1DHSEQR(1) LAPACK driver routine (version 3.1) DHSEQR(1)
2
6 DHSEQR - 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**T,
8 where T is an upper quasi-triangular matrix (the Schur form), and Z is
9 the orthogonal matrix of Schur vectors
10
12 SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ,
13 WORK, LWORK, INFO )
14 INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
16 CHARACTER COMPZ, JOB
18 DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
20 Z( LDZ, * )
21
23 DHSEQR computes the eigenvalues of a Hessenberg matrix H
24 and, optionally, the matrices T and Z from the Schur decomposition
25 H = Z T Z**T, where T is an upper quasi-triangular matrix (the
26 Schur form), and Z is the orthogonal matrix of Schur vectors.
27 Optionally Z may be postmultiplied into an input orthogonal
29 matrix Q so that this routine can give the Schur factorization
30 of a matrix A which has been reduced to the Hessenberg form H
31 by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
32
35 JOB (input) CHARACTER*1
36 = 'E': compute eigenvalues only;
37 = 'S': compute eigenvalues and the Schur form T.
38 COMPZ (input) CHARACTER*1
40 = 'N': no Schur vectors are computed;
41 = 'I': Z is initialized to the unit matrix and the matrix Z of
42 Schur vectors of H is returned; = 'V': Z must contain an orthog‐
43 onal matrix Q on entry, and the product Q*Z is returned.
44 N (input) INTEGER
46 The order of the matrix H. N .GE. 0.
47 ILO (input) INTEGER
49 IHI (input) INTEGER It is assumed that H is already upper tri‐
50 angular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
51 normally set by a previous call to DGEBAL, and then passed to
52 DGEHRD when the matrix output by DGEBAL is reduced to Hessenberg
53 form. Otherwise ILO and IHI should be set to 1 and N respec‐
54 tively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then
55 ILO = 1 and IHI = 0.
56 H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
58 On entry, the upper Hessenberg matrix H. On exit, if INFO = 0
59 and JOB = 'S', then H contains the upper quasi-triangular matrix
60 T from the Schur decomposition (the Schur form); 2-by-2 diagonal
61 blocks (corresponding to complex conjugate pairs of eigenvalues)
62 are returned in standard form, with H(i,i) = H(i+1,i+1) and
63 H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the contents
64 of H are unspecified on exit. (The output value of H when
65 INFO.GT.0 is given under the description of INFO below.)
66 Unlike earlier versions of DHSEQR, this subroutine may explicitly
68 H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1,
69 IHI+2, ... N.
70 LDH (input) INTEGER
72 The leading dimension of the array H. LDH .GE. max(1,N).
73 WR (output) DOUBLE PRECISION array, dimension (N)
75 WI (output) DOUBLE PRECISION array, dimension (N) The real and
76 imaginary parts, respectively, of the computed eigenvalues. If
77 two eigenvalues are computed as a complex conjugate pair, they
78 are stored in consecutive elements of WR and WI, say the i-th and
79 (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If JOB = 'S', the
80 eigenvalues are stored in the same order as on the diagonal of
81 the Schur form returned in H, with WR(i) = H(i,i) and, if
82 H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) =
83 sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
84 Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
86 If COMPZ = 'N', Z is not referenced. If COMPZ = 'I', on entry Z
87 need not be set and on exit, if INFO = 0, Z contains the orthogo‐
88 nal matrix Z of the Schur vectors of H. If COMPZ = 'V', on entry
89 Z must contain an N-by-N matrix Q, which is assumed to be equal
90 to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI).
91 On exit, if INFO = 0, Z contains Q*Z. Normally Q is the orthogo‐
92 nal matrix generated by DORGHR after the call to DGEHRD which
93 formed the Hessenberg matrix H. (The output value of Z when
94 INFO.GT.0 is given under the description of INFO below.)
95 LDZ (input) INTEGER
97 The leading dimension of the array Z. if COMPZ = 'I' or COMPZ =
98 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
99 WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
101 On exit, if INFO = 0, WORK(1) returns an estimate of the optimal
102 value for LWORK.
103 LWORK (input) INTEGER The dimension of the array WORK. LWORK
105 .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N
106 may be required for optimal performance. A workspace query to
107 determine the optimal workspace size is recommended.
108 If LWORK = -1, then DHSEQR does a workspace query. In this case,
110 DHSEQR checks the input parameters and estimates the optimal
111 workspace size for the given values of N, ILO and IHI. The esti‐
112 mate is returned in WORK(1). No error message related to LWORK
113 is issued by XERBLA. Neither H nor Z are accessed.
114 INFO (output) INTEGER
116 = 0: successful exit
117 value
118 the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain
119 those eigenvalues which have been successfully computed. (Fail‐
120 ures are rare.)
121 If INFO .GT. 0 and JOB = 'E', then on exit, the remaining uncon‐
123 verged eigenvalues are the eigen- values of the upper Hessenberg
124 matrix rows and columns ILO through INFO of the final, output
125 value of H.
126 If INFO .GT. 0 and JOB = 'S', then on exit
128 (*) (initial value of H)*U = U*(final value of H)
130 where U is an orthogonal matrix. The final value of H is upper
132 Hessenberg and quasi-triangular in rows and columns INFO+1 through
133 IHI.
134 If INFO .GT. 0 and COMPZ = 'V', then on exit
136 (final value of Z) = (initial value of Z)*U
138 where U is the orthogonal matrix in (*) (regard- less of the value
140 of JOB.)
141 If INFO .GT. 0 and COMPZ = 'I', then on exit (final value of Z) =
143 U where U is the orthogonal matrix in (*) (regard- less of the
144 value of JOB.)
145 If INFO .GT. 0 and COMPZ = 'N', then Z is not accessed.
147 LAPACK driver routine (version 3.N1o)vember 2006 DHSEQR(1)