 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ spbtf2()

 subroutine spbtf2 ( character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, integer INFO )

SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

Purpose:
``` SPBTF2 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 REAL 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.```
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 141 of file spbtf2.f.

142*
143* -- LAPACK computational routine --
144* -- LAPACK is a software package provided by Univ. of Tennessee, --
145* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146*
147* .. Scalar Arguments ..
148 CHARACTER UPLO
149 INTEGER INFO, KD, LDAB, N
150* ..
151* .. Array Arguments ..
152 REAL AB( LDAB, * )
153* ..
154*
155* =====================================================================
156*
157* .. Parameters ..
158 REAL ONE, ZERO
159 parameter( one = 1.0e+0, zero = 0.0e+0 )
160* ..
161* .. Local Scalars ..
162 LOGICAL UPPER
163 INTEGER J, KLD, KN
164 REAL AJJ
165* ..
166* .. External Functions ..
167 LOGICAL LSAME
168 EXTERNAL lsame
169* ..
170* .. External Subroutines ..
171 EXTERNAL sscal, ssyr, xerbla
172* ..
173* .. Intrinsic Functions ..
174 INTRINSIC max, min, sqrt
175* ..
176* .. Executable Statements ..
177*
178* Test the input parameters.
179*
180 info = 0
181 upper = lsame( uplo, 'U' )
182 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
183 info = -1
184 ELSE IF( n.LT.0 ) THEN
185 info = -2
186 ELSE IF( kd.LT.0 ) THEN
187 info = -3
188 ELSE IF( ldab.LT.kd+1 ) THEN
189 info = -5
190 END IF
191 IF( info.NE.0 ) THEN
192 CALL xerbla( 'SPBTF2', -info )
193 RETURN
194 END IF
195*
196* Quick return if possible
197*
198 IF( n.EQ.0 )
199 \$ RETURN
200*
201 kld = max( 1, ldab-1 )
202*
203 IF( upper ) THEN
204*
205* Compute the Cholesky factorization A = U**T*U.
206*
207 DO 10 j = 1, n
208*
209* Compute U(J,J) and test for non-positive-definiteness.
210*
211 ajj = ab( kd+1, j )
212 IF( ajj.LE.zero )
213 \$ GO TO 30
214 ajj = sqrt( ajj )
215 ab( kd+1, j ) = ajj
216*
217* Compute elements J+1:J+KN of row J and update the
218* trailing submatrix within the band.
219*
220 kn = min( kd, n-j )
221 IF( kn.GT.0 ) THEN
222 CALL sscal( kn, one / ajj, ab( kd, j+1 ), kld )
223 CALL ssyr( 'Upper', kn, -one, ab( kd, j+1 ), kld,
224 \$ ab( kd+1, j+1 ), kld )
225 END IF
226 10 CONTINUE
227 ELSE
228*
229* Compute the Cholesky factorization A = L*L**T.
230*
231 DO 20 j = 1, n
232*
233* Compute L(J,J) and test for non-positive-definiteness.
234*
235 ajj = ab( 1, j )
236 IF( ajj.LE.zero )
237 \$ GO TO 30
238 ajj = sqrt( ajj )
239 ab( 1, j ) = ajj
240*
241* Compute elements J+1:J+KN of column J and update the
242* trailing submatrix within the band.
243*
244 kn = min( kd, n-j )
245 IF( kn.GT.0 ) THEN
246 CALL sscal( kn, one / ajj, ab( 2, j ), 1 )
247 CALL ssyr( 'Lower', kn, -one, ab( 2, j ), 1,
248 \$ ab( 1, j+1 ), kld )
249 END IF
250 20 CONTINUE
251 END IF
252 RETURN
253*
254 30 CONTINUE
255 info = j
256 RETURN
257*
258* End of SPBTF2
259*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine ssyr(UPLO, N, ALPHA, X, INCX, A, LDA)
SSYR
Definition: ssyr.f:132
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