LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ ztbt03()

subroutine ztbt03 ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
integer  KD,
integer  NRHS,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
double precision  SCALE,
double precision, dimension( * )  CNORM,
double precision  TSCAL,
complex*16, dimension( ldx, * )  X,
integer  LDX,
complex*16, dimension( ldb, * )  B,
integer  LDB,
complex*16, dimension( * )  WORK,
double precision  RESID 
)

ZTBT03

Purpose:
 ZTBT03 computes the residual for the solution to a scaled triangular
 system of equations  A*x = s*b,  A**T *x = s*b,  or  A**H *x = s*b
 when A is a triangular band matrix.  Here A**T  denotes the transpose
 of A, A**H denotes the conjugate transpose of A, s is a scalar, and
 x and b are N by NRHS matrices.  The test ratio is the maximum over
 the number of right hand sides of
    norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
 where op(A) denotes A, A**T, or A**H, and EPS is the machine epsilon.
Parameters
[in]UPLO
          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular
[in]TRANS
          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  A *x = s*b     (No transpose)
          = 'T':  A**T *x = s*b  (Transpose)
          = 'C':  A**H *x = s*b  (Conjugate transpose)
[in]DIAG
          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in]KD
          KD is INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.
[in]NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices X and B.  NRHS >= 0.
[in]AB
          AB is COMPLEX*16 array, dimension (LDAB,N)
          The upper or lower triangular 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).
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.
[in]SCALE
          SCALE is DOUBLE PRECISION
          The scaling factor s used in solving the triangular system.
[in]CNORM
          CNORM is DOUBLE PRECISION array, dimension (N)
          The 1-norms of the columns of A, not counting the diagonal.
[in]TSCAL
          TSCAL is DOUBLE PRECISION
          The scaling factor used in computing the 1-norms in CNORM.
          CNORM actually contains the column norms of TSCAL*A.
[in]X
          X is COMPLEX*16 array, dimension (LDX,NRHS)
          The computed solution vectors for the system of linear
          equations.
[in]LDX
          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).
[in]B
          B is COMPLEX*16 array, dimension (LDB,NRHS)
          The right hand side vectors for the system of linear
          equations.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
[out]WORK
          WORK is COMPLEX*16 array, dimension (N)
[out]RESID
          RESID is DOUBLE PRECISION
          The maximum over the number of right hand sides of
          norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016

Definition at line 179 of file ztbt03.f.

179 *
180 * -- LAPACK test routine (version 3.7.0) --
181 * -- LAPACK is a software package provided by Univ. of Tennessee, --
182 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183 * December 2016
184 *
185 * .. Scalar Arguments ..
186  CHARACTER diag, trans, uplo
187  INTEGER kd, ldab, ldb, ldx, n, nrhs
188  DOUBLE PRECISION resid, scale, tscal
189 * ..
190 * .. Array Arguments ..
191  DOUBLE PRECISION cnorm( * )
192  COMPLEX*16 ab( ldab, * ), b( ldb, * ), work( * ),
193  $ x( ldx, * )
194 * ..
195 *
196 * =====================================================================
197 *
198 *
199 * .. Parameters ..
200  DOUBLE PRECISION one, zero
201  parameter( one = 1.0d+0, zero = 0.0d+0 )
202 * ..
203 * .. Local Scalars ..
204  INTEGER ix, j
205  DOUBLE PRECISION eps, err, smlnum, tnorm, xnorm, xscal
206 * ..
207 * .. External Functions ..
208  LOGICAL lsame
209  INTEGER izamax
210  DOUBLE PRECISION dlamch
211  EXTERNAL lsame, izamax, dlamch
212 * ..
213 * .. External Subroutines ..
214  EXTERNAL zaxpy, zcopy, zdscal, ztbmv
215 * ..
216 * .. Intrinsic Functions ..
217  INTRINSIC abs, dble, dcmplx, max
218 * ..
219 * .. Executable Statements ..
220 *
221 * Quick exit if N = 0
222 *
223  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
224  resid = zero
225  RETURN
226  END IF
227  eps = dlamch( 'Epsilon' )
228  smlnum = dlamch( 'Safe minimum' )
229 *
230 * Compute the norm of the triangular matrix A using the column
231 * norms already computed by ZLATBS.
232 *
233  tnorm = zero
234  IF( lsame( diag, 'N' ) ) THEN
235  IF( lsame( uplo, 'U' ) ) THEN
236  DO 10 j = 1, n
237  tnorm = max( tnorm, tscal*abs( ab( kd+1, j ) )+
238  $ cnorm( j ) )
239  10 CONTINUE
240  ELSE
241  DO 20 j = 1, n
242  tnorm = max( tnorm, tscal*abs( ab( 1, j ) )+cnorm( j ) )
243  20 CONTINUE
244  END IF
245  ELSE
246  DO 30 j = 1, n
247  tnorm = max( tnorm, tscal+cnorm( j ) )
248  30 CONTINUE
249  END IF
250 *
251 * Compute the maximum over the number of right hand sides of
252 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
253 *
254  resid = zero
255  DO 40 j = 1, nrhs
256  CALL zcopy( n, x( 1, j ), 1, work, 1 )
257  ix = izamax( n, work, 1 )
258  xnorm = max( one, abs( x( ix, j ) ) )
259  xscal = ( one / xnorm ) / dble( kd+1 )
260  CALL zdscal( n, xscal, work, 1 )
261  CALL ztbmv( uplo, trans, diag, n, kd, ab, ldab, work, 1 )
262  CALL zaxpy( n, dcmplx( -scale*xscal ), b( 1, j ), 1, work, 1 )
263  ix = izamax( n, work, 1 )
264  err = tscal*abs( work( ix ) )
265  ix = izamax( n, x( 1, j ), 1 )
266  xnorm = abs( x( ix, j ) )
267  IF( err*smlnum.LE.xnorm ) THEN
268  IF( xnorm.GT.zero )
269  $ err = err / xnorm
270  ELSE
271  IF( err.GT.zero )
272  $ err = one / eps
273  END IF
274  IF( err*smlnum.LE.tnorm ) THEN
275  IF( tnorm.GT.zero )
276  $ err = err / tnorm
277  ELSE
278  IF( err.GT.zero )
279  $ err = one / eps
280  END IF
281  resid = max( resid, err )
282  40 CONTINUE
283 *
284  RETURN
285 *
286 * End of ZTBT03
287 *
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:90
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
subroutine ztbmv(UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
ZTBMV
Definition: ztbmv.f:188
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:73
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