.TH "SRC/zlaqr0.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/zlaqr0.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBzlaqr0\fP (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)" .br .RI "\fBZLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "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)" .PP \fBZLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIWANTT\fP .PP .nf WANTT is LOGICAL = \&.TRUE\&. : the full Schur form T is required; = \&.FALSE\&.: only eigenvalues are required\&. .fi .PP .br \fIWANTZ\fP .PP .nf WANTZ is LOGICAL = \&.TRUE\&. : the matrix of Schur vectors Z is required; = \&.FALSE\&.: Schur vectors are not required\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix H\&. N >= 0\&. .fi .PP .br \fIILO\fP .PP .nf ILO is INTEGER .fi .PP .br \fIIHI\fP .PP .nf IHI is 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\&. .fi .PP .br \fIH\fP .PP .nf H is 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\&. .fi .PP .br \fILDH\fP .PP .nf LDH is INTEGER The leading dimension of the array H\&. LDH >= max(1,N)\&. .fi .PP .br \fIW\fP .PP .nf W is 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)\&. .fi .PP .br \fIILOZ\fP .PP .nf ILOZ is INTEGER .fi .PP .br \fIIHIZ\fP .PP .nf IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is \&.TRUE\&.\&. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N\&. .fi .PP .br \fIZ\fP .PP .nf Z is 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\&.) .fi .PP .br \fILDZ\fP .PP .nf LDZ is INTEGER The leading dimension of the array Z\&. if WANTZ is \&.TRUE\&. then LDZ >= MAX(1,IHIZ)\&. Otherwise, LDZ >= 1\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX*16 array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK\&. .fi .PP .br \fILWORK\fP .PP .nf LWORK is 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\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is 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 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 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 (*) (regard- less of the value of WANTT\&.) If INFO > 0 and WANTZ is \&.FALSE\&., then Z is not accessed\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA .RE .PP \fBReferences:\fP .RS 4 .PP .nf 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\&. .fi .PP .br 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\&. .RE .PP .PP Definition at line \fB239\fP of file \fBzlaqr0\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.