LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine dpbtf2 ( character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, integer INFO )

DPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

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Purpose:
``` DPBTF2 computes the Cholesky factorization of a real symmetric
positive definite band matrix A.

The factorization has the form
A = U**T * U ,  if UPLO = 'U', or
A = L  * L**T,  if UPLO = 'L',
where U is an upper triangular matrix, U**T is the 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 symmetric 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 DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric 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**T*U or A = L*L**T 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
September 2012
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 dpbtf2.f.

144 *
145 * -- LAPACK computational routine (version 3.4.2) --
146 * -- LAPACK is a software package provided by Univ. of Tennessee, --
147 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
148 * September 2012
149 *
150 * .. Scalar Arguments ..
151  CHARACTER uplo
152  INTEGER info, kd, ldab, n
153 * ..
154 * .. Array Arguments ..
155  DOUBLE PRECISION 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 dscal, dsyr, xerbla
175 * ..
176 * .. Intrinsic Functions ..
177  INTRINSIC 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( 'DPBTF2', -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**T*U.
209 *
210  DO 10 j = 1, n
211 *
212 * Compute U(J,J) and test for non-positive-definiteness.
213 *
214  ajj = ab( kd+1, j )
215  IF( ajj.LE.zero )
216  \$ GO TO 30
217  ajj = sqrt( ajj )
218  ab( kd+1, j ) = ajj
219 *
220 * Compute elements J+1:J+KN of row J and update the
221 * trailing submatrix within the band.
222 *
223  kn = min( kd, n-j )
224  IF( kn.GT.0 ) THEN
225  CALL dscal( kn, one / ajj, ab( kd, j+1 ), kld )
226  CALL dsyr( 'Upper', kn, -one, ab( kd, j+1 ), kld,
227  \$ ab( kd+1, j+1 ), kld )
228  END IF
229  10 CONTINUE
230  ELSE
231 *
232 * Compute the Cholesky factorization A = L*L**T.
233 *
234  DO 20 j = 1, n
235 *
236 * Compute L(J,J) and test for non-positive-definiteness.
237 *
238  ajj = ab( 1, j )
239  IF( ajj.LE.zero )
240  \$ GO TO 30
241  ajj = sqrt( ajj )
242  ab( 1, j ) = ajj
243 *
244 * Compute elements J+1:J+KN of column J and update the
245 * trailing submatrix within the band.
246 *
247  kn = min( kd, n-j )
248  IF( kn.GT.0 ) THEN
249  CALL dscal( kn, one / ajj, ab( 2, j ), 1 )
250  CALL dsyr( 'Lower', kn, -one, ab( 2, j ), 1,
251  \$ ab( 1, j+1 ), kld )
252  END IF
253  20 CONTINUE
254  END IF
255  RETURN
256 *
257  30 CONTINUE
258  info = j
259  RETURN
260 *
261 * End of DPBTF2
262 *
subroutine dsyr(UPLO, N, ALPHA, X, INCX, A, LDA)
DSYR
Definition: dsyr.f:134
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:55
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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