LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
ssbt21.f
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00001 *> \brief \b SSBT21
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 SSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
00012 *                          RESULT )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          UPLO
00016 *       INTEGER            KA, KS, LDA, LDU, N
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       REAL               A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
00020 *      $                   U( LDU, * ), WORK( * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *> SSBT21  generally checks a decomposition of the form
00030 *>
00031 *>         A = U S U'
00032 *>
00033 *> where ' means transpose, A is symmetric banded, U is
00034 *> orthogonal, and S is diagonal (if KS=0) or symmetric
00035 *> tridiagonal (if KS=1).
00036 *>
00037 *> Specifically:
00038 *>
00039 *>         RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC>         RESULT(2) = | I - UU' | / ( n ulp )
00040 *> \endverbatim
00041 *
00042 *  Arguments:
00043 *  ==========
00044 *
00045 *> \param[in] UPLO
00046 *> \verbatim
00047 *>          UPLO is CHARACTER
00048 *>          If UPLO='U', the upper triangle of A and V will be used and
00049 *>          the (strictly) lower triangle will not be referenced.
00050 *>          If UPLO='L', the lower triangle of A and V will be used and
00051 *>          the (strictly) upper triangle will not be referenced.
00052 *> \endverbatim
00053 *>
00054 *> \param[in] N
00055 *> \verbatim
00056 *>          N is INTEGER
00057 *>          The size of the matrix.  If it is zero, SSBT21 does nothing.
00058 *>          It must be at least zero.
00059 *> \endverbatim
00060 *>
00061 *> \param[in] KA
00062 *> \verbatim
00063 *>          KA is INTEGER
00064 *>          The bandwidth of the matrix A.  It must be at least zero.  If
00065 *>          it is larger than N-1, then max( 0, N-1 ) will be used.
00066 *> \endverbatim
00067 *>
00068 *> \param[in] KS
00069 *> \verbatim
00070 *>          KS is INTEGER
00071 *>          The bandwidth of the matrix S.  It may only be zero or one.
00072 *>          If zero, then S is diagonal, and E is not referenced.  If
00073 *>          one, then S is symmetric tri-diagonal.
00074 *> \endverbatim
00075 *>
00076 *> \param[in] A
00077 *> \verbatim
00078 *>          A is REAL array, dimension (LDA, N)
00079 *>          The original (unfactored) matrix.  It is assumed to be
00080 *>          symmetric, and only the upper (UPLO='U') or only the lower
00081 *>          (UPLO='L') will be referenced.
00082 *> \endverbatim
00083 *>
00084 *> \param[in] LDA
00085 *> \verbatim
00086 *>          LDA is INTEGER
00087 *>          The leading dimension of A.  It must be at least 1
00088 *>          and at least min( KA, N-1 ).
00089 *> \endverbatim
00090 *>
00091 *> \param[in] D
00092 *> \verbatim
00093 *>          D is REAL array, dimension (N)
00094 *>          The diagonal of the (symmetric tri-) diagonal matrix S.
00095 *> \endverbatim
00096 *>
00097 *> \param[in] E
00098 *> \verbatim
00099 *>          E is REAL array, dimension (N-1)
00100 *>          The off-diagonal of the (symmetric tri-) diagonal matrix S.
00101 *>          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
00102 *>          (3,2) element, etc.
00103 *>          Not referenced if KS=0.
00104 *> \endverbatim
00105 *>
00106 *> \param[in] U
00107 *> \verbatim
00108 *>          U is REAL array, dimension (LDU, N)
00109 *>          The orthogonal matrix in the decomposition, expressed as a
00110 *>          dense matrix (i.e., not as a product of Householder
00111 *>          transformations, Givens transformations, etc.)
00112 *> \endverbatim
00113 *>
00114 *> \param[in] LDU
00115 *> \verbatim
00116 *>          LDU is INTEGER
00117 *>          The leading dimension of U.  LDU must be at least N and
00118 *>          at least 1.
00119 *> \endverbatim
00120 *>
00121 *> \param[out] WORK
00122 *> \verbatim
00123 *>          WORK is REAL array, dimension (N**2+N)
00124 *> \endverbatim
00125 *>
00126 *> \param[out] RESULT
00127 *> \verbatim
00128 *>          RESULT is REAL array, dimension (2)
00129 *>          The values computed by the two tests described above.  The
00130 *>          values are currently limited to 1/ulp, to avoid overflow.
00131 *> \endverbatim
00132 *
00133 *  Authors:
00134 *  ========
00135 *
00136 *> \author Univ. of Tennessee 
00137 *> \author Univ. of California Berkeley 
00138 *> \author Univ. of Colorado Denver 
00139 *> \author NAG Ltd. 
