SRC/zlaqr0.f(3)

🌸 NAME 🌸

Okay bestie, this page is SRC/zlaqr0.f(3) from the Library Functions Manual — literally the source-file page for zlaqr0, and it's giving iconic LAPACK documentation energy fr fr. 💅

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

• subroutine zlaqr0 — ZLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

subroutine zlaqr0 (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z,
    ldz, work, lwork, info)

🧚 Function/Subroutine Documentation 🧚

subroutine zlaqr0 (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)

ZLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. No cap, that's the whole main-character job. 👑

🎀 Purpose 🎀

ZLAQR0 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

It's giving decomposition, it's giving unitary, it's giving slay. ✨

💅 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.
• 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.
• 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 ZLAQR0 does a workspace query. In this case, ZLAQR0 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.
  • = 0: successful exit.
  • > 0: if INFO = i, ZLAQR0 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 239 of file zlaqr0.f.

📝 Author 📝

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