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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DSYT22 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 DSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, 00012 * V, LDV, TAU, WORK, RESULT ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER UPLO 00016 * INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N 00017 * .. 00018 * .. Array Arguments .. 00019 * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ), 00020 * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) 00021 * .. 00022 * 00023 * 00024 *> \par Purpose: 00025 * ============= 00026 *> 00027 *> \verbatim 00028 *> 00029 *> DSYT22 generally checks a decomposition of the form 00030 *> 00031 *> A U = U S 00032 *> 00033 *> where A is symmetric, the columns of U are orthonormal, and S 00034 *> is diagonal (if KBAND=0) or symmetric tridiagonal (if 00035 *> KBAND=1). If ITYPE=1, then U is represented as a dense matrix, 00036 *> otherwise the U is expressed as a product of Householder 00037 *> transformations, whose vectors are stored in the array "V" and 00038 *> whose scaling constants are in "TAU"; we shall use the letter 00039 *> "V" to refer to the product of Householder transformations 00040 *> (which should be equal to U). 00041 *> 00042 *> Specifically, if ITYPE=1, then: 00043 *> 00044 *> RESULT(1) = | U' A U - S | / ( |A| m ulp ) *andC> RESULT(2) = | I - U'U | / ( m ulp ) 00045 *> \endverbatim 00046 * 00047 * Arguments: 00048 * ========== 00049 * 00050 *> \verbatim 00051 *> ITYPE INTEGER 00052 *> Specifies the type of tests to be performed. 00053 *> 1: U expressed as a dense orthogonal matrix: 00054 *> RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( n ulp ) 00055 *> 00056 *> UPLO CHARACTER 00057 *> If UPLO='U', the upper triangle of A will be used and the 00058 *> (strictly) lower triangle will not be referenced. If 00059 *> UPLO='L', the lower triangle of A will be used and the 00060 *> (strictly) upper triangle will not be referenced. 00061 *> Not modified. 00062 *> 00063 *> N INTEGER 00064 *> The size of the matrix. If it is zero, DSYT22 does nothing. 00065 *> It must be at least zero. 00066 *> Not modified. 00067 *> 00068 *> M INTEGER 00069 *> The number of columns of U. If it is zero, DSYT22 does 00070 *> nothing. It must be at least zero. 00071 *> Not modified. 00072 *> 00073 *> KBAND INTEGER 00074 *> The bandwidth of the matrix. It may only be zero or one. 00075 *> If zero, then S is diagonal, and E is not referenced. If 00076 *> one, then S is symmetric tri-diagonal. 00077 *> Not modified. 00078 *> 00079 *> A DOUBLE PRECISION array, dimension (LDA , N) 00080 *> The original (unfactored) matrix. It is assumed to be 00081 *> symmetric, and only the upper (UPLO='U') or only the lower 00082 *> (UPLO='L') will be referenced. 00083 *> Not modified. 00084 *> 00085 *> LDA INTEGER 00086 *> The leading dimension of A. It must be at least 1 00087 *> and at least N. 00088 *> Not modified. 00089 *> 00090 *> D DOUBLE PRECISION array, dimension (N) 00091 *> The diagonal of the (symmetric tri-) diagonal matrix. 00092 *> Not modified. 00093 *> 00094 *> E DOUBLE PRECISION array, dimension (N) 00095 *> The off-diagonal of the (symmetric tri-) diagonal matrix. 00096 *> E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc. 00097 *> Not referenced if KBAND=0. 00098 *> Not modified. 00099 *> 00100 *> U DOUBLE PRECISION array, dimension (LDU, N) 00101 *> If ITYPE=1 or 3, this contains the orthogonal matrix in 00102 *> the decomposition, expressed as a dense matrix. If ITYPE=2, 00103 *> then it is not referenced. 