LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
spbstf.f
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1 *> \brief \b SPBSTF
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPBSTF( UPLO, N, KD, AB, LDAB, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, KD, LDAB, N
26 * ..
27 * .. Array Arguments ..
28 * REAL AB( LDAB, * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SPBSTF computes a split Cholesky factorization of a real
38 *> symmetric positive definite band matrix A.
39 *>
40 *> This routine is designed to be used in conjunction with SSBGST.
41 *>
42 *> The factorization has the form A = S**T*S where S is a band matrix
43 *> of the same bandwidth as A and the following structure:
44 *>
45 *> S = ( U )
46 *> ( M L )
47 *>
48 *> where U is upper triangular of order m = (n+kd)/2, and L is lower
49 *> triangular of order n-m.
50 *> \endverbatim
51 *
52 * Arguments:
53 * ==========
54 *
55 *> \param[in] UPLO
56 *> \verbatim
57 *> UPLO is CHARACTER*1
58 *> = 'U': Upper triangle of A is stored;
59 *> = 'L': Lower triangle of A is stored.
60 *> \endverbatim
61 *>
62 *> \param[in] N
63 *> \verbatim
64 *> N is INTEGER
65 *> The order of the matrix A. N >= 0.
66 *> \endverbatim
67 *>
68 *> \param[in] KD
69 *> \verbatim
70 *> KD is INTEGER
71 *> The number of superdiagonals of the matrix A if UPLO = 'U',
72 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
73 *> \endverbatim
74 *>
75 *> \param[in,out] AB
76 *> \verbatim
77 *> AB is REAL array, dimension (LDAB,N)
78 *> On entry, the upper or lower triangle of the symmetric band
79 *> matrix A, stored in the first kd+1 rows of the array. The
80 *> j-th column of A is stored in the j-th column of the array AB
81 *> as follows:
82 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
83 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
84 *>
85 *> On exit, if INFO = 0, the factor S from the split Cholesky
86 *> factorization A = S**T*S. See Further Details.
87 *> \endverbatim
88 *>
89 *> \param[in] LDAB
90 *> \verbatim
91 *> LDAB is INTEGER
92 *> The leading dimension of the array AB. LDAB >= KD+1.
93 *> \endverbatim
94 *>
95 *> \param[out] INFO
96 *> \verbatim
97 *> INFO is INTEGER
98 *> = 0: successful exit
99 *> < 0: if INFO = -i, the i-th argument had an illegal value
100 *> > 0: if INFO = i, the factorization could not be completed,
101 *> because the updated element a(i,i) was negative; the
102 *> matrix A is not positive definite.
103 *> \endverbatim
104 *
105 * Authors:
106 * ========
107 *
108 *> \author Univ. of Tennessee
109 *> \author Univ. of California Berkeley
110 *> \author Univ. of Colorado Denver
111 *> \author NAG Ltd.
112 *
113 *> \date December 2016
114 *
115 *> \ingroup realOTHERcomputational
116 *
117 *> \par Further Details:
118 * =====================
119 *>
120 *> \verbatim
121 *>
122 *> The band storage scheme is illustrated by the following example, when
123 *> N = 7, KD = 2:
124 *>
125 *> S = ( s11 s12 s13 )
126 *> ( s22 s23 s24 )
127 *> ( s33 s34 )
128 *> ( s44 )
129 *> ( s53 s54 s55 )
130 *> ( s64 s65 s66 )
131 *> ( s75 s76 s77 )
132 *>
133 *> If UPLO = 'U', the array AB holds:
134 *>
135 *> on entry: on exit:
136 *>
137 *> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
138 *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
139 *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
140 *>
141 *> If UPLO = 'L', the array AB holds:
142 *>
143 *> on entry: on exit:
144 *>
145 *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
146 *> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
147 *> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
148 *>
149 *> Array elements marked * are not used by the routine.
150 *> \endverbatim
151 *>
152 * =====================================================================
153  SUBROUTINE spbstf( UPLO, N, KD, AB, LDAB, INFO )
154 *
155 * -- LAPACK computational routine (version 3.7.0) --
156 * -- LAPACK is a software package provided by Univ. of Tennessee, --
157 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
158 * December 2016
159 *
160 * .. Scalar Arguments ..
161  CHARACTER UPLO
162  INTEGER INFO, KD, LDAB, N
163 * ..
164 * .. Array Arguments ..
165  REAL AB( ldab, * )
166 * ..
167 *
168 * =====================================================================
169 *
170 * .. Parameters ..
171  REAL ONE, ZERO
172  parameter( one = 1.0e+0, zero = 0.0e+0 )
173 * ..
174 * .. Local Scalars ..
175  LOGICAL UPPER
176  INTEGER J, KLD, KM, M
177  REAL AJJ
178 * ..
179 * .. External Functions ..
180  LOGICAL LSAME
181  EXTERNAL lsame
182 * ..
