 LAPACK  3.10.1 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 ).```

Definition at line 174 of file ztbt03.f.

177 *
178 * -- LAPACK test routine --
179 * -- LAPACK is a software package provided by Univ. of Tennessee, --
180 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181 *
182 * .. Scalar Arguments ..
183  CHARACTER DIAG, TRANS, UPLO
184  INTEGER KD, LDAB, LDB, LDX, N, NRHS
185  DOUBLE PRECISION RESID, SCALE, TSCAL
186 * ..
187 * .. Array Arguments ..
188  DOUBLE PRECISION CNORM( * )
189  COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ),
190  \$ X( LDX, * )
191 * ..
192 *
193 * =====================================================================
194 *
195 *
196 * .. Parameters ..
197  DOUBLE PRECISION ONE, ZERO
198  parameter( one = 1.0d+0, zero = 0.0d+0 )
199 * ..
200 * .. Local Scalars ..
201  INTEGER IX, J
202  DOUBLE PRECISION EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
203 * ..
204 * .. External Functions ..
205  LOGICAL LSAME
206  INTEGER IZAMAX
207  DOUBLE PRECISION DLAMCH
208  EXTERNAL lsame, izamax, dlamch
209 * ..
210 * .. External Subroutines ..
211  EXTERNAL zaxpy, zcopy, zdscal, ztbmv
212 * ..
213 * .. Intrinsic Functions ..
214  INTRINSIC abs, dble, dcmplx, max
215 * ..
216 * .. Executable Statements ..
217 *
218 * Quick exit if N = 0
219 *
220  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
221  resid = zero
222  RETURN
223  END IF
224  eps = dlamch( 'Epsilon' )
225  smlnum = dlamch( 'Safe minimum' )
226 *
227 * Compute the norm of the triangular matrix A using the column
228 * norms already computed by ZLATBS.
229 *
230  tnorm = zero
231  IF( lsame( diag, 'N' ) ) THEN
232  IF( lsame( uplo, 'U' ) ) THEN
233  DO 10 j = 1, n
234  tnorm = max( tnorm, tscal*abs( ab( kd+1, j ) )+
235  \$ cnorm( j ) )
236  10 CONTINUE
237  ELSE
238  DO 20 j = 1, n
239  tnorm = max( tnorm, tscal*abs( ab( 1, j ) )+cnorm( j ) )
240  20 CONTINUE
241  END IF
242  ELSE
243  DO 30 j = 1, n
244  tnorm = max( tnorm, tscal+cnorm( j ) )
245  30 CONTINUE
246  END IF
247 *
248 * Compute the maximum over the number of right hand sides of
249 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
250 *
251  resid = zero
252  DO 40 j = 1, nrhs
253  CALL zcopy( n, x( 1, j ), 1, work, 1 )
254  ix = izamax( n, work, 1 )
255  xnorm = max( one, abs( x( ix, j ) ) )
256  xscal = ( one / xnorm ) / dble( kd+1 )
257  CALL zdscal( n, xscal, work, 1 )
258  CALL ztbmv( uplo, trans, diag, n, kd, ab, ldab, work, 1 )
259  CALL zaxpy( n, dcmplx( -scale*xscal ), b( 1, j ), 1, work, 1 )
260  ix = izamax( n, work, 1 )
261  err = tscal*abs( work( ix ) )
262  ix = izamax( n, x( 1, j ), 1 )
263  xnorm = abs( x( ix, j ) )
264  IF( err*smlnum.LE.xnorm ) THEN
265  IF( xnorm.GT.zero )
266  \$ err = err / xnorm
267  ELSE
268  IF( err.GT.zero )
269  \$ err = one / eps
270  END IF
271  IF( err*smlnum.LE.tnorm ) THEN
272  IF( tnorm.GT.zero )
273  \$ err = err / tnorm
274  ELSE
275  IF( err.GT.zero )
276  \$ err = one / eps
277  END IF
278  resid = max( resid, err )
279  40 CONTINUE
280 *
281  RETURN
282 *
283 * End of ZTBT03
284 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine ztbmv(UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
ZTBMV
Definition: ztbmv.f:186
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