LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dtbt03()

subroutine dtbt03 ( character  uplo,
character  trans,
character  diag,
integer  n,
integer  kd,
integer  nrhs,
double precision, dimension( ldab, * )  ab,
integer  ldab,
double precision  scale,
double precision, dimension( * )  cnorm,
double precision  tscal,
double precision, dimension( ldx, * )  x,
integer  ldx,
double precision, dimension( ldb, * )  b,
integer  ldb,
double precision, dimension( * )  work,
double precision  resid 
)

DTBT03

Purpose:
 DTBT03 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.

Definition at line 172 of file dtbt03.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 DOUBLE PRECISION RESID, SCALE, TSCAL
184* ..
185* .. Array Arguments ..
186 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), CNORM( * ),
187 $ WORK( * ), X( LDX, * )
188* ..
189*
190* =====================================================================
191*
192* .. Parameters ..
193 DOUBLE PRECISION ONE, ZERO
194 parameter( one = 1.0d+0, zero = 0.0d+0 )
195* ..
196* .. Local Scalars ..
197 INTEGER IX, J
198 DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
199* ..
200* .. External Functions ..
201 LOGICAL LSAME
202 INTEGER IDAMAX
203 DOUBLE PRECISION DLAMCH
204 EXTERNAL lsame, idamax, dlamch
205* ..
206* .. External Subroutines ..
207 EXTERNAL daxpy, dcopy, dscal, dtbmv
208* ..
209* .. Intrinsic Functions ..
210 INTRINSIC abs, dble, max
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 = dlamch( 'Epsilon' )
221 smlnum = dlamch( 'Safe minimum' )
222 bignum = one / smlnum
223*
224* Compute the norm of the triangular matrix A using the column
225* norms already computed by DLATBS.
226*
227 tnorm = zero
228 IF( lsame( diag, 'N' ) ) THEN
229 IF( lsame( uplo, 'U' ) ) THEN
230 DO 10 j = 1, n
231 tnorm = max( tnorm, tscal*abs( ab( kd+1, j ) )+
232 $ cnorm( j ) )
233 10 CONTINUE
234 ELSE
235 DO 20 j = 1, n
236 tnorm = max( tnorm, tscal*abs( ab( 1, j ) )+cnorm( j ) )
237 20 CONTINUE
238 END IF
239 ELSE
240 DO 30 j = 1, n
241 tnorm = max( tnorm, tscal+cnorm( j ) )
242 30 CONTINUE
243 END IF
244*
245* Compute the maximum over the number of right hand sides of
246* norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
247*
248 resid = zero
249 DO 40 j = 1, nrhs
250 CALL dcopy( n, x( 1, j ), 1, work, 1 )
251 ix = idamax( n, work, 1 )
252 xnorm = max( one, abs( x( ix, j ) ) )
253 xscal = ( one / xnorm ) / dble( kd+1 )
254 CALL dscal( n, xscal, work, 1 )
255 CALL dtbmv( uplo, trans, diag, n, kd, ab, ldab, work, 1 )
256 CALL daxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
257 ix = idamax( n, work, 1 )
258 err = tscal*abs( work( ix ) )
259 ix = idamax( n, x( 1, j ), 1 )
260 xnorm = abs( x( ix, j ) )
261 IF( err*smlnum.LE.xnorm ) THEN
262 IF( xnorm.GT.zero )
263 $ err = err / xnorm
264 ELSE
265 IF( err.GT.zero )
266 $ err = one / eps
267 END IF
268 IF( err*smlnum.LE.tnorm ) THEN
269 IF( tnorm.GT.zero )
270 $ err = err / tnorm
271 ELSE
272 IF( err.GT.zero )
273 $ err = one / eps
274 END IF
275 resid = max( resid, err )
276 40 CONTINUE
277*
278 RETURN
279*
280* End of DTBT03
281*
daxpy
subroutine daxpy(n, da, dx, incx, dy, incy)
DAXPY
Definition daxpy.f:89
dcopy
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
idamax
integer function idamax(n, dx, incx)
IDAMAX
Definition idamax.f:71
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
dscal
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
dtbmv
subroutine dtbmv(uplo, trans, diag, n, k, a, lda, x, incx)
DTBMV
Definition dtbmv.f:186
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