NAME¶

CHSEQR - CHSEQR 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 CHSEQR(

JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO )

INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N CHARACTER COMPZ, JOB COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE¶

CHSEQR 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¶

JOB (input) CHARACTER*1

= 'E': compute eigenvalues only;
= 'S': compute eigenvalues and the Schur form T. COMPZ (input) CHARACTER*1
= 'N': no Schur vectors are computed;
= 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned; = 'V': Z must contain an unitary matrix Q on entry, and the product Q*Z is returned.

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. 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.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.

H (input/output) COMPLEX array, dimension (LDH,N)

On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and JOB = 'S', H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and JOB = 'E', 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.) Unlike earlier versions of CHSEQR, this subroutine may explicitly 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 array, dimension (N)

The computed eigenvalues. If JOB = 'S', 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 array, dimension (LDZ,N)

If COMPZ = 'N', Z is not referenced. If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the unitary matrix Z of the Schur vectors of H. If COMPZ = 'V', on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z. Normally Q is the unitary matrix generated by CUNGHR after the call to CGEHRD which formed the Hessenberg matrix H. (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 COMPZ = 'I' or COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.

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

On exit, if INFO = 0, 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 and delivers very good and sometimes optimal performance. However, LWORK as large as 11*N may be required for optimal performance. A workspace query is recommended to determine the optimal workspace size. If LWORK = -1, then CHSEQR does a workspace query. In this case, CHSEQR 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
value
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 JOB = 'E', 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 JOB = 'S', 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 COMPZ = 'V', then on exit (final value of Z) = (initial value of Z)*U where U is the unitary matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'I', then on exit (final value of Z) = U where U is the unitary matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'N', then Z is not accessed.