LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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dpbstf.f
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1*> \brief \b DPBSTF
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbstf.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
22*
23* .. Scalar Arguments ..
24* CHARACTER UPLO
25* INTEGER INFO, KD, LDAB, N
26* ..
27* .. Array Arguments ..
28* DOUBLE PRECISION AB( LDAB, * )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> DPBSTF 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 DSBGST.
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 DOUBLE PRECISION 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*> \ingroup pbstf
114*
115*> \par Further Details:
116* =====================
117*>
118*> \verbatim
119*>
120*> The band storage scheme is illustrated by the following example, when
121*> N = 7, KD = 2:
122*>
123*> S = ( s11 s12 s13 )
124*> ( s22 s23 s24 )
125*> ( s33 s34 )
126*> ( s44 )
127*> ( s53 s54 s55 )
128*> ( s64 s65 s66 )
129*> ( s75 s76 s77 )
130*>
131*> If UPLO = 'U', the array AB holds:
132*>
133*> on entry: on exit:
134*>
135*> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
136*> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
137*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
138*>
139*> If UPLO = 'L', the array AB holds:
140*>
141*> on entry: on exit:
142*>
143*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
144*> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
145*> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
146*>
147*> Array elements marked * are not used by the routine.
148*> \endverbatim
149*>
150* =====================================================================
151 SUBROUTINE dpbstf( UPLO, N, KD, AB, LDAB, INFO )
152*
153* -- LAPACK computational routine --
154* -- LAPACK is a software package provided by Univ. of Tennessee, --
155* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
156*
157* .. Scalar Arguments ..
158 CHARACTER UPLO
159 INTEGER INFO, KD, LDAB, N
160* ..
161* .. Array Arguments ..
162 DOUBLE PRECISION AB( LDAB, * )
163* ..
164*
165* =====================================================================
166*
167* .. Parameters ..
168 DOUBLE PRECISION ONE, ZERO
169 parameter( one = 1.0d+0, zero = 0.0d+0 )
170* ..
171* .. Local Scalars ..
172 LOGICAL UPPER
173 INTEGER J, KLD, KM, M
174 DOUBLE PRECISION AJJ
175* ..
176* .. External Functions ..
177 LOGICAL LSAME
178 EXTERNAL lsame
179* ..
180* .. External Subroutines ..
181 EXTERNAL dscal, dsyr, xerbla
182* ..
183* .. Intrinsic Functions ..
184 INTRINSIC max, min, sqrt
185* ..
186* .. Executable Statements ..
187*
188* Test the input parameters.
189*
190 info = 0
191 upper = lsame( uplo, 'U' )
192 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
193 info = -1
194 ELSE IF( n.LT.0 ) THEN
195 info = -2
196 ELSE IF( kd.LT.0 ) THEN
197 info = -3
198 ELSE IF( ldab.LT.kd+1 ) THEN
199 info = -5
200 END IF
201 IF( info.NE.0 ) THEN
202 CALL xerbla( 'DPBSTF', -info )
203 RETURN
204 END IF
205*
206* Quick return if possible
207*
208 IF( n.EQ.0 )
209 \$ RETURN
210*
211 kld = max( 1, ldab-1 )
212*
213* Set the splitting point m.
214*
215 m = ( n+kd ) / 2
216*
217 IF( upper ) THEN
218*
219* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
220*
221 DO 10 j = n, m + 1, -1
222*
223* Compute s(j,j) and test for non-positive-definiteness.
224*
225 ajj = ab( kd+1, j )
226 IF( ajj.LE.zero )
227 \$ GO TO 50
228 ajj = sqrt( ajj )
229 ab( kd+1, j ) = ajj
230 km = min( j-1, kd )
231*
232* Compute elements j-km:j-1 of the j-th column and update the
233* the leading submatrix within the band.
234*
235 CALL dscal( km, one / ajj, ab( kd+1-km, j ), 1 )
236 CALL dsyr( 'Upper', km, -one, ab( kd+1-km, j ), 1,
237 \$ ab( kd+1, j-km ), kld )
238 10 CONTINUE
239*
240* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
241*
242 DO 20 j = 1, m
243*
244* Compute s(j,j) and test for non-positive-definiteness.
245*
246 ajj = ab( kd+1, j )
247 IF( ajj.LE.zero )
248 \$ GO TO 50
249 ajj = sqrt( ajj )
250 ab( kd+1, j ) = ajj
251 km = min( kd, m-j )
252*
253* Compute elements j+1:j+km of the j-th row and update the
254* trailing submatrix within the band.
255*
256 IF( km.GT.0 ) THEN
257 CALL dscal( km, one / ajj, ab( kd, j+1 ), kld )
258 CALL dsyr( 'Upper', km, -one, ab( kd, j+1 ), kld,
259 \$ ab( kd+1, j+1 ), kld )
260 END IF
261 20 CONTINUE
262 ELSE
263*
264* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
265*
266 DO 30 j = n, m + 1, -1
267*
268* Compute s(j,j) and test for non-positive-definiteness.
269*
270 ajj = ab( 1, j )
271 IF( ajj.LE.zero )
272 \$ GO TO 50
273 ajj = sqrt( ajj )
274 ab( 1, j ) = ajj
275 km = min( j-1, kd )
276*
277* Compute elements j-km:j-1 of the j-th row and update the
278* trailing submatrix within the band.
279*
280 CALL dscal( km, one / ajj, ab( km+1, j-km ), kld )
281 CALL dsyr( 'Lower', km, -one, ab( km+1, j-km ), kld,
282 \$ ab( 1, j-km ), kld )
283 30 CONTINUE
284*
285* Factorize the updated submatrix A(1:m,1:m) as U**T*U.
286*
287 DO 40 j = 1, m
288*
289* Compute s(j,j) and test for non-positive-definiteness.
290*
291 ajj = ab( 1, j )
292 IF( ajj.LE.zero )
293 \$ GO TO 50
294 ajj = sqrt( ajj )
295 ab( 1, j ) = ajj
296 km = min( kd, m-j )
297*
298* Compute elements j+1:j+km of the j-th column and update the
299* trailing submatrix within the band.
300*
301 IF( km.GT.0 ) THEN
302 CALL dscal( km, one / ajj, ab( 2, j ), 1 )
303 CALL dsyr( 'Lower', km, -one, ab( 2, j ), 1,
304 \$ ab( 1, j+1 ), kld )
305 END IF
306 40 CONTINUE
307 END IF
308 RETURN
309*
310 50 CONTINUE
311 info = j
312 RETURN
313*
314* End of DPBSTF
315*
316 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dsyr(uplo, n, alpha, x, incx, a, lda)
DSYR
Definition dsyr.f:132
subroutine dpbstf(uplo, n, kd, ab, ldab, info)
DPBSTF
Definition dpbstf.f:152
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79