LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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cpbstf.f
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1*> \brief \b CPBSTF
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/cpbstf.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CPBSTF( UPLO, N, KD, AB, LDAB, INFO )
22*
23* .. Scalar Arguments ..
24* CHARACTER UPLO
25* INTEGER INFO, KD, LDAB, N
26* ..
27* .. Array Arguments ..
28* COMPLEX AB( LDAB, * )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> CPBSTF computes a split Cholesky factorization of a complex
38*> Hermitian positive definite band matrix A.
39*>
40*> This routine is designed to be used in conjunction with CHBGST.
41*>
42*> The factorization has the form A = S**H*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 COMPLEX array, dimension (LDAB,N)
78*> On entry, the upper or lower triangle of the Hermitian 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**H*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**H s64**H s75**H
136*> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
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**H s23**H s34**H s54 s65 s76 *
145*> a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
146*>
147*> Array elements marked * are not used by the routine; s12**H denotes
148*> conjg(s12); the diagonal elements of S are real.
149*> \endverbatim
150*>
151* =====================================================================
152 SUBROUTINE cpbstf( UPLO, N, KD, AB, LDAB, INFO )
153*
154* -- LAPACK computational routine --
155* -- LAPACK is a software package provided by Univ. of Tennessee, --
156* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157*
158* .. Scalar Arguments ..
159 CHARACTER UPLO
160 INTEGER INFO, KD, LDAB, N
161* ..
162* .. Array Arguments ..
163 COMPLEX AB( LDAB, * )
164* ..
165*
166* =====================================================================
167*
168* .. Parameters ..
169 REAL ONE, ZERO
170 parameter( one = 1.0e+0, zero = 0.0e+0 )
171* ..
172* .. Local Scalars ..
173 LOGICAL UPPER
174 INTEGER J, KLD, KM, M
175 REAL AJJ
176* ..
177* .. External Functions ..
178 LOGICAL LSAME
179 EXTERNAL lsame
180* ..
181* .. External Subroutines ..
182 EXTERNAL cher, clacgv, csscal, xerbla
183* ..
184* .. Intrinsic Functions ..
185 INTRINSIC max, min, real, sqrt
186* ..
187* .. Executable Statements ..
188*
189* Test the input parameters.
190*
191 info = 0
192 upper = lsame( uplo, 'U' )
193 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
194 info = -1
195 ELSE IF( n.LT.0 ) THEN
196 info = -2
197 ELSE IF( kd.LT.0 ) THEN
198 info = -3
199 ELSE IF( ldab.LT.kd+1 ) THEN
200 info = -5
201 END IF
202 IF( info.NE.0 ) THEN
203 CALL xerbla( 'CPBSTF', -info )
204 RETURN
205 END IF
206*
207* Quick return if possible
208*
209 IF( n.EQ.0 )
210 \$ RETURN
211*
212 kld = max( 1, ldab-1 )
213*
214* Set the splitting point m.
215*
216 m = ( n+kd ) / 2
217*
218 IF( upper ) THEN
219*
220* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
221*
222 DO 10 j = n, m + 1, -1
223*
224* Compute s(j,j) and test for non-positive-definiteness.
225*
226 ajj = real( ab( kd+1, j ) )
227 IF( ajj.LE.zero ) THEN
228 ab( kd+1, j ) = ajj
229 GO TO 50
230 END IF
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 csscal( km, one / ajj, ab( kd+1-km, j ), 1 )
239 CALL cher( '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**H*U.
244*
245 DO 20 j = 1, m
246*
247* Compute s(j,j) and test for non-positive-definiteness.
248*
249 ajj = real( ab( kd+1, j ) )
250 IF( ajj.LE.zero ) THEN
251 ab( kd+1, j ) = ajj
252 GO TO 50
253 END IF
254 ajj = sqrt( ajj )
255 ab( kd+1, j ) = ajj
256 km = min( kd, m-j )
257*
258* Compute elements j+1:j+km of the j-th row and update the
259* trailing submatrix within the band.
260*
261 IF( km.GT.0 ) THEN
262 CALL csscal( km, one / ajj, ab( kd, j+1 ), kld )
263 CALL clacgv( km, ab( kd, j+1 ), kld )
264 CALL cher( 'Upper', km, -one, ab( kd, j+1 ), kld,
265 \$ ab( kd+1, j+1 ), kld )
266 CALL clacgv( km, ab( kd, j+1 ), kld )
267 END IF
268 20 CONTINUE
269 ELSE
270*
271* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
272*
273 DO 30 j = n, m + 1, -1
274*
275* Compute s(j,j) and test for non-positive-definiteness.
276*
277 ajj = real( ab( 1, j ) )
278 IF( ajj.LE.zero ) THEN
279 ab( 1, j ) = ajj
280 GO TO 50
281 END IF
282 ajj = sqrt( ajj )
283 ab( 1, j ) = ajj
284 km = min( j-1, kd )
285*
286* Compute elements j-km:j-1 of the j-th row and update the
287* trailing submatrix within the band.
288*
289 CALL csscal( km, one / ajj, ab( km+1, j-km ), kld )
290 CALL clacgv( km, ab( km+1, j-km ), kld )
291 CALL cher( 'Lower', km, -one, ab( km+1, j-km ), kld,
292 \$ ab( 1, j-km ), kld )
293 CALL clacgv( km, ab( km+1, j-km ), kld )
294 30 CONTINUE
295*
296* Factorize the updated submatrix A(1:m,1:m) as U**H*U.
297*
298 DO 40 j = 1, m
299*
300* Compute s(j,j) and test for non-positive-definiteness.
301*
302 ajj = real( ab( 1, j ) )
303 IF( ajj.LE.zero ) THEN
304 ab( 1, j ) = ajj
305 GO TO 50
306 END IF
307 ajj = sqrt( ajj )
308 ab( 1, j ) = ajj
309 km = min( kd, m-j )
310*
311* Compute elements j+1:j+km of the j-th column and update the
312* trailing submatrix within the band.
313*
314 IF( km.GT.0 ) THEN
315 CALL csscal( km, one / ajj, ab( 2, j ), 1 )
316 CALL cher( 'Lower', km, -one, ab( 2, j ), 1,
317 \$ ab( 1, j+1 ), kld )
318 END IF
319 40 CONTINUE
320 END IF
321 RETURN
322*
323 50 CONTINUE
324 info = j
325 RETURN
326*
327* End of CPBSTF
328*
329 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cher(uplo, n, alpha, x, incx, a, lda)
CHER
Definition cher.f:135
subroutine clacgv(n, x, incx)
CLACGV conjugates a complex vector.
Definition clacgv.f:74
subroutine cpbstf(uplo, n, kd, ab, ldab, info)
CPBSTF
Definition cpbstf.f:153
subroutine csscal(n, sa, cx, incx)
CSSCAL
Definition csscal.f:78