LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
zpbstf.f
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1 *> \brief \b ZPBSTF
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZPBSTF( 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*16 AB( LDAB, * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> ZPBSTF 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 ZHBGST.
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*16 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 *> \date December 2016
114 *
115 *> \ingroup complex16OTHERcomputational
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**H s64**H s75**H
138 *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
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**H s23**H s34**H s54 s65 s76 *
147 *> a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
148 *>
149 *> Array elements marked * are not used by the routine; s12**H denotes
150 *> conjg(s12); the diagonal elements of S are real.
151 *> \endverbatim
152 *>
153 * =====================================================================
154  SUBROUTINE zpbstf( UPLO, N, KD, AB, LDAB, INFO )
155 *
156 * -- LAPACK computational routine (version 3.7.0) --
157 * -- LAPACK is a software package provided by Univ. of Tennessee, --
158 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
159 * December 2016
160 *
161 * .. Scalar Arguments ..
162  CHARACTER UPLO
163  INTEGER INFO, KD, LDAB, N
164 * ..
165 * .. Array Arguments ..
166  COMPLEX*16 AB( ldab, * )
167 * ..
168 *
169 * =====================================================================
170 *
171 * .. Parameters ..
172  DOUBLE PRECISION ONE, ZERO
173  parameter( one = 1.0d+0, zero = 0.0d+0 )
174 * ..
175 * .. Local Scalars ..
176  LOGICAL UPPER
177  INTEGER J, KLD, KM, M
178  DOUBLE PRECISION AJJ
179 * ..
180 * .. External Functions ..
181  LOGICAL LSAME
182  EXTERNAL lsame
183 * ..
184 * .. External Subroutines ..
185  EXTERNAL xerbla, zdscal, zher, zlacgv
186 * ..
187 * .. Intrinsic Functions ..
188  INTRINSIC dble, max, min, sqrt
189 * ..
190 * .. Executable Statements ..
191 *
192 * Test the input parameters.
193 *
194  info = 0
195  upper = lsame( uplo, 'U' )
196  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
197  info = -1
198  ELSE IF( n.LT.0 ) THEN
199  info = -2
200  ELSE IF( kd.LT.0 ) THEN
201  info = -3
202  ELSE IF( ldab.LT.kd+1 ) THEN
203  info = -5
204  END IF
205  IF( info.NE.0 ) THEN
206  CALL xerbla( 'ZPBSTF', -info )
207  RETURN
208  END IF
209 *
210 * Quick return if possible
211 *
212  IF( n.EQ.0 )
213  $ RETURN
214 *
215  kld = max( 1, ldab-1 )
216 *
217 * Set the splitting point m.
218 *
219  m = ( n+kd ) / 2
220 *
221  IF( upper ) THEN
222 *
223 * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
224 *
225  DO 10 j = n, m + 1, -1
226 *
227 * Compute s(j,j) and test for non-positive-definiteness.
228 *
229  ajj = dble( ab( kd+1, j ) )
230  IF( ajj.LE.zero ) THEN
231  ab( kd+1, j ) = ajj
232  GO TO 50
233  END IF
234  ajj = sqrt( ajj )
235  ab( kd+1, j ) = ajj
236  km = min( j-1, kd )
237 *
238 * Compute elements j-km:j-1 of the j-th column and update the
239 * the leading submatrix within the band.
240 *
241  CALL zdscal( km, one / ajj, ab( kd+1-km, j ), 1 )
242  CALL zher( 'Upper', km, -one, ab( kd+1-km, j ), 1,
243  $ ab( kd+1, j-km ), kld )
244  10 CONTINUE
245 *
246 * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
247 *
248  DO 20 j = 1, m
249 *
250 * Compute s(j,j) and test for non-positive-definiteness.
251 *
252  ajj = dble( ab( kd+1, j ) )
253  IF( ajj.LE.zero ) THEN
254  ab( kd+1, j ) = ajj
255  GO TO 50
256  END IF
257  ajj = sqrt( ajj )
258  ab( kd+1, j ) = ajj
259  km = min( kd, m-j )
260 *
261 * Compute elements j+1:j+km of the j-th row and update the
262 * trailing submatrix within the band.
263 *
264  IF( km.GT.0 ) THEN
265  CALL zdscal( km, one / ajj, ab( kd, j+1 ), kld )
266  CALL zlacgv( km, ab( kd, j+1 ), kld )
267  CALL zher( 'Upper', km, -one, ab( kd, j+1 ), kld,
268  $ ab( kd+1, j+1 ), kld )
269  CALL zlacgv( km, ab( kd, j+1 ), kld )
270  END IF
271  20 CONTINUE
272  ELSE
273 *
274 * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
275 *
276  DO 30 j = n, m + 1, -1
277 *
278 * Compute s(j,j) and test for non-positive-definiteness.
279 *
280  ajj = dble( ab( 1, j ) )
281  IF( ajj.LE.zero ) THEN
282  ab( 1, j ) = ajj
283  GO TO 50
284  END IF
285  ajj = sqrt( ajj )
286  ab( 1, j ) = ajj
287  km = min( j-1, kd )
288 *
289 * Compute elements j-km:j-1 of the j-th row and update the
290 * trailing submatrix within the band.
291 *
292  CALL zdscal( km, one / ajj, ab( km+1, j-km ), kld )
293  CALL zlacgv( km, ab( km+1, j-km ), kld )
294  CALL zher( 'Lower', km, -one, ab( km+1, j-km ), kld,
295  $ ab( 1, j-km ), kld )
296  CALL zlacgv( km, ab( km+1, j-km ), kld )
297  30 CONTINUE
298 *
299 * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
300 *
301  DO 40 j = 1, m
302 *
303 * Compute s(j,j) and test for non-positive-definiteness.
304 *
305  ajj = dble( ab( 1, j ) )
306  IF( ajj.LE.zero ) THEN
307  ab( 1, j ) = ajj
308  GO TO 50
309  END IF
310  ajj = sqrt( ajj )
311  ab( 1, j ) = ajj
312  km = min( kd, m-j )
313 *
314 * Compute elements j+1:j+km of the j-th column and update the
315 * trailing submatrix within the band.
316 *
317  IF( km.GT.0 ) THEN
318  CALL zdscal( km, one / ajj, ab( 2, j ), 1 )
319  CALL zher( 'Lower', km, -one, ab( 2, j ), 1,
320  $ ab( 1, j+1 ), kld )
321  END IF
322  40 CONTINUE
323  END IF
324  RETURN
325 *
326  50 CONTINUE
327  info = j
328  RETURN
329 *
330 * End of ZPBSTF
331 *
332  END
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76
subroutine zher(UPLO, N, ALPHA, X, INCX, A, LDA)
ZHER
Definition: zher.f:137
subroutine zpbstf(UPLO, N, KD, AB, LDAB, INFO)
ZPBSTF
Definition: zpbstf.f:155
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62