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
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
|
SUBROUTINE SSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
$ IWORK, LIWORK, INFO )
*
*
* -- LAPACK driver routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDZ, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL AP( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* SSPEVD computes all the eigenvalues and, optionally, eigenvectors
* of a real symmetric matrix A in packed storage. If eigenvectors are
* desired, it uses a divide and conquer algorithm.
*
* The divide and conquer algorithm makes very mild assumptions about
* floating point arithmetic. It will work on machines with a guard
* digit in add/subtract, or on those binary machines without guard
* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
* Cray-2. It could conceivably fail on hexadecimal or decimal machines
* without guard digits, but we know of none.
*
* Arguments
* =========
*
* JOBZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only;
* = 'V': Compute eigenvalues and eigenvectors.
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
* On exit, AP is overwritten by values generated during the
* reduction to tridiagonal form. If UPLO = 'U', the diagonal
* and first superdiagonal of the tridiagonal matrix T overwrite
* the corresponding elements of A, and if UPLO = 'L', the
* diagonal and first subdiagonal of T overwrite the
* corresponding elements of A.
*
* W (output) REAL array, dimension (N)
* If INFO = 0, the eigenvalues in ascending order.
*
* Z (output) REAL array, dimension (LDZ, N)
* If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
* eigenvectors of the matrix A, with the i-th column of Z
* holding the eigenvector associated with W(i).
* If JOBZ = 'N', then Z is not referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* JOBZ = 'V', LDZ >= max(1,N).
*
* WORK (workspace/output) REAL array,
* dimension (LWORK)
* On exit, if LWORK > 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* If N <= 1, LWORK must be at least 1.
* If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
* If JOBZ = 'V' and N > 1, LWORK must be at least
* ( 1 + 5*N + 2*N*lg N + 2*N**2 ),
* where lg( N ) = smallest integer k such
* that 2**k >= N.
*
* IWORK (workspace/output) INTEGER array, dimension (LIWORK)
* On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK.
* If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
* If JOBZ = 'V' and N > 1, LIWORK must be at least 2 + 5*N.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, the algorithm failed to converge; i
* off-diagonal elements of an intermediate tridiagonal
* form did not converge to zero.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL WANTZ
INTEGER IINFO, INDE, INDTAU, INDWRK, ISCALE, LGN,
$ LIWMIN, LLWORK, LWMIN
REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANSP
EXTERNAL LSAME, SLAMCH, SLANSP
* ..
* .. External Subroutines ..
EXTERNAL SOPMTR, SSCAL, SSPTRD, SSTEDC, SSTERF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, LOG, REAL, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
*
INFO = 0
IF( N.LE.1 ) THEN
LGN = 0
LIWMIN = 1
LWMIN = 1
ELSE
LGN = INT( LOG( REAL( N ) ) / LOG( TWO ) )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( WANTZ ) THEN
LIWMIN = 2 + 5*N
LWMIN = 1 + 5*N + 2*N*LGN + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N
END IF
END IF
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -7
ELSE IF( LWORK.LT.LWMIN ) THEN
INFO = -9
ELSE IF( LIWORK.LT.LIWMIN ) THEN
INFO = -11
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSPEVD ', -INFO )
GO TO 10
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ GO TO 10
*
IF( N.EQ.1 ) THEN
W( 1 ) = AP( 1 )
IF( WANTZ )
$ Z( 1, 1 ) = ONE
GO TO 10
END IF
*
* Get machine constants.
*
SAFMIN = SLAMCH( 'Safe minimum' )
EPS = SLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = SLANSP( 'M', UPLO, N, AP, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL SSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
END IF
*
* Call SSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
CALL SSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ), IINFO )
*
* For eigenvalues only, call SSTERF. For eigenvectors, first call
* SSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
* tridiagonal matrix, then call SOPMTR to multiply it by the
* Householder transformations represented in AP.
*
IF( .NOT.WANTZ ) THEN
CALL SSTERF( N, W, WORK( INDE ), INFO )
ELSE
INDWRK = INDTAU + N
LLWORK = LWORK - INDWRK + 1
CALL SSTEDC( 'I', N, W, WORK( INDE ), Z, LDZ, WORK( INDWRK ),
$ LLWORK, IWORK, LIWORK, INFO )
CALL SOPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z, LDZ,
$ WORK( INDWRK ), IINFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 )
$ CALL SSCAL( N, ONE / SIGMA, W, 1 )
*
10 CONTINUE
IF( LWORK.GT.0 )
$ WORK( 1 ) = LWMIN
IF( LIWORK.GT.0 )
$ IWORK( 1 ) = LIWMIN
RETURN
*
* End of SSPEVD
*
END
|
