NAME¶

ZLAQR4 - ZLAQR4 compute 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

SYNOPSIS¶

SUBROUTINE ZLAQR4(

WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )

INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N LOGICAL WANTT, WANTZ COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE¶

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.

ARGUMENTS¶

WANTT (input) LOGICAL

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

WANTZ (input) LOGICAL

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

N (input) INTEGER

The order of the matrix H. N .GE. 0.

ILO (input) INTEGER

IHI (input) INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.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.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.

H (input/output) 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.GT.0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

LDH (input) INTEGER

The leading dimension of the array H. LDH .GE. max(1,N).

W (output) 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).

Z (input/output) 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.GT.0 is given under the description of INFO below.)

LDZ (input) INTEGER

The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.

WORK (workspace/output) COMPLEX*16 array, dimension LWORK

On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK .GE. 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 (output) INTEGER

= 0: successful exit
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 .GT. 0 and WANT is .FALSE., 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 .GT. 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 .GT. 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 (*) (regard- less of the value of WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed.