LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zpbsv()

 subroutine zpbsv ( character UPLO, integer N, integer KD, integer NRHS, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO )

ZPBSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:
``` ZPBSV computes the solution to a complex system of linear equations
A * X = B,
where A is an N-by-N Hermitian positive definite band matrix and X
and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as
A = U**H * U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',
where U is an upper triangular band matrix, and L is a lower
triangular band matrix, with the same number of superdiagonals or
subdiagonals as A.  The factored form of A is then used to solve the
system of equations A * X = B.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 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). See below for further details. 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.``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.```
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 163 of file zpbsv.f.

164 *
165 * -- LAPACK driver routine --
166 * -- LAPACK is a software package provided by Univ. of Tennessee, --
167 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
168 *
169 * .. Scalar Arguments ..
170  CHARACTER UPLO
171  INTEGER INFO, KD, LDAB, LDB, N, NRHS
172 * ..
173 * .. Array Arguments ..
174  COMPLEX*16 AB( LDAB, * ), B( LDB, * )
175 * ..
176 *
177 * =====================================================================
178 *
179 * .. External Functions ..
180  LOGICAL LSAME
181  EXTERNAL lsame
182 * ..
183 * .. External Subroutines ..
184  EXTERNAL xerbla, zpbtrf, zpbtrs
185 * ..
186 * .. Intrinsic Functions ..
187  INTRINSIC max
188 * ..
189 * .. Executable Statements ..
190 *
191 * Test the input parameters.
192 *
193  info = 0
194  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
195  info = -1
196  ELSE IF( n.LT.0 ) THEN
197  info = -2
198  ELSE IF( kd.LT.0 ) THEN
199  info = -3
200  ELSE IF( nrhs.LT.0 ) THEN
201  info = -4
202  ELSE IF( ldab.LT.kd+1 ) THEN
203  info = -6
204  ELSE IF( ldb.LT.max( 1, n ) ) THEN
205  info = -8
206  END IF
207  IF( info.NE.0 ) THEN
208  CALL xerbla( 'ZPBSV ', -info )
209  RETURN
210  END IF
211 *
212 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
213 *
214  CALL zpbtrf( uplo, n, kd, ab, ldab, info )
215  IF( info.EQ.0 ) THEN
216 *
217 * Solve the system A*X = B, overwriting B with X.
218 *
219  CALL zpbtrs( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )
220 *
221  END IF
222  RETURN
223 *
224 * End of ZPBSV
225 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zpbtrf(UPLO, N, KD, AB, LDAB, INFO)
ZPBTRF
Definition: zpbtrf.f:142
subroutine zpbtrs(UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
ZPBTRS
Definition: zpbtrs.f:121
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