ZLAQR4(3)

💖 ZLAQR4(3) 💖

okay bestie, welcome to ZLAQR4(3) from the Library Functions Manual ✨. This is the LAPACK page for the Hessenberg eigenvalue + optional Schur decomposition routine, and the source file path is literally /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlaqr4.f. Cute. Slay. 💅

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlaqr4.f — yes, that’s the whole NAME section: the path to the Fortran source. 🎀

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine ZLAQR4 (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z,
    ldz, work, lwork, info)

ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. 🦋

✨ Function/Subroutine Documentation ✨

subroutine ZLAQR4 (logical wantt, logical wantz, integer n, integer ilo,
    integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16,
    dimension( * ) w, integer iloz, integer ihiz, complex*16, dimension(
    ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork,
    integer info)

ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. 💖

💅 Purpose 💅

ZLAQR4 implements one level of recursion for ZLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by ZLAQR0 and, for large enough deflation window size, it may be called by ZLAQR3. This subroutine is identical to ZLAQR0 except that it calls ZLAQR2 instead of ZLAQR3. 👑

ZLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors.

Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. 🌈

🦋 Parameters 🦋

• WANTT (LOGICAL) — = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.
• WANTZ (LOGICAL) — = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.
• N (INTEGER) — the order of the matrix H. N >= 0.
• ILO (INTEGER) — just the type here, bestie. The constraint tea is under IHI.
• IHI (INTEGER) — it is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to ZGEBAL, and then passed to ZGEHRD when the matrix output by ZGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N > 0, then 1 <= ILO <= IHI <= N. If N = 0, then ILO = 1 and IHI = 0.
• H (COMPLEX*16 array, dimension (LDH,N)) — on entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and WANT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
• LDH (INTEGER) — the leading dimension of the array H. LDH >= max(1,N).
• W (COMPLEX*16 array, dimension (N)) — the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in W(ILO:IHI). If WANTT is .TRUE., then 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 (INTEGER) — just the type here too. 💅
• IHIZ (INTEGER) — specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
• Z (COMPLEX*16 array, dimension (LDZ,IHI)) — if WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO > 0 is given under the description of INFO below.)
• LDZ (INTEGER) — the leading dimension of the array Z. if WANTZ is .TRUE. then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
• WORK (COMPLEX*16 array, dimension LWORK) — on exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK.
• LWORK (INTEGER) — the dimension of the array WORK. LWORK >= max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then ZLAQR4 does a workspace query. In this case, ZLAQR4 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.
• INFO (INTEGER) — exit status tea:
  • = 0: successful exit
  • > 0: if INFO = i, ZLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.)
  • If INFO > 0 and WANT 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 a unitary 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(ILO:IHI,ILOZ:IHIZ)
 =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

where U is the unitary matrix in (*) (regardless of the value of WANTT.)

• If INFO > 0 and WANTZ is .FALSE., then Z is not accessed.

👑 Author 👑

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

🌸 Contributors: 🌸

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA. 🫶

🌈 References: 🌈

• K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.
• K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 245 of file zlaqr4.f. ⭐

👑 Author (auto-generated) 👑

Generated automatically by Doxygen for LAPACK from the source code.

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okay that’s the whole page, bestie 💖. Signing off from LAPACK, Version 3.12.0, page path /home/abui...RC/zlaqr4.f(3). Mwah 💋✨