1CLAQR4(1) LAPACK auxiliary routine (version 3.1) CLAQR4(1)
2
6 CLAQR4 - compute the eigenvalues of a Hessenberg matrix H and, option‐
7 ally, the matrices T and Z from the Schur decomposition H = Z T Z**H,
8 where T is an upper triangular matrix (the Schur form), and Z is the
9 unitary matrix of Schur vectors
10
12 SUBROUTINE CLAQR4( 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 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
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22 CLAQR4 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
28 matrix Q so that this routine can give the Schur factorization
29 of a matrix A which has been reduced to the Hessenberg form H
30 by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
31
34 WANTT (input) LOGICAL
35 = .TRUE. : the full Schur form T is required;
36 = .FALSE.: only eigenvalues are required.
37 WANTZ (input) LOGICAL
39 = .TRUE. : the matrix of Schur vectors Z is required;
40 = .FALSE.: Schur vectors are not required.
41 N (input) INTEGER
43 The order of the matrix H. N .GE. 0.
44 ILO (input) INTEGER
46 IHI (input) INTEGER It is assumed that H is already upper tri‐
47 angular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
48 H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous
49 call to CGEBAL, and then passed to CGEHRD when the matrix output
50 by CGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI
51 should be set to 1 and N, respectively. If N.GT.0, then
52 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.
53 H (input/output) COMPLEX array, dimension (LDH,N)
55 On entry, the upper Hessenberg matrix H. On exit, if INFO = 0
56 and WANTT is .TRUE., then H contains the upper triangular matrix
57 T from the Schur decomposition (the Schur form). If INFO = 0 and
58 WANT is (The output value of H when INFO.GT.0 is given under the
59 description of INFO below.)
60 This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j =
62 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
63 LDH (input) INTEGER
65 The leading dimension of the array H. LDH .GE. max(1,N).
66 W (output) COMPLEX array, dimension (N)
68 The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
69 in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
70 stored in the same order as on the diagonal of the Schur form
71 returned in H, with W(i) = H(i,i).
72 Z (input/output) COMPLEX array, dimension (LDZ,IHI)
74 If WANTZ is .FALSE., then Z is not referenced. If WANTZ is
75 .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
76 replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
77 orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value
78 of Z when INFO.GT.0 is given under the description of INFO
79 below.)
80 LDZ (input) INTEGER
82 The leading dimension of the array Z. if WANTZ is .TRUE. then
83 LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
84 WORK (workspace/output) COMPLEX array, dimension LWORK
86 On exit, if LWORK = -1, WORK(1) returns an estimate of the opti‐
87 mal value for LWORK.
88 LWORK (input) INTEGER The dimension of the array WORK. LWORK
90 .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N
91 may be required for optimal performance. A workspace query to
92 determine the optimal workspace size is recommended.
93 If LWORK = -1, then CLAQR4 does a workspace query. In this case,
95 CLAQR4 checks the input parameters and estimates the optimal
96 workspace size for the given values of N, ILO and IHI. The esti‐
97 mate is returned in WORK(1). No error message related to LWORK
98 is issued by XERBLA. Neither H nor Z are accessed.
99 INFO (output) INTEGER
101 = 0: successful exit
102 the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain
103 those eigenvalues which have been successfully computed. (Fail‐
104 ures are rare.)
105 If INFO .GT. 0 and WANT is .FALSE., then on exit, the remaining
107 unconverged eigenvalues are the eigen- values of the upper Hes‐
108 senberg matrix rows and columns ILO through INFO of the final,
109 output value of H.
110 If INFO .GT. 0 and WANTT is .TRUE., then on exit
112 (*) (initial value of H)*U = U*(final value of H)
114 where U is a unitary matrix. The final value of H is upper Hes‐
116 senberg and triangular in rows and columns INFO+1 through IHI.
117 If INFO .GT. 0 and WANTZ is .TRUE., then on exit
119 (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of
121 Z(ILO:IHI,ILOZ:IHIZ)*U
122 where U is the unitary matrix in (*) (regard- less of the value of
124 WANTT.)
125 If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed.
127 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 CLAQR4(1)