 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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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