00140 *
00141 *> \date November 2011
00142 *
00143 *> \ingroup single_eig
00144 *
00145 *  =====================================================================
00146       SUBROUTINE SSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
00147      $                   RESULT )
00148 *
00149 *  -- LAPACK test routine (version 3.4.0) --
00150 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00151 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00152 *     November 2011
00153 *
00154 *     .. Scalar Arguments ..
00155       CHARACTER          UPLO
00156       INTEGER            KA, KS, LDA, LDU, N
00157 *     ..
00158 *     .. Array Arguments ..
00159       REAL               A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
00160      $                   U( LDU, * ), WORK( * )
00161 *     ..
00162 *
00163 *  =====================================================================
00164 *
00165 *     .. Parameters ..
00166       REAL               ZERO, ONE
00167       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
00168 *     ..
00169 *     .. Local Scalars ..
00170       LOGICAL            LOWER
00171       CHARACTER          CUPLO
00172       INTEGER            IKA, J, JC, JR, LW
00173       REAL               ANORM, ULP, UNFL, WNORM
00174 *     ..
00175 *     .. External Functions ..
00176       LOGICAL            LSAME
00177       REAL               SLAMCH, SLANGE, SLANSB, SLANSP
00178       EXTERNAL           LSAME, SLAMCH, SLANGE, SLANSB, SLANSP
00179 *     ..
00180 *     .. External Subroutines ..
00181       EXTERNAL           SGEMM, SSPR, SSPR2
00182 *     ..
00183 *     .. Intrinsic Functions ..
00184       INTRINSIC          MAX, MIN, REAL
00185 *     ..
00186 *     .. Executable Statements ..
00187 *
00188 *     Constants
00189 *
00190       RESULT( 1 ) = ZERO
00191       RESULT( 2 ) = ZERO
00192       IF( N.LE.0 )
00193      $   RETURN
00194 *
00195       IKA = MAX( 0, MIN( N-1, KA ) )
00196       LW = ( N*( N+1 ) ) / 2
00197 *
00198       IF( LSAME( UPLO, 'U' ) ) THEN
00199          LOWER = .FALSE.
00200          CUPLO = 'U'
00201       ELSE
00202          LOWER = .TRUE.
00203          CUPLO = 'L'
00204       END IF
00205 *
00206       UNFL = SLAMCH( 'Safe minimum' )
00207       ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' )
00208 *
00209 *     Some Error Checks
00210 *
00211 *     Do Test 1
00212 *
00213 *     Norm of A:
00214 *
00215       ANORM = MAX( SLANSB( '1', CUPLO, N, IKA, A, LDA, WORK ), UNFL )
00216 *
00217 *     Compute error matrix:    Error = A - U S U'
00218 *
00219 *     Copy A from SB to SP storage format.
00220 *
00221       J = 0
00222       DO 50 JC = 1, N
00223          IF( LOWER ) THEN
00224             DO 10 JR = 1, MIN( IKA+1, N+1-JC )
00225                J = J + 1
00226                WORK( J ) = A( JR, JC )
00227    10       CONTINUE
00228             DO 20 JR = IKA + 2, N + 1 - JC
00229                J = J + 1
00230                WORK( J ) = ZERO
00231    20       CONTINUE
00232          ELSE
00233             DO 30 JR = IKA + 2, JC
00234                J = J + 1
00235                WORK( J ) = ZERO
00236    30       CONTINUE
00237             DO 40 JR = MIN( IKA, JC-1 ), 0, -1
00238                J = J + 1
00239                WORK( J ) = A( IKA+1-JR, JC )
00240    40       CONTINUE
00241          END IF
00242    50 CONTINUE
00243 *
00244       DO 60 J = 1, N
00245          CALL SSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
00246    60 CONTINUE
00247 *
00248       IF( N.GT.1 .AND. KS.EQ.1 ) THEN
00249          DO 70 J = 1, N - 1
00250             CALL SSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
00251      $                  WORK )
00252    70    CONTINUE
00253       END IF
00254       WNORM = SLANSP( '1', CUPLO, N, WORK, WORK( LW+1 ) )
00255 *
00256       IF( ANORM.GT.WNORM ) THEN
00257          RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
00258       ELSE
00259          IF( ANORM.LT.ONE ) THEN
00260             RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
00261          ELSE
00262             RESULT( 1 ) = MIN( WNORM / ANORM, REAL( N ) ) / ( N*ULP )
00263          END IF
00264       END IF
00265 *
00266 *     Do Test 2
00267 *
00268 *     Compute  UU' - I
00269 *
00270       CALL SGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
00271      $            N )
00272 *
00273       DO 80 J = 1, N
00274          WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
00275    80 CONTINUE
00276 *
00277       RESULT( 2 ) = MIN( SLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ),
00278      $              REAL( N ) ) / ( N*ULP )
00279 *
00280       RETURN
00281 *
00282 *     End of SSBT21
00283 *
00284       END
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