.TH "laqr4" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME laqr4 \- laqr4: eig of Hessenberg, step in hseqr .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBclaqr4\fP (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)" .br .RI "\fBCLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. " .ti -1c .RI "subroutine \fBdlaqr4\fP (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info)" .br .RI "\fBDLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. " .ti -1c .RI "subroutine \fBslaqr4\fP (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info)" .br .RI "\fBSLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. " .ti -1c .RI "subroutine \fBzlaqr4\fP (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)" .br .RI "\fBZLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine claqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) w, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, integer info)" .PP \fBCLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CLAQR4 implements one level of recursion for CLAQR0\&. It is a complete implementation of the small bulge multi-shift QR algorithm\&. It may be called by CLAQR0 and, for large enough deflation window size, it may be called by CLAQR3\&. This subroutine is identical to CLAQR0 except that it calls CLAQR2 instead of CLAQR3\&. CLAQR4 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 CGEBAL, and then passed to CGEHRD when the matrix output by CGEBAL 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 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 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 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 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 CLAQR4 does a workspace query\&. In this case, CLAQR4 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, CLAQR4 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 .SS "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)" .PP \fBDLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. 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\&. 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\&. .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 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\&. .fi .PP .br \fIH\fP .PP .nf H is 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\&. (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 \fIWR\fP .PP .nf WR is DOUBLE PRECISION array, dimension (IHI) .fi .PP .br \fIWI\fP .PP .nf WI is 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)\&. .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 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\&.) .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 DOUBLE PRECISION 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 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\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is 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\&., 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 (*) (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 .SS "subroutine slaqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)" .PP \fBSLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. .PP \fBPurpose:\fP .RS 4 .PP .nf SLAQR4 implements one level of recursion for SLAQR0\&. It is a complete implementation of the small bulge multi-shift QR algorithm\&. It may be called by SLAQR0 and, for large enough deflation window size, it may be called by SLAQR3\&. This subroutine is identical to SLAQR0 except that it calls SLAQR2 instead of SLAQR3\&. SLAQR4 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\&. 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\&. .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 SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL 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 REAL 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\&. (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 \fIWR\fP .PP .nf WR is REAL array, dimension (IHI) .fi .PP .br \fIWI\fP .PP .nf WI is REAL 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)\&. .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 REAL 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 REAL 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 SLAQR4 does a workspace query\&. In this case, SLAQR4 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 \\verbatim INFO is INTEGER = 0: successful exit > 0: if INFO = i, SLAQR4 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 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 (*) (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 .SS "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)" .PP \fBZLAQR4\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .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 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\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 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 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 .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.