LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ stbt03()

subroutine stbt03 ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
integer  KD,
integer  NRHS,
real, dimension( ldab, * )  AB,
integer  LDAB,
real  SCALE,
real, dimension( * )  CNORM,
real  TSCAL,
real, dimension( ldx, * )  X,
integer  LDX,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( * )  WORK,
real  RESID 
)

STBT03

Purpose:
 STBT03 computes the residual for the solution to a scaled triangular
 system of equations  A*x = s*b  or  A'*x = s*b  when A is a
 triangular band matrix. Here A' is the 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 or A' 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 = b  (No transpose)
          = 'T':  A'*x = b  (Transpose)
          = 'C':  A'*x = b  (Conjugate transpose = 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 REAL 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 REAL
          The scaling factor s used in solving the triangular system.
[in]CNORM
          CNORM is REAL array, dimension (N)
          The 1-norms of the columns of A, not counting the diagonal.
[in]TSCAL
          TSCAL is REAL
          The scaling factor used in computing the 1-norms in CNORM.
          CNORM actually contains the column norms of TSCAL*A.
[in]X
          X is REAL 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 REAL 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 REAL array, dimension (N)
[out]RESID
          RESID is REAL
          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.

Definition at line 172 of file stbt03.f.

175 *
176 * -- LAPACK test routine --
177 * -- LAPACK is a software package provided by Univ. of Tennessee, --
178 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
179 *
180 * .. Scalar Arguments ..
181  CHARACTER DIAG, TRANS, UPLO
182  INTEGER KD, LDAB, LDB, LDX, N, NRHS
183  REAL RESID, SCALE, TSCAL
184 * ..
185 * .. Array Arguments ..
186  REAL AB( LDAB, * ), B( LDB, * ), CNORM( * ),
187  $ WORK( * ), X( LDX, * )
188 * ..
189 *
190 * =====================================================================
191 *
192 * .. Parameters ..
193  REAL ONE, ZERO
194  parameter( one = 1.0e+0, zero = 0.0e+0 )
195 * ..
196 * .. Local Scalars ..
197  INTEGER IX, J
198  REAL BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
199 * ..
200 * .. External Functions ..
201  LOGICAL LSAME
202  INTEGER ISAMAX
203  REAL SLAMCH
204  EXTERNAL lsame, isamax, slamch
205 * ..
206 * .. External Subroutines ..
207  EXTERNAL saxpy, scopy, slabad, sscal, stbmv
208 * ..
209 * .. Intrinsic Functions ..
210  INTRINSIC abs, max, real
211 * ..
212 * .. Executable Statements ..
213 *
214 * Quick exit if N = 0
215 *
216  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
217  resid = zero
218  RETURN
219  END IF
220  eps = slamch( 'Epsilon' )
221  smlnum = slamch( 'Safe minimum' )
222  bignum = one / smlnum
223  CALL slabad( smlnum, bignum )
224 *
225 * Compute the norm of the triangular matrix A using the column
226 * norms already computed by SLATBS.
227 *
228  tnorm = zero
229  IF( lsame( diag, 'N' ) ) THEN
230  IF( lsame( uplo, 'U' ) ) THEN
231  DO 10 j = 1, n
232  tnorm = max( tnorm, tscal*abs( ab( kd+1, j ) )+
233  $ cnorm( j ) )
234  10 CONTINUE
235  ELSE
236  DO 20 j = 1, n
237  tnorm = max( tnorm, tscal*abs( ab( 1, j ) )+cnorm( j ) )
238  20 CONTINUE
239  END IF
240  ELSE
241  DO 30 j = 1, n
242  tnorm = max( tnorm, tscal+cnorm( j ) )
243  30 CONTINUE
244  END IF
245 *
246 * Compute the maximum over the number of right hand sides of
247 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
248 *
249  resid = zero
250  DO 40 j = 1, nrhs
251  CALL scopy( n, x( 1, j ), 1, work, 1 )
252  ix = isamax( n, work, 1 )
253  xnorm = max( one, abs( x( ix, j ) ) )
254  xscal = ( one / xnorm ) / real( kd+1 )
255  CALL sscal( n, xscal, work, 1 )
256  CALL stbmv( uplo, trans, diag, n, kd, ab, ldab, work, 1 )
257  CALL saxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
258  ix = isamax( n, work, 1 )
259  err = tscal*abs( work( ix ) )
260  ix = isamax( n, x( 1, j ), 1 )
261  xnorm = abs( x( ix, j ) )
262  IF( err*smlnum.LE.xnorm ) THEN
263  IF( xnorm.GT.zero )
264  $ err = err / xnorm
265  ELSE
266  IF( err.GT.zero )
267  $ err = one / eps
268  END IF
269  IF( err*smlnum.LE.tnorm ) THEN
270  IF( tnorm.GT.zero )
271  $ err = err / tnorm
272  ELSE
273  IF( err.GT.zero )
274  $ err = one / eps
275  END IF
276  resid = max( resid, err )
277  40 CONTINUE
278 *
279  RETURN
280 *
281 * End of STBT03
282 *
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine stbmv(UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
STBMV
Definition: stbmv.f:186
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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