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
|
*> \brief \b SPPTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SPPTRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spptrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/spptrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spptrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SPPTRF( UPLO, N, AP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* REAL AP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SPPTRF computes the Cholesky factorization of a real symmetric
*> positive definite matrix A stored in packed format.
*>
*> The factorization has the form
*> A = U**T * U, if UPLO = 'U', or
*> A = L * L**T, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is 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.
*> See below for further details.
*>
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**T*U or A = L*L**T, in the same
*> storage format as A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, the leading minor of order i is not
*> positive definite, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup realOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The packed storage scheme is illustrated by the following example
*> when N = 4, UPLO = 'U':
*>
*> Two-dimensional storage of the symmetric matrix A:
*>
*> a11 a12 a13 a14
*> a22 a23 a24
*> a33 a34 (aij = aji)
*> a44
*>
*> Packed storage of the upper triangle of A:
*>
*> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SPPTRF( UPLO, N, AP, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
REAL AP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, JC, JJ
REAL AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SDOT
EXTERNAL LSAME, SDOT
* ..
* .. External Subroutines ..
EXTERNAL SSCAL, SSPR, STPSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPPTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**T*U.
*
JJ = 0
DO 10 J = 1, N
JC = JJ + 1
JJ = JJ + J
*
* Compute elements 1:J-1 of column J.
*
IF( J.GT.1 )
$ CALL STPSV( 'Upper', 'Transpose', 'Non-unit', J-1, AP,
$ AP( JC ), 1 )
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = AP( JJ ) - SDOT( J-1, AP( JC ), 1, AP( JC ), 1 )
IF( AJJ.LE.ZERO ) THEN
AP( JJ ) = AJJ
GO TO 30
END IF
AP( JJ ) = SQRT( AJJ )
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L**T.
*
JJ = 1
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = AP( JJ )
IF( AJJ.LE.ZERO ) THEN
AP( JJ ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
AP( JJ ) = AJJ
*
* Compute elements J+1:N of column J and update the trailing
* submatrix.
*
IF( J.LT.N ) THEN
CALL SSCAL( N-J, ONE / AJJ, AP( JJ+1 ), 1 )
CALL SSPR( 'Lower', N-J, -ONE, AP( JJ+1 ), 1,
$ AP( JJ+N-J+1 ) )
JJ = JJ + N - J + 1
END IF
20 CONTINUE
END IF
GO TO 40
*
30 CONTINUE
INFO = J
*
40 CONTINUE
RETURN
*
* End of SPPTRF
*
END
|
