1SLAHQR(1) LAPACK auxiliary routine (version 3.2) SLAHQR(1)
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6 SLAHQR - SLAHQR i an auxiliary routine called by SHSEQR to update the
7 eigenvalues and Schur decomposition already computed by SHSEQR, by
8 dealing with the Hessenberg submatrix in rows and columns ILO to IHI
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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
27 WANTT (input) LOGICAL
28 = .TRUE. : the full Schur form T is required;
29 = .FALSE.: only eigenvalues are required.
30 WANTZ (input) LOGICAL
32 = .TRUE. : the matrix of Schur vectors Z is required;
33 = .FALSE.: Schur vectors are not required.
34 N (input) INTEGER
36 The order of the matrix H. N >= 0.
37 ILO (input) INTEGER
39 IHI (input) INTEGER It is assumed that H is already upper
40 quasi-triangular in rows and columns IHI+1:N, and that
41 H(ILO,ILO-1) = 0 (unless ILO = 1). SLAHQR works primarily with
42 the Hessenberg submatrix in rows and columns ILO to IHI, but
43 applies transformations to all of H if WANTT is .TRUE.. 1 <=
44 ILO <= max(1,IHI); IHI <= N.
45 H (input/output) REAL array, dimension (LDH,N)
47 On entry, the upper Hessenberg matrix H. On exit, if INFO is
48 zero and if WANTT is .TRUE., H is upper quasi-triangular in
49 rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in
50 standard form. If INFO is zero and WANTT is .FALSE., the con‐
51 tents of H are unspecified on exit. The output state of H if
52 INFO is nonzero is given below under the description of INFO.
53 LDH (input) INTEGER
55 The leading dimension of the array H. LDH >= max(1,N).
56 WR (output) REAL array, dimension (N)
58 WI (output) REAL array, dimension (N) The real and imagi‐
59 nary parts, respectively, of the computed eigenvalues ILO to
60 IHI are stored in the corresponding elements of WR and WI. If
61 two eigenvalues are computed as a complex conjugate pair, they
62 are stored in consecutive elements of WR and WI, say the i-th
63 and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If WANTT is
64 .TRUE., the eigenvalues are stored in the same order as on the
65 diagonal of the Schur form returned in H, with WR(i) = H(i,i),
66 and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) =
67 sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
68 ILOZ (input) INTEGER
70 IHIZ (input) INTEGER Specify the rows of Z to which trans‐
71 formations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <=
72 ILO; IHI <= IHIZ <= N.
73 Z (input/output) REAL array, dimension (LDZ,N)
75 If WANTZ is .TRUE., on entry Z must contain the current matrix
76 Z of transformations accumulated by SHSEQR, and on exit Z has
77 been updated; transformations are applied only to the submatrix
78 Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not refer‐
79 enced.
80 LDZ (input) INTEGER
82 The leading dimension of the array Z. LDZ >= max(1,N).
83 INFO (output) INTEGER
85 = 0: successful exit
86 eigenvalues ILO to IHI in a total of 30 iterations per eigen‐
87 value; elements i+1:ihi of WR and WI contain those eigenvalues
88 which have been successfully computed. If INFO .GT. 0 and
89 WANTT is .FALSE., then on exit, the remaining unconverged ei‐
90 genvalues are the eigenvalues of the upper Hessenberg matrix
91 rows and columns ILO thorugh INFO of the final, output value of
92 H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*)
93 (initial value of H)*U = U*(final value of H) where U is an
94 orthognal matrix. The final value of H is upper Hessenberg
95 and triangular in rows and columns INFO+1 through IHI. If INFO
96 .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z) =
97 (initial value of Z)*U where U is the orthogonal matrix in (*)
98 (regardless of the value of WANTT.)
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101 02-96 Based on modifications by
102 David Day, Sandia National Laboratory, USA
103 12-04 Further modifications by
104 Ralph Byers, University of Kansas, USA
105 This is a modified version of SLAHQR from LAPACK version 3.0.
106 It is (1) more robust against overflow and underflow and
107 (2) adopts the more conservative Ahues & Tisseur stopping
108 criterion (LAWN 122, 1997).
109 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 SLAHQR(1)