1ZLAQR0(1) LAPACK auxiliary routine (version 3.2) ZLAQR0(1)
2
6 ZLAQR0 - ZLAQR0 compute the eigenvalues of a Hessenberg matrix H and,
7 optionally, the matrices T and Z from the Schur decomposition H = Z T
8 Z**H, where T is an upper triangular matrix (the Schur form), and Z is
9 the unitary matrix of Schur vectors
10
12 SUBROUTINE ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z,
13 LDZ, WORK, LWORK, INFO )
14 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
16 LOGICAL WANTT, WANTZ
18 COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
20
22 ZLAQR0 computes the eigenvalues of a Hessenberg matrix H
23 and, optionally, the matrices T and Z from the Schur decomposition
24 H = Z T Z**H, where T is an upper triangular matrix (the
25 Schur form), and Z is the unitary matrix of Schur vectors.
26 Optionally Z may be postmultiplied into an input unitary
27 matrix Q so that this routine can give the Schur factorization
28 of a matrix A which has been reduced to the Hessenberg form H
29 by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
30
32 WANTT (input) LOGICAL
33 = .TRUE. : the full Schur form T is required;
34 = .FALSE.: only eigenvalues are required.
35 WANTZ (input) LOGICAL
37 = .TRUE. : the matrix of Schur vectors Z is required;
38 = .FALSE.: Schur vectors are not required.
39 N (input) INTEGER
41 The order of the matrix H. N .GE. 0.
42 ILO (input) INTEGER
44 IHI (input) INTEGER It is assumed that H is already upper tri‐
45 angular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
46 H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous
47 call to ZGEBAL, and then passed to ZGEHRD when the matrix output
48 by ZGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI
49 should be set to 1 and N, respectively. If N.GT.0, then
50 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.
51 H (input/output) COMPLEX*16 array, dimension (LDH,N)
53 On entry, the upper Hessenberg matrix H. On exit, if INFO = 0
54 and WANTT is .TRUE., then H contains the upper triangular matrix
55 T from the Schur decomposition (the Schur form). If INFO = 0 and
56 WANT is .FALSE., then the contents of H are unspecified on exit.
57 (The output value of H when INFO.GT.0 is given under the descrip‐
58 tion of INFO below.) This subroutine may explicitly set H(i,j) =
59 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
60 LDH (input) INTEGER
62 The leading dimension of the array H. LDH .GE. max(1,N).
63 W (output) COMPLEX*16 array, dimension (N)
65 The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
66 in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
67 stored in the same order as on the diagonal of the Schur form
68 returned in H, with W(i) = H(i,i).
69 Z (input/output) COMPLEX*16 array, dimension (LDZ,IHI)
71 If WANTZ is .FALSE., then Z is not referenced. If WANTZ is
72 .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
73 replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
74 orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value
75 of Z when INFO.GT.0 is given under the description of INFO
76 below.)
77 LDZ (input) INTEGER
79 The leading dimension of the array Z. if WANTZ is .TRUE. then
80 LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
81 WORK (workspace/output) COMPLEX*16 array, dimension LWORK
83 On exit, if LWORK = -1, WORK(1) returns an estimate of the opti‐
84 mal value for LWORK. LWORK (input) INTEGER The dimension of the
85 array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typi‐
86 cally as large as 6*N may be required for optimal performance. A
87 workspace query to determine the optimal workspace size is recom‐
88 mended. If LWORK = -1, then ZLAQR0 does a workspace query. In
89 this case, ZLAQR0 checks the input parameters and estimates the
90 optimal workspace size for the given values of N, ILO and IHI.
91 The estimate is returned in WORK(1). No error message related to
92 LWORK is issued by XERBLA. Neither H nor Z are accessed.
93 INFO (output) INTEGER
95 = 0: successful exit
96 the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain
97 those eigenvalues which have been successfully computed. (Fail‐
98 ures are rare.) If INFO .GT. 0 and WANT is .FALSE., then on
99 exit, the remaining unconverged eigenvalues are the eigen- values
100 of the upper Hessenberg matrix rows and columns ILO through INFO
101 of the final, output value of H. If INFO .GT. 0 and WANTT is
102 .TRUE., then on exit
103 (*) (initial value of H)*U = U*(final value of H)
105 where U is a unitary matrix. The final value of H is upper Hes‐
106 senberg and triangular in rows and columns INFO+1 through IHI. If
107 INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of
108 Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
109 where U is the unitary matrix in (*) (regard- less of the value of
110 WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
111 accessed.
112 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 ZLAQR0(1)