183 * .. External Subroutines ..
184  EXTERNAL sscal, ssyr, xerbla
185 * ..
186 * .. Intrinsic Functions ..
187  INTRINSIC max, min, sqrt
188 * ..
189 * .. Executable Statements ..
190 *
191 * Test the input parameters.
192 *
193  info = 0
194  upper = lsame( uplo, 'U' )
195  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
196  info = -1
197  ELSE IF( n.LT.0 ) THEN
198  info = -2
199  ELSE IF( kd.LT.0 ) THEN
200  info = -3
201  ELSE IF( ldab.LT.kd+1 ) THEN
202  info = -5
203  END IF
204  IF( info.NE.0 ) THEN
205  CALL xerbla( 'SPBSTF', -info )
206  RETURN
207  END IF
208 *
209 * Quick return if possible
210 *
211  IF( n.EQ.0 )
212  $ RETURN
213 *
214  kld = max( 1, ldab-1 )
215 *
216 * Set the splitting point m.
217 *
218  m = ( n+kd ) / 2
219 *
220  IF( upper ) THEN
221 *
222 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
223 *
224  DO 10 j = n, m + 1, -1
225 *
226 * Compute s(j,j) and test for non-positive-definiteness.
227 *
228  ajj = ab( kd+1, j )
229  IF( ajj.LE.zero )
230  $ GO TO 50
231  ajj = sqrt( ajj )
232  ab( kd+1, j ) = ajj
233  km = min( j-1, kd )
234 *
235 * Compute elements j-km:j-1 of the j-th column and update the
236 * the leading submatrix within the band.
237 *
238  CALL sscal( km, one / ajj, ab( kd+1-km, j ), 1 )
239  CALL ssyr( 'Upper', km, -one, ab( kd+1-km, j ), 1,
240  $ ab( kd+1, j-km ), kld )
241  10 CONTINUE
242 *
243 * Factorize the updated submatrix A(1:m,1:m) as U**T*U.
244 *
245  DO 20 j = 1, m
246 *
247 * Compute s(j,j) and test for non-positive-definiteness.
248 *
249  ajj = ab( kd+1, j )
250  IF( ajj.LE.zero )
251  $ GO TO 50
252  ajj = sqrt( ajj )
253  ab( kd+1, j ) = ajj
254  km = min( kd, m-j )
255 *
256 * Compute elements j+1:j+km of the j-th row and update the
257 * trailing submatrix within the band.
258 *
259  IF( km.GT.0 ) THEN
260  CALL sscal( km, one / ajj, ab( kd, j+1 ), kld )
261  CALL ssyr( 'Upper', km, -one, ab( kd, j+1 ), kld,
262  $ ab( kd+1, j+1 ), kld )
263  END IF
264  20 CONTINUE
265  ELSE
266 *
267 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
268 *
269  DO 30 j = n, m + 1, -1
270 *
271 * Compute s(j,j) and test for non-positive-definiteness.
272 *
273  ajj = ab( 1, j )
274  IF( ajj.LE.zero )
275  $ GO TO 50
276  ajj = sqrt( ajj )
277  ab( 1, j ) = ajj
278  km = min( j-1, kd )
279 *
280 * Compute elements j-km:j-1 of the j-th row and update the
281 * trailing submatrix within the band.
282 *
283  CALL sscal( km, one / ajj, ab( km+1, j-km ), kld )
284  CALL ssyr( 'Lower', km, -one, ab( km+1, j-km ), kld,
285  $ ab( 1, j-km ), kld )
286  30 CONTINUE
287 *
288 * Factorize the updated submatrix A(1:m,1:m) as U**T*U.
289 *
290  DO 40 j = 1, m
291 *
292 * Compute s(j,j) and test for non-positive-definiteness.
293 *
294  ajj = ab( 1, j )
295  IF( ajj.LE.zero )
296  $ GO TO 50
297  ajj = sqrt( ajj )
298  ab( 1, j ) = ajj
299  km = min( kd, m-j )
300 *
301 * Compute elements j+1:j+km of the j-th column and update the
302 * trailing submatrix within the band.
303 *
304  IF( km.GT.0 ) THEN
305  CALL sscal( km, one / ajj, ab( 2, j ), 1 )
306  CALL ssyr( 'Lower', km, -one, ab( 2, j ), 1,
307  $ ab( 1, j+1 ), kld )
308  END IF
309  40 CONTINUE
310  END IF
311  RETURN
312 *
313  50 CONTINUE
314  info = j
315  RETURN
316 *
317 * End of SPBSTF
318 *
319  END
subroutine ssyr(UPLO, N, ALPHA, X, INCX, A, LDA)
SSYR
Definition: ssyr.f:134
subroutine spbstf(UPLO, N, KD, AB, LDAB, INFO)
SPBSTF
Definition: spbstf.f:154
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81