1STREVC(1) LAPACK routine (version 3.2) STREVC(1)
2
6 STREVC - computes some or all of the right and/or left eigenvectors of
7 a real upper quasi-triangular matrix T
8
10 SUBROUTINE STREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR,
11 MM, M, WORK, INFO )
12 CHARACTER HOWMNY, SIDE
14 INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
16 LOGICAL SELECT( * )
18 REAL T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
20
22 STREVC computes some or all of the right and/or left eigenvectors of a
23 real upper quasi-triangular matrix T. Matrices of this type are pro‐
24 duced by the Schur factorization of a real general matrix: A =
25 Q*T*Q**T, as computed by SHSEQR.
26 The right eigenvector x and the left eigenvector y of T corresponding
27 to an eigenvalue w are defined by:
28 T*x = w*x, (y**H)*T = w*(y**H)
29 where y**H denotes the conjugate transpose of y.
30 The eigenvalues are not input to this routine, but are read directly
31 from the diagonal blocks of T.
32 This routine returns the matrices X and/or Y of right and left eigen‐
33 vectors of T, or the products Q*X and/or Q*Y, where Q is an input
34 matrix. If Q is the orthogonal factor that reduces a matrix A to Schur
35 form T, then Q*X and Q*Y are the matrices of right and left eigenvec‐
36 tors of A.
37
39 SIDE (input) CHARACTER*1
40 = 'R': compute right eigenvectors only;
41 = 'L': compute left eigenvectors only;
42 = 'B': compute both right and left eigenvectors.
43 HOWMNY (input) CHARACTER*1
45 = 'A': compute all right and/or left eigenvectors;
46 = 'B': compute all right and/or left eigenvectors, backtrans‐
47 formed by the matrices in VR and/or VL; = 'S': compute
48 selected right and/or left eigenvectors, as indicated by the
49 logical array SELECT.
50 SELECT (input/output) LOGICAL array, dimension (N)
52 If HOWMNY = 'S', SELECT specifies the eigenvectors to be com‐
53 puted. If w(j) is a real eigenvalue, the corresponding real
54 eigenvector is computed if SELECT(j) is .TRUE.. If w(j) and
55 w(j+1) are the real and imaginary parts of a complex eigenval‐
56 ue, the corresponding complex eigenvector is computed if either
57 SELECT(j) or SELECT(j+1) is .TRUE., and on exit SELECT(j) is
58 set to .TRUE. and SELECT(j+1) is set to .FALSE.. Not refer‐
59 enced if HOWMNY = 'A' or 'B'.
60 N (input) INTEGER
62 The order of the matrix T. N >= 0.
63 T (input) REAL array, dimension (LDT,N)
65 The upper quasi-triangular matrix T in Schur canonical form.
66 LDT (input) INTEGER
68 The leading dimension of the array T. LDT >= max(1,N).
69 VL (input/output) REAL array, dimension (LDVL,MM)
71 On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must con‐
72 tain an N-by-N matrix Q (usually the orthogonal matrix Q of
73 Schur vectors returned by SHSEQR). On exit, if SIDE = 'L' or
74 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigen‐
75 vectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S',
76 the left eigenvectors of T specified by SELECT, stored consecu‐
77 tively in the columns of VL, in the same order as their eigen‐
78 values. A complex eigenvector corresponding to a complex ei‐
79 genvalue is stored in two consecutive columns, the first hold‐
80 ing the real part, and the second the imaginary part. Not ref‐
81 erenced if SIDE = 'R'.
82 LDVL (input) INTEGER
84 The leading dimension of the array VL. LDVL >= 1, and if SIDE
85 = 'L' or 'B', LDVL >= N.
86 VR (input/output) REAL array, dimension (LDVR,MM)
88 On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must con‐
89 tain an N-by-N matrix Q (usually the orthogonal matrix Q of
90 Schur vectors returned by SHSEQR). On exit, if SIDE = 'R' or
91 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigen‐
92 vectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S',
93 the right eigenvectors of T specified by SELECT, stored consec‐
94 utively in the columns of VR, in the same order as their eigen‐
95 values. A complex eigenvector corresponding to a complex ei‐
96 genvalue is stored in two consecutive columns, the first hold‐
97 ing the real part and the second the imaginary part. Not ref‐
98 erenced if SIDE = 'L'.
99 LDVR (input) INTEGER
101 The leading dimension of the array VR. LDVR >= 1, and if SIDE
102 = 'R' or 'B', LDVR >= N.
103 MM (input) INTEGER
105 The number of columns in the arrays VL and/or VR. MM >= M.
106 M (output) INTEGER
108 The number of columns in the arrays VL and/or VR actually used
109 to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to
110 N. Each selected real eigenvector occupies one column and each
111 selected complex eigenvector occupies two columns.
112 WORK (workspace) REAL array, dimension (3*N)
114 INFO (output) INTEGER
116 = 0: successful exit
117 < 0: if INFO = -i, the i-th argument had an illegal value
118
120 The algorithm used in this program is basically backward (forward) sub‐
121 stitution, with scaling to make the the code robust against possible
122 overflow.
123 Each eigenvector is normalized so that the element of largest magnitude
124 has magnitude 1; here the magnitude of a complex number (x,y) is taken
125 to be |x| + |y|.
126 LAPACK routine (version 3.2) November 2008 STREVC(1)