SRC/dlaqr4.f(3)

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version 3.12.1 · 7 versions
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🌸 NAME 🌸

okay so bestie, welcome to the LAPACK Library Functions Manual page for SRC/dlaqr4.f — yes, that’s the whole NAME section, she’s short and iconic 💅.

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

subroutine dlaqr4 (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz,
    ihiz, z, ldz, work, lwork, info)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

💖 Function/Subroutine Documentation 💖

subroutine dlaqr4 (logical wantt, logical wantz, integer n, integer ilo,
    integer ihi, double precision, dimension( ldh, * ) h, integer ldh,
    double precision, dimension( * ) wr, double precision, dimension( * )
    wi, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z,
    integer ldz, double precision, dimension( * ) work, integer lwork,
    integer info)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

🔥 Purpose 🔥

okay so here’s the tea, DLAQR4 implements one level of recursion for DLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by DLAQR0 and, for large enough deflation window size, it may be called by DLAQR3. This subroutine is identical to DLAQR0 except that it calls DLAQR2 instead of DLAQR3. slay.

DLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition:

H = Z T Z**T

where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. iconic behavior honestly 🦋.

Optionally Z may be postmultiplied into an input orthogonal 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 orthogonal matrix Q:

A = Q*H*Q**T = (QZ)*T*(QZ)**T

🎀 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

that’s literally all the page gives us for ILO, bestie. ILO is INTEGER. no cap.

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 DGEBAL, and then passed to DGEHRD when the matrix output by DGEBAL 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 — DOUBLE PRECISION 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 quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.

If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. 🫠 like, don’t trust it. (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).

WR — DOUBLE PRECISION array, dimension (IHI)

WR is exactly that: a DOUBLE PRECISION array with dimension (IHI).

WI — DOUBLE PRECISION array, dimension (IHI)

The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI).

If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and WI(i+1) < 0.

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 WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

ILOZ — INTEGER

ILOZ is INTEGER. that’s the whole mood. 🎀

IHIZ — INTEGER

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

Z — DOUBLE PRECISION 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 — DOUBLE PRECISION 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 DLAQR4 does a workspace query. In this case, DLAQR4 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, DLAQR4 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. — yes, it says WANT here, i’m just the messenger — then on exit, the remaining unconverged eigenvalues are the eigen-values 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 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(ILO:IHI,ILOZ:IHIZ)
   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

where U is the orthogonal 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: Definition at line 261 of file dlaqr4.f.

🧾 Author 🧾

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