LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zpbtf2()

 subroutine zpbtf2 ( character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, integer INFO )

ZPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

Purpose:
``` ZPBTF2 computes the Cholesky factorization of a complex Hermitian
positive definite band matrix A.

The factorization has the form
A = U**H * U ,  if UPLO = 'U', or
A = L  * L**H,  if UPLO = 'L',
where U is an upper triangular matrix, U**H is the conjugate transpose
of U, and L is lower triangular.

This is the unblocked version of the algorithm, calling Level 2 BLAS.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0.``` [in,out] AB ``` AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the band matrix A, in the same storage format as A.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, the leading minor of order k is not positive definite, and the factorization could not be completed.```
Date
December 2016
Further Details:
```  The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':

On entry:                       On exit:

*    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
*   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry:                       On exit:

a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine.```

Definition at line 144 of file zpbtf2.f.

144 *
145 * -- LAPACK computational routine (version 3.7.0) --
146 * -- LAPACK is a software package provided by Univ. of Tennessee, --
147 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
148 * December 2016
149 *
150 * .. Scalar Arguments ..
151  CHARACTER uplo
152  INTEGER info, kd, ldab, n
153 * ..
154 * .. Array Arguments ..
155  COMPLEX*16 ab( ldab, * )
156 * ..
157 *
158 * =====================================================================
159 *
160 * .. Parameters ..
161  DOUBLE PRECISION one, zero
162  parameter( one = 1.0d+0, zero = 0.0d+0 )
163 * ..
164 * .. Local Scalars ..
165  LOGICAL upper
166  INTEGER j, kld, kn
167  DOUBLE PRECISION ajj
168 * ..
169 * .. External Functions ..
170  LOGICAL lsame
171  EXTERNAL lsame
172 * ..
173 * .. External Subroutines ..
174  EXTERNAL xerbla, zdscal, zher, zlacgv
175 * ..
176 * .. Intrinsic Functions ..
177  INTRINSIC dble, max, min, sqrt
178 * ..
179 * .. Executable Statements ..
180 *
181 * Test the input parameters.
182 *
183  info = 0
184  upper = lsame( uplo, 'U' )
185  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
186  info = -1
187  ELSE IF( n.LT.0 ) THEN
188  info = -2
189  ELSE IF( kd.LT.0 ) THEN
190  info = -3
191  ELSE IF( ldab.LT.kd+1 ) THEN
192  info = -5
193  END IF
194  IF( info.NE.0 ) THEN
195  CALL xerbla( 'ZPBTF2', -info )
196  RETURN
197  END IF
198 *
199 * Quick return if possible
200 *
201  IF( n.EQ.0 )
202  \$ RETURN
203 *
204  kld = max( 1, ldab-1 )
205 *
206  IF( upper ) THEN
207 *
208 * Compute the Cholesky factorization A = U**H * U.
209 *
210  DO 10 j = 1, n
211 *
212 * Compute U(J,J) and test for non-positive-definiteness.
213 *
214  ajj = dble( ab( kd+1, j ) )
215  IF( ajj.LE.zero ) THEN
216  ab( kd+1, j ) = ajj
217  GO TO 30
218  END IF
219  ajj = sqrt( ajj )
220  ab( kd+1, j ) = ajj
221 *
222 * Compute elements J+1:J+KN of row J and update the
223 * trailing submatrix within the band.
224 *
225  kn = min( kd, n-j )
226  IF( kn.GT.0 ) THEN
227  CALL zdscal( kn, one / ajj, ab( kd, j+1 ), kld )
228  CALL zlacgv( kn, ab( kd, j+1 ), kld )
229  CALL zher( 'Upper', kn, -one, ab( kd, j+1 ), kld,
230  \$ ab( kd+1, j+1 ), kld )
231  CALL zlacgv( kn, ab( kd, j+1 ), kld )
232  END IF
233  10 CONTINUE
234  ELSE
235 *
236 * Compute the Cholesky factorization A = L*L**H.
237 *
238  DO 20 j = 1, n
239 *
240 * Compute L(J,J) and test for non-positive-definiteness.
241 *
242  ajj = dble( ab( 1, j ) )
243  IF( ajj.LE.zero ) THEN
244  ab( 1, j ) = ajj
245  GO TO 30
246  END IF
247  ajj = sqrt( ajj )
248  ab( 1, j ) = ajj
249 *
250 * Compute elements J+1:J+KN of column J and update the
251 * trailing submatrix within the band.
252 *
253  kn = min( kd, n-j )
254  IF( kn.GT.0 ) THEN
255  CALL zdscal( kn, one / ajj, ab( 2, j ), 1 )
256  CALL zher( 'Lower', kn, -one, ab( 2, j ), 1,
257  \$ ab( 1, j+1 ), kld )
258  END IF
259  20 CONTINUE
260  END IF
261  RETURN
262 *
263  30 CONTINUE
264  info = j
265  RETURN
266 *
267 * End of ZPBTF2
268 *
subroutine zher(UPLO, N, ALPHA, X, INCX, A, LDA)
ZHER
Definition: zher.f:137
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
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