1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
|
SUBROUTINE SSPGST( ITYPE, UPLO, N, AP, BP, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, N
* ..
* .. Array Arguments ..
REAL AP( * ), BP( * )
* ..
*
* Purpose
* =======
*
* SSPGST reduces a real symmetric-definite generalized eigenproblem
* to standard form, using packed storage.
*
* If ITYPE = 1, the problem is A*x = lambda*B*x,
* and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*
* If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
* B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*
* B must have been previously factorized as U**T*U or L*L**T by SPPTRF.
*
* Arguments
* =========
*
* ITYPE (input) INTEGER
* = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
* = 2 or 3: compute U*A*U**T or L**T*A*L.
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored and B is factored as
* U**T*U;
* = 'L': Lower triangle of A is stored and B is factored as
* L*L**T.
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* AP (input/output) REAL array, dimension (N*(N+1)/2)
* On entry, the upper or lower triangle of the symmetric matrix
* A, packed columnwise in a linear array. The j-th column of A
* is stored in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*
* On exit, if INFO = 0, the transformed matrix, stored in the
* same format as A.
*
* BP (input) REAL array, dimension (N*(N+1)/2)
* The triangular factor from the Cholesky factorization of B,
* stored in the same format as A, as returned by SPPTRF.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, HALF
PARAMETER ( ONE = 1.0, HALF = 0.5 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, J1, J1J1, JJ, K, K1, K1K1, KK
REAL AJJ, AKK, BJJ, BKK, CT
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SSCAL, SSPMV, SSPR2, STPMV, STPSV,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SDOT
EXTERNAL LSAME, SDOT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSPGST', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
* J1 and JJ are the indices of A(1,j) and A(j,j)
*
JJ = 0
DO 10 J = 1, N
J1 = JJ + 1
JJ = JJ + J
*
* Compute the j-th column of the upper triangle of A
*
BJJ = BP( JJ )
CALL STPSV( UPLO, 'Transpose', 'Nonunit', J, BP,
$ AP( J1 ), 1 )
CALL SSPMV( UPLO, J-1, -ONE, AP, BP( J1 ), 1, ONE,
$ AP( J1 ), 1 )
CALL SSCAL( J-1, ONE / BJJ, AP( J1 ), 1 )
AP( JJ ) = ( AP( JJ )-SDOT( J-1, AP( J1 ), 1, BP( J1 ),
$ 1 ) ) / BJJ
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
* KK and K1K1 are the indices of A(k,k) and A(k+1,k+1)
*
KK = 1
DO 20 K = 1, N
K1K1 = KK + N - K + 1
*
* Update the lower triangle of A(k:n,k:n)
*
AKK = AP( KK )
BKK = BP( KK )
AKK = AKK / BKK**2
AP( KK ) = AKK
IF( K.LT.N ) THEN
CALL SSCAL( N-K, ONE / BKK, AP( KK+1 ), 1 )
CT = -HALF*AKK
CALL SAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
CALL SSPR2( UPLO, N-K, -ONE, AP( KK+1 ), 1,
$ BP( KK+1 ), 1, AP( K1K1 ) )
CALL SAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
CALL STPSV( UPLO, 'No transpose', 'Non-unit', N-K,
$ BP( K1K1 ), AP( KK+1 ), 1 )
END IF
KK = K1K1
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
* K1 and KK are the indices of A(1,k) and A(k,k)
*
KK = 0
DO 30 K = 1, N
K1 = KK + 1
KK = KK + K
*
* Update the upper triangle of A(1:k,1:k)
*
AKK = AP( KK )
BKK = BP( KK )
CALL STPMV( UPLO, 'No transpose', 'Non-unit', K-1, BP,
$ AP( K1 ), 1 )
CT = HALF*AKK
CALL SAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
CALL SSPR2( UPLO, K-1, ONE, AP( K1 ), 1, BP( K1 ), 1,
$ AP )
CALL SAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
CALL SSCAL( K-1, BKK, AP( K1 ), 1 )
AP( KK ) = AKK*BKK**2
30 CONTINUE
ELSE
*
* Compute L**T *A*L
*
* JJ and J1J1 are the indices of A(j,j) and A(j+1,j+1)
*
JJ = 1
DO 40 J = 1, N
J1J1 = JJ + N - J + 1
*
* Compute the j-th column of the lower triangle of A
*
AJJ = AP( JJ )
BJJ = BP( JJ )
AP( JJ ) = AJJ*BJJ + SDOT( N-J, AP( JJ+1 ), 1,
$ BP( JJ+1 ), 1 )
CALL SSCAL( N-J, BJJ, AP( JJ+1 ), 1 )
CALL SSPMV( UPLO, N-J, ONE, AP( J1J1 ), BP( JJ+1 ), 1,
$ ONE, AP( JJ+1 ), 1 )
CALL STPMV( UPLO, 'Transpose', 'Non-unit', N-J+1,
$ BP( JJ ), AP( JJ ), 1 )
JJ = J1J1
40 CONTINUE
END IF
END IF
RETURN
*
* End of SSPGST
*
END
|
