SRC/clahqr.f(3)

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version 3.12.1 · 7 versions
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✨ SRC/clahqr.f ✨

okay bestie, welcome to the Library Functions Manual page for SRC/clahqr.f — this is the one, the iconic, the main character of Hessenberg eigenvalue computing. no cap, this routine is lowkey carrying the whole QR algorithm on its back 💅

✨ 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 "I do the math so you don't have to" energy 🔥

🎀 Function/Subroutine Documentation 🎀

subroutine clahqr

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. yes bestie, it's that girl. 🦋

💖 Purpose 💖

!> !> 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. !>

so basically it's the helper routine — it comes in after CHSEQR has done the preliminary work and handles the specific Hessenberg submatrix chunk from ILO to IHI. teamwork makes the dream work fr fr 🤝

🌸 Parameters 🌸

  • WANTT — is LOGICAL. = .TRUE. means the full Schur form T is required; = .FALSE. means only eigenvalues are required. choose your fighter bestie 😌
!> WANTT is LOGICAL !> = .TRUE. : the full Schur form T is required; !> = .FALSE.: only eigenvalues are required.
  • WANTZ — is LOGICAL. = .TRUE. means the matrix of Schur vectors Z is required; = .FALSE. means Schur vectors are not required. sometimes you need Z, sometimes you don't, it's giving low-maintenance ✨
!> WANTZ is LOGICAL !> = .TRUE. : the matrix of Schur vectors Z is required; !> = .FALSE.: Schur vectors are not required.
  • N — is INTEGER. the order of the matrix H. N >= 0. that's the size, the vibe check of dimensions 💅
!> N is INTEGER !> The order of the matrix H. N >= 0.
  • ILO — is INTEGER. one of the boundary indices for the submatrix. bounds go hand in hand with IHI 🫶
!> ILO is INTEGER
  • IHI — is INTEGER. it's 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.. constraints: 1 <= ILO <= max(1,IHI); IHI <= N. don't @ me, those are the rules 😤
!> 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 — 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. so like, H goes in one way and comes out another — glow-up or unspecified, depends on the flags 🦋
!> 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 — is INTEGER. the leading dimension of the array H. LDH >= max(1,N). keep it big enough, bestie 📐
!> LDH is INTEGER !> The leading dimension of the array H. LDH >= max(1,N).
  • 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). eigenvalues live here, periodt 💅
!> 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 — is INTEGER. part of the row range for Z transformations 🎀
!> ILOZ is INTEGER
  • IHIZ — is INTEGER. specifies the rows of Z to which transformations must be applied if WANTZ is .TRUE.. constraints: 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. it's giving "know your boundaries" energy ✨
!> 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 — 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. so if you don't need Z, Z doesn't even get looked at — it's giving ignored 💀
!> 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 — is INTEGER. the leading dimension of the array Z. LDZ >= max(1,N). same energy as LDH, keep it roomy 🫶
!> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= max(1,N).
  • INFO — is INTEGER. the exit code, the vibe report:
    • = 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).

!> 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 👑

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

the academic royalty of numerical computing, no cap 🎓✨

💜 Contributors 💜

!> !> 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). !>

David Day came through in '96, Ralph Byers said "let me make it even better" in '04. this is a modified version of CLAHQR from LAPACK version 3.0 — and bestie, it's (1) more robust against overflow and underflow and (2) adopts the more conservative Ahues & Tisseur stopping criterion (LAWN 122, 1997). iconic behavior only 🔥

Definition at line 193 of file clahqr.f. yes, line 193, remember it for the test 📝

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this page was generated automatically by Doxygen for LAPACK from the source code — shoutout to the bots doing the Lord's work 🤖💖 it's LAPACK Version 3.12.0, page SRC/clahqr.f(3). that's the whole tea, bestie. slay, read, and go compute some eigenvalues. periodt 💅✨

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