00104 *> Not modified. 00105 *> 00106 *> LDU INTEGER 00107 *> The leading dimension of U. LDU must be at least N and 00108 *> at least 1. 00109 *> Not modified. 00110 *> 00111 *> V DOUBLE PRECISION array, dimension (LDV, N) 00112 *> If ITYPE=2 or 3, the lower triangle of this array contains 00113 *> the Householder vectors used to describe the orthogonal 00114 *> matrix in the decomposition. If ITYPE=1, then it is not 00115 *> referenced. 00116 *> Not modified. 00117 *> 00118 *> LDV INTEGER 00119 *> The leading dimension of V. LDV must be at least N and 00120 *> at least 1. 00121 *> Not modified. 00122 *> 00123 *> TAU DOUBLE PRECISION array, dimension (N) 00124 *> If ITYPE >= 2, then TAU(j) is the scalar factor of 00125 *> v(j) v(j)' in the Householder transformation H(j) of 00126 *> the product U = H(1)...H(n-2) 00127 *> If ITYPE < 2, then TAU is not referenced. 00128 *> Not modified. 00129 *> 00130 *> WORK DOUBLE PRECISION array, dimension (2*N**2) 00131 *> Workspace. 00132 *> Modified. 00133 *> 00134 *> RESULT DOUBLE PRECISION array, dimension (2) 00135 *> The values computed by the two tests described above. The 00136 *> values are currently limited to 1/ulp, to avoid overflow. 00137 *> RESULT(1) is always modified. RESULT(2) is modified only 00138 *> if LDU is at least N. 00139 *> Modified. 00140 *> \endverbatim 00141 * 00142 * Authors: 00143 * ======== 00144 * 00145 *> \author Univ. of Tennessee 00146 *> \author Univ. of California Berkeley 00147 *> \author Univ. of Colorado Denver 00148 *> \author NAG Ltd. 00149 * 00150 *> \date November 2011 00151 * 00152 *> \ingroup double_eig 00153 * 00154 * ===================================================================== 00155 SUBROUTINE DSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, 00156 $ V, LDV, TAU, WORK, RESULT ) 00157 * 00158 * -- LAPACK test routine (version 3.4.0) -- 00159 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00161 * November 2011 00162 * 00163 * .. Scalar Arguments .. 00164 CHARACTER UPLO 00165 INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N 00166 * .. 00167 * .. Array Arguments .. 00168 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ), 00169 $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) 00170 * .. 00171 * 00172 * ===================================================================== 00173 * 00174 * .. Parameters .. 00175 DOUBLE PRECISION ZERO, ONE 00176 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 00177 * .. 00178 * .. Local Scalars .. 00179 INTEGER J, JJ, JJ1, JJ2, NN, NNP1 00180 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM 00181 * .. 00182 * .. External Functions .. 00183 DOUBLE PRECISION DLAMCH, DLANSY 00184 EXTERNAL DLAMCH, DLANSY 00185 * .. 00186 * .. External Subroutines .. 00187 EXTERNAL DGEMM, DORT01, DSYMM 00188 * .. 00189 * .. Intrinsic Functions .. 00190 INTRINSIC DBLE, MAX, MIN 00191 * .. 00192 * .. Executable Statements .. 00193 * 00194 RESULT( 1 ) = ZERO 00195 RESULT( 2 ) = ZERO 00196 IF( N.LE.0 .OR. M.LE.0 ) 00197 $ RETURN 00198 * 00199 UNFL = DLAMCH( 'Safe minimum' ) 00200 ULP = DLAMCH( 'Precision' ) 00201 * 00202 * Do Test 1 00203 * 00204 * Norm of A: 00205 * 00206 ANORM = MAX( DLANSY( '1', UPLO, N, A, LDA, WORK ), UNFL ) 00207 * 00208 * Compute error matrix: 00209 * 00210 * ITYPE=1: error = U' A U - S 00211 * 00212 CALL DSYMM( 'L', UPLO, N, M, ONE, A, LDA, U, LDU, ZERO, WORK, N ) 00213 NN = N*N 00214 NNP1 = NN + 1 00215 CALL DGEMM( 'T', 'N', M, M, N, ONE, U, LDU, WORK, N, ZERO, 00216 $ WORK( NNP1 ), N ) 00217 DO 10 J = 1, M 00218 JJ = NN + ( J-1 )*N + J 00219 WORK( JJ ) = WORK( JJ ) - D( J ) 00220 10 CONTINUