1SLAHQR(1) LAPACK auxiliary routine (version 3.1) SLAHQR(1)
2
6 SLAHQR - i an auxiliary routine called by SHSEQR to update the eigen‐
7 values and Schur decomposition already computed by SHSEQR, by dealing
8 with the Hessenberg submatrix in rows and columns ILO to IHI
9
11 SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ,
12 IHIZ, Z, LDZ, INFO )
13 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
15 LOGICAL WANTT, WANTZ
17 REAL H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
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21 SLAHQR is an auxiliary routine called by SHSEQR to update the
22 eigenvalues and Schur decomposition already computed by SHSEQR, by
23 dealing with the Hessenberg submatrix in rows and columns ILO to
24 IHI.
25
28 WANTT (input) LOGICAL
29 = .TRUE. : the full Schur form T is required;
30 = .FALSE.: only eigenvalues are required.
31 WANTZ (input) LOGICAL
33 = .TRUE. : the matrix of Schur vectors Z is required;
34 = .FALSE.: Schur vectors are not required.
35 N (input) INTEGER
37 The order of the matrix H. N >= 0.
38 ILO (input) INTEGER
40 IHI (input) INTEGER It is assumed that H is already upper
41 quasi-triangular in rows and columns IHI+1:N, and that
42 H(ILO,ILO-1) = 0 (unless ILO = 1). SLAHQR works primarily with
43 the Hessenberg submatrix in rows and columns ILO to IHI, but
44 applies transformations to all of H if WANTT is .TRUE.. 1 <=
45 ILO <= max(1,IHI); IHI <= N.
46 H (input/output) REAL array, dimension (LDH,N)
48 On entry, the upper Hessenberg matrix H. On exit, if INFO is
49 zero and if WANTT is .TRUE., H is upper quasi-triangular in
50 rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in
51 standard form. If INFO is zero and WANTT is .FALSE., the con‐
52 tents of H are unspecified on exit. The output state of H if
53 INFO is nonzero is given below under the description of INFO.
54 LDH (input) INTEGER
56 The leading dimension of the array H. LDH >= max(1,N).
57 WR (output) REAL array, dimension (N)
59 WI (output) REAL array, dimension (N) The real and imagi‐
60 nary parts, respectively, of the computed eigenvalues ILO to
61 IHI are stored in the corresponding elements of WR and WI. If
62 two eigenvalues are computed as a complex conjugate pair, they
63 are stored in consecutive elements of WR and WI, say the i-th
64 and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If WANTT is
65 .TRUE., the eigenvalues are stored in the same order as on the
66 diagonal of the Schur form returned in H, with WR(i) = H(i,i),
67 and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) =
68 sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
69 ILOZ (input) INTEGER
71 IHIZ (input) INTEGER Specify the rows of Z to which trans‐
72 formations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <=
73 ILO; IHI <= IHIZ <= N.
74 Z (input/output) REAL array, dimension (LDZ,N)
76 If WANTZ is .TRUE., on entry Z must contain the current matrix
77 Z of transformations accumulated by SHSEQR, and on exit Z has
78 been updated; transformations are applied only to the submatrix
79 Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not refer‐
80 enced.
81 LDZ (input) INTEGER
83 The leading dimension of the array Z. LDZ >= max(1,N).
84 INFO (output) INTEGER
86 = 0: successful exit
87 eigenvalues ILO to IHI in a total of 30 iterations per eigen‐
88 value; elements i+1:ihi of WR and WI contain those eigenvalues
89 which have been successfully computed.
90 If INFO .GT. 0 and WANTT is .FALSE., then on exit, the remain‐
92 ing unconverged eigenvalues are the eigenvalues of the upper
93 Hessenberg matrix rows and columns ILO thorugh INFO of the
94 final, output value of H.
95 If INFO .GT. 0 and WANTT is .TRUE., then on exit (*)
97 (initial value of H)*U = U*(final value of H) where U is an
98 orthognal matrix. The final value of H is upper Hessenberg
99 and triangular in rows and columns INFO+1 through IHI.
100 If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value
102 of Z) = (initial value of Z)*U where U is the orthogonal
103 matrix in (*) (regardless of the value of WANTT.)
104
106 02-96 Based on modifications by
107 David Day, Sandia National Laboratory, USA
108 12-04 Further modifications by
110 Ralph Byers, University of Kansas, USA
111 This is a modified version of SLAHQR from LAPACK version 3.0.
113 It is (1) more robust against overflow and underflow and
114 (2) adopts the more conservative Ahues & Tisseur stopping
115 criterion (LAWN 122, 1997).
116 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 SLAHQR(1)