LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dtbt03.f
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00001 *> \brief \b DTBT03
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *  Definition:
00009 *  ===========
00010 *
00011 *       SUBROUTINE DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
00012 *                          SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
00013 *                          RESID )
00014 * 
00015 *       .. Scalar Arguments ..
00016 *       CHARACTER          DIAG, TRANS, UPLO
00017 *       INTEGER            KD, LDAB, LDB, LDX, N, NRHS
00018 *       DOUBLE PRECISION   RESID, SCALE, TSCAL
00019 *       ..
00020 *       .. Array Arguments ..
00021 *       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), CNORM( * ),
00022 *      $                   WORK( * ), X( LDX, * )
00023 *       ..
00024 *  
00025 *
00026 *> \par Purpose:
00027 *  =============
00028 *>
00029 *> \verbatim
00030 *>
00031 *> DTBT03 computes the residual for the solution to a scaled triangular
00032 *> system of equations  A*x = s*b  or  A'*x = s*b  when A is a
00033 *> triangular band matrix. Here A' is the transpose of A, s is a scalar,
00034 *> and x and b are N by NRHS matrices.  The test ratio is the maximum
00035 *> over the number of right hand sides of
00036 *>    norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
00037 *> where op(A) denotes A or A' and EPS is the machine epsilon.
00038 *> \endverbatim
00039 *
00040 *  Arguments:
00041 *  ==========
00042 *
00043 *> \param[in] UPLO
00044 *> \verbatim
00045 *>          UPLO is CHARACTER*1
00046 *>          Specifies whether the matrix A is upper or lower triangular.
00047 *>          = 'U':  Upper triangular
00048 *>          = 'L':  Lower triangular
00049 *> \endverbatim
00050 *>
00051 *> \param[in] TRANS
00052 *> \verbatim
00053 *>          TRANS is CHARACTER*1
00054 *>          Specifies the operation applied to A.
00055 *>          = 'N':  A *x = b  (No transpose)
00056 *>          = 'T':  A'*x = b  (Transpose)
00057 *>          = 'C':  A'*x = b  (Conjugate transpose = Transpose)
00058 *> \endverbatim
00059 *>
00060 *> \param[in] DIAG
00061 *> \verbatim
00062 *>          DIAG is CHARACTER*1
00063 *>          Specifies whether or not the matrix A is unit triangular.
00064 *>          = 'N':  Non-unit triangular
00065 *>          = 'U':  Unit triangular
00066 *> \endverbatim
00067 *>
00068 *> \param[in] N
00069 *> \verbatim
00070 *>          N is INTEGER
00071 *>          The order of the matrix A.  N >= 0.
00072 *> \endverbatim
00073 *>
00074 *> \param[in] KD
00075 *> \verbatim
00076 *>          KD is INTEGER
00077 *>          The number of superdiagonals or subdiagonals of the
00078 *>          triangular band matrix A.  KD >= 0.
00079 *> \endverbatim
00080 *>
00081 *> \param[in] NRHS
00082 *> \verbatim
00083 *>          NRHS is INTEGER
00084 *>          The number of right hand sides, i.e., the number of columns
00085 *>          of the matrices X and B.  NRHS >= 0.
00086 *> \endverbatim
00087 *>
00088 *> \param[in] AB
00089 *> \verbatim
00090 *>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
00091 *>          The upper or lower triangular band matrix A, stored in the
00092 *>          first kd+1 rows of the array. The j-th column of A is stored
00093 *>          in the j-th column of the array AB as follows:
00094 *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
00095 *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
00096 *> \endverbatim
00097 *>
00098 *> \param[in] LDAB
00099 *> \verbatim
00100 *>          LDAB is INTEGER
00101 *>          The leading dimension of the array AB.  LDAB >= KD+1.
00102 *> \endverbatim
00103 *>
00104 *> \param[in] SCALE
00105 *> \verbatim
00106 *>          SCALE is DOUBLE PRECISION
00107 *>          The scaling factor s used in solving the triangular system.
00108 *> \endverbatim
00109 *>
00110 *> \param[in] CNORM
00111 *> \verbatim
00112 *>          CNORM is DOUBLE PRECISION array, dimension (N)
00113 *>          The 1-norms of the columns of A, not counting the diagonal.
00114 *> \endverbatim
00115 *>
00116 *> \param[in] TSCAL
00117 *> \verbatim
00118 *>          TSCAL is DOUBLE PRECISION
00119 *>          The scaling factor used in computing the 1-norms in CNORM.
00120 *>          CNORM actually contains the column norms of TSCAL*A.
00121 *> \endverbatim
00122 *>
00123 *> \param[in] X
00124 *> \verbatim
00125 *>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
00126 *>          The computed solution vectors for the system of linear
00127 *>          equations.
00128 *> \endverbatim
00129 *>
00130 *> \param[in] LDX
00131 *> \verbatim
00132 *>          LDX is INTEGER
00133 *>          The leading dimension of the array X.  LDX >= max(1,N).
00134 *> \endverbatim
00135 *>
00136 *> \param[in] B
00137 *> \verbatim
00138 *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
00139 *>          The right hand side vectors for the system of linear
00140 *>          equations.
00141 *> \endverbatim
00142 *>
00143 *> \param[in] LDB
00144 *> \verbatim
00145 *>          LDB is INTEGER
00146 *>          The leading dimension of the array B.  LDB >= max(1,N).
00147 *> \endverbatim
00148 *>
00149 *> \param[out] WORK
00150 *> \verbatim
00151 *>          WORK is DOUBLE PRECISION array, dimension (N)
00152 *> \endverbatim
00153 *>
00154 *> \param[out] RESID
00155 *> \verbatim
00156 *>          RESID is DOUBLE PRECISION
00157 *>          The maximum over the number of right hand sides of
00158 *>          norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
00159 *> \endverbatim
00160 *
00161 *  Authors:
00162 *  ========
00163 *
00164 *> \author Univ. of Tennessee 
00165 *> \author Univ. of California Berkeley 
00166 *> \author Univ. of Colorado Denver 
00167 *> \author NAG Ltd. 
00168 *
00169 *> \date November 2011
00170 *
00171 *> \ingroup double_lin
00172 *
00173 *  =====================================================================
00174       SUBROUTINE DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
00175      $                   SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
00176      $                   RESID )
00177 *
00178 *  -- LAPACK test routine (version 3.4.0) --
00179 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00180 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00181 *     November 2011
00182 *
00183 *     .. Scalar Arguments ..
00184       CHARACTER          DIAG, TRANS, UPLO
00185       INTEGER            KD, LDAB, LDB, LDX, N, NRHS
00186       DOUBLE PRECISION   RESID, SCALE, TSCAL
00187 *     ..
00188 *     .. Array Arguments ..
00189       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), CNORM( * ),
00190      $                   WORK( * ), X( LDX, * )
00191 *     ..
00192 *
00193 *  =====================================================================
00194 *
00195 *     .. Parameters ..
00196       DOUBLE PRECISION   ONE, ZERO
00197       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
00198 *     ..
00199 *     .. Local Scalars ..
00200       INTEGER            IX, J
00201       DOUBLE PRECISION   BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
00202 *     ..
00203 *     .. External Functions ..
00204       LOGICAL            LSAME
00205       INTEGER            IDAMAX
00206       DOUBLE PRECISION   DLAMCH
00207       EXTERNAL           LSAME, IDAMAX, DLAMCH
00208 *     ..
00209 *     .. External Subroutines ..
00210       EXTERNAL           DAXPY, DCOPY, DLABAD, DSCAL, DTBMV
00211 *     ..
00212 *     .. Intrinsic Functions ..
00213       INTRINSIC          ABS, DBLE, MAX
00214 *     ..
00215 *     .. Executable Statements ..
00216 *
00217 *     Quick exit if N = 0
00218 *
00219       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00220          RESID = ZERO
00221          RETURN
00222       END IF
00223       EPS = DLAMCH( 'Epsilon' )
00224       SMLNUM = DLAMCH( 'Safe minimum' )
00225       BIGNUM = ONE / SMLNUM
00226       CALL DLABAD( SMLNUM, BIGNUM )
00227 *
00228 *     Compute the norm of the triangular matrix A using the column
00229 *     norms already computed by DLATBS.
00230 *
00231       TNORM = ZERO
00232       IF( LSAME( DIAG, 'N' ) ) THEN
00233          IF( LSAME( UPLO, 'U' ) ) THEN
00234             DO 10 J = 1, N
00235                TNORM = MAX( TNORM, TSCAL*ABS( AB( KD+1, J ) )+
00236      $                 CNORM( J ) )
00237    10       CONTINUE
00238          ELSE
00239             DO 20 J = 1, N
00240                TNORM = MAX( TNORM, TSCAL*ABS( AB( 1, J ) )+CNORM( J ) )
00241    20       CONTINUE
00242          END IF
00243       ELSE
00244          DO 30 J = 1, N
00245             TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
00246    30    CONTINUE
00247       END IF
00248 *
00249 *     Compute the maximum over the number of right hand sides of
00250 *        norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
00251 *
00252       RESID = ZERO
00253       DO 40 J = 1, NRHS
00254          CALL DCOPY( N, X( 1, J ), 1, WORK, 1 )
00255          IX = IDAMAX( N, WORK, 1 )
00256          XNORM = MAX( ONE, ABS( X( IX, J ) ) )
00257          XSCAL = ( ONE / XNORM ) / DBLE( KD+1 )
00258          CALL DSCAL( N, XSCAL, WORK, 1 )
00259          CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
00260          CALL DAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 )
00261          IX = IDAMAX( N, WORK, 1 )
00262          ERR = TSCAL*ABS( WORK( IX ) )
00263          IX = IDAMAX( N, X( 1, J ), 1 )
00264          XNORM = ABS( X( IX, J ) )
00265          IF( ERR*SMLNUM.LE.XNORM ) THEN
00266             IF( XNORM.GT.ZERO )
00267      $         ERR = ERR / XNORM
00268          ELSE
00269             IF( ERR.GT.ZERO )
00270      $         ERR = ONE / EPS
00271          END IF
00272          IF( ERR*SMLNUM.LE.TNORM ) THEN
00273             IF( TNORM.GT.ZERO )
00274      $         ERR = ERR / TNORM
00275          ELSE
00276             IF( ERR.GT.ZERO )
00277      $         ERR = ONE / EPS
00278          END IF
00279          RESID = MAX( RESID, ERR )
00280    40 CONTINUE
00281 *
00282       RETURN
00283 *
00284 *     End of DTBT03
00285 *
00286       END
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