1SLAQR4(1) LAPACK auxiliary routine (version 3.1) SLAQR4(1)
2
6 SLAQR4 - 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 SLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ,
13 IHIZ, Z, LDZ, WORK, LWORK, INFO )
14 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
16 LOGICAL WANTT, WANTZ
18 REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, *
20 )
21
23 SLAQR4 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 WANTT (input) LOGICAL
36 = .TRUE. : the full Schur form T is required;
37 = .FALSE.: only eigenvalues are required.
38 WANTZ (input) LOGICAL
40 = .TRUE. : the matrix of Schur vectors Z is required;
41 = .FALSE.: Schur vectors are not required.
42 N (input) INTEGER
44 The order of the matrix H. N .GE. 0.
45 ILO (input) INTEGER
47 IHI (input) INTEGER It is assumed that H is already upper tri‐
48 angular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
49 H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous
50 call to SGEBAL, and then passed to SGEHRD when the matrix output
51 by SGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI
52 should be set to 1 and N, respectively. If N.GT.0, then
53 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.
54 H (input/output) REAL array, dimension (LDH,N)
56 On entry, the upper Hessenberg matrix H. On exit, if INFO = 0
57 and WANTT is .TRUE., then H contains the upper quasi-triangular
58 matrix T from the Schur decomposition (the Schur form); 2-by-2
59 diagonal blocks (corresponding to complex conjugate pairs of ei‐
60 genvalues) are returned in standard form, with H(i,i) =
61 H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
62 (The output value of H when INFO.GT.0 is given under the descrip‐
63 tion of INFO below.)
64 This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j =
66 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
67 LDH (input) INTEGER
69 The leading dimension of the array H. LDH .GE. max(1,N).
70 WR (output) REAL array, dimension (IHI)
72 WI (output) REAL array, dimension (IHI) The real and imaginary
73 parts, respectively, of the computed eigenvalues of
74 H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI)
75 and WI(ILO:IHI). If two eigenvalues are computed as a complex
76 conjugate pair, they are stored in consecutive elements of WR and
77 WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT.
78 0. If WANTT is .TRUE., then the eigenvalues are stored in the
79 same order as on the diagonal of the Schur form returned in H,
80 with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
81 block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
82 ILOZ (input) INTEGER
84 IHIZ (input) INTEGER Specify the rows of Z to which trans‐
85 formations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ
86 .LE. ILO; IHI .LE. IHIZ .LE. N.
87 Z (input/output) REAL array, dimension (LDZ,IHI)
89 If WANTZ is .FALSE., then Z is not referenced. If WANTZ is
90 .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
91 replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
92 orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value
93 of Z when INFO.GT.0 is given under the description of INFO
94 below.)
95 LDZ (input) INTEGER
97 The leading dimension of the array Z. if WANTZ is .TRUE. then
98 LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
99 WORK (workspace/output) REAL array, dimension LWORK
101 On exit, if LWORK = -1, WORK(1) returns an estimate of the opti‐
102 mal 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 SLAQR4 does a workspace query. In this case,
110 SLAQR4 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 the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain
118 those eigenvalues which have been successfully computed. (Fail‐
119 ures are rare.)
120 If INFO .GT. 0 and WANT is .FALSE., then on exit, the remaining
122 unconverged eigenvalues are the eigen- values of the upper Hes‐
123 senberg matrix rows and columns ILO through INFO of the final,
124 output value of H.
125 If INFO .GT. 0 and WANTT is .TRUE., then on exit
127 (*) (initial value of H)*U = U*(final value of H)
129 where U is an orthogonal matrix. The final value of H is upper
131 Hessenberg and quasi-triangular in rows and columns INFO+1 through
132 IHI.
133 If INFO .GT. 0 and WANTZ is .TRUE., then on exit
135 (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of
137 Z(ILO:IHI,ILOZ:IHIZ)*U
138 where U is the orthogonal matrix in (*) (regard- less of the value
140 of WANTT.)
141 If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed.
143 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 SLAQR4(1)