay! 🎉
• > 0: if INFO = i, CLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of W contain those eigenvalues which have been successfully computed. (not every queen converges in 30 tries, and that's okay 😭)

okay so if INFO > 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H.

If INFO > 0 and WANTT is .TRUE., then on exit:

!>          (*)       (initial value of H)*U  = U*(final value of H)

where U is an orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI.

If INFO > 0 and WANTZ is .TRUE., then on exit:

!>                      (final value of Z)  = (initial value of Z)*U

where U is the orthogonal matrix in (*) (regardless of the value of WANTT). The way that's a conditional within a conditional within a conditional… nested queen behavior. 👑

Here's the exact block for the record:

!>          INFO is INTEGER
!>           = 0:  successful exit
!>           > 0:  if INFO = i, CLAHQR failed to compute all the
!>                  eigenvalues ILO to IHI in a total of 30 iterations
!>                  per eigenvalue; elements i+1:ihi of W contain
!>                  those eigenvalues which have been successfully
!>                  computed.
!>
!>                  If INFO > 0 and WANTT is .FALSE., then on exit,
!>                  the remaining unconverged eigenvalues are the
!>                  eigenvalues of the upper Hessenberg matrix
!>                  rows and columns ILO through INFO of the final,
!>                  output value of H.
!>
!>                  If INFO > 0 and WANTT is .TRUE., then on exit
!>          (*)       (initial value of H)*U  = U*(final value of H)
!>                  where U is an orthogonal matrix.    The final
!>                  value of H is upper Hessenberg and triangular in
!>                  rows and columns INFO+1 through IHI.
!>
!>                  If INFO > 0 and WANTZ is .TRUE., then on exit
!>                      (final value of Z)  = (initial value of Z)*U
!>                  where U is the orthogonal matrix in (*)
!>                  (regardless of the value of WANTT.)
!>

Author

Like, the whole academic squad wrote this and we're grateful:

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

Contributors:

okay so the contribs are giving history lesson and honestly? iconic:

!>
!>     02-96 Based on modifications by
!>     David Day, Sandia National Laboratory, USA
!>
!>     12-04 Further modifications by
!>     Ralph Byers, University of Kansas, USA
!>     This is a modified version of CLAHQR from LAPACK version 3.0.
!>     It is (1) more robust against overflow and underflow and
!>     (2) adopts the more conservative Ahues & Tisseur stopping
!>     criterion (LAWN 122, 1997).
!>

And real talk: this is a modified version of CLAHQR from LAPACK version 3.0. It's (1) more robust against overflow and underflow and (2) adopts the more conservative Ahues & Tisseur stopping criterion (LAWN 122, 1997). Facts only. 💅

Definition at line 193 of file clahqr.f. (yes bestie, line 193 specifically, go off 🎀)

🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code. (Not a human, bestie. A robot did this. And the robot slayed. 💀✨)

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