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

 subroutine spbcon ( character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SPBCON

Purpose:
``` SPBCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite band matrix using the
Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangular factor stored in AB; = 'L': Lower triangular factor stored in AB.``` [in] N ``` N is INTEGER 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] AB ``` AB is REAL array, dimension (LDAB,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, stored in the first KD+1 rows of the array. The j-th column of U or L is stored in the j-th column of the array AB as follows: if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j; if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.``` [in] ANORM ``` ANORM is REAL The 1-norm (or infinity-norm) of the symmetric band matrix A.``` [out] RCOND ``` RCOND is REAL The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.``` [out] WORK ` WORK is REAL array, dimension (3*N)` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```

Definition at line 130 of file spbcon.f.

132*
133* -- LAPACK computational routine --
134* -- LAPACK is a software package provided by Univ. of Tennessee, --
135* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136*
137* .. Scalar Arguments ..
138 CHARACTER UPLO
139 INTEGER INFO, KD, LDAB, N
140 REAL ANORM, RCOND
141* ..
142* .. Array Arguments ..
143 INTEGER IWORK( * )
144 REAL AB( LDAB, * ), WORK( * )
145* ..
146*
147* =====================================================================
148*
149* .. Parameters ..
150 REAL ONE, ZERO
151 parameter( one = 1.0e+0, zero = 0.0e+0 )
152* ..
153* .. Local Scalars ..
154 LOGICAL UPPER
155 CHARACTER NORMIN
156 INTEGER IX, KASE
157 REAL AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
158* ..
159* .. Local Arrays ..
160 INTEGER ISAVE( 3 )
161* ..
162* .. External Functions ..
163 LOGICAL LSAME
164 INTEGER ISAMAX
165 REAL SLAMCH
166 EXTERNAL lsame, isamax, slamch
167* ..
168* .. External Subroutines ..
169 EXTERNAL slacn2, slatbs, srscl, xerbla
170* ..
171* .. Intrinsic Functions ..
172 INTRINSIC abs
173* ..
174* .. Executable Statements ..
175*
176* Test the input parameters.
177*
178 info = 0
179 upper = lsame( uplo, 'U' )
180 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
181 info = -1
182 ELSE IF( n.LT.0 ) THEN
183 info = -2
184 ELSE IF( kd.LT.0 ) THEN
185 info = -3
186 ELSE IF( ldab.LT.kd+1 ) THEN
187 info = -5
188 ELSE IF( anorm.LT.zero ) THEN
189 info = -6
190 END IF
191 IF( info.NE.0 ) THEN
192 CALL xerbla( 'SPBCON', -info )
193 RETURN
194 END IF
195*
196* Quick return if possible
197*
198 rcond = zero
199 IF( n.EQ.0 ) THEN
200 rcond = one
201 RETURN
202 ELSE IF( anorm.EQ.zero ) THEN
203 RETURN
204 END IF
205*
206 smlnum = slamch( 'Safe minimum' )
207*
208* Estimate the 1-norm of the inverse.
209*
210 kase = 0
211 normin = 'N'
212 10 CONTINUE
213 CALL slacn2( n, work( n+1 ), work, iwork, ainvnm, kase, isave )
214 IF( kase.NE.0 ) THEN
215 IF( upper ) THEN
216*
217* Multiply by inv(U**T).
218*
219 CALL slatbs( 'Upper', 'Transpose', 'Non-unit', normin, n,
220 \$ kd, ab, ldab, work, scalel, work( 2*n+1 ),
221 \$ info )
222 normin = 'Y'
223*
224* Multiply by inv(U).
225*
226 CALL slatbs( 'Upper', 'No transpose', 'Non-unit', normin, n,
227 \$ kd, ab, ldab, work, scaleu, work( 2*n+1 ),
228 \$ info )
229 ELSE
230*
231* Multiply by inv(L).
232*
233 CALL slatbs( 'Lower', 'No transpose', 'Non-unit', normin, n,
234 \$ kd, ab, ldab, work, scalel, work( 2*n+1 ),
235 \$ info )
236 normin = 'Y'
237*
238* Multiply by inv(L**T).
239*
240 CALL slatbs( 'Lower', 'Transpose', 'Non-unit', normin, n,
241 \$ kd, ab, ldab, work, scaleu, work( 2*n+1 ),
242 \$ info )
243 END IF
244*
245* Multiply by 1/SCALE if doing so will not cause overflow.
246*
247 scale = scalel*scaleu
248 IF( scale.NE.one ) THEN
249 ix = isamax( n, work, 1 )
250 IF( scale.LT.abs( work( ix ) )*smlnum .OR. scale.EQ.zero )
251 \$ GO TO 20
252 CALL srscl( n, scale, work, 1 )
253 END IF
254 GO TO 10
255 END IF
256*
257* Compute the estimate of the reciprocal condition number.
258*
259 IF( ainvnm.NE.zero )
260 \$ rcond = ( one / ainvnm ) / anorm
261*
262 20 CONTINUE
263*
264 RETURN
265*
266* End of SPBCON
267*
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine slacn2(N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: slacn2.f:136
subroutine slatbs(UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
SLATBS solves a triangular banded system of equations.
Definition: slatbs.f:242
subroutine srscl(N, SA, SX, INCX)
SRSCL multiplies a vector by the reciprocal of a real scalar.
Definition: srscl.f:84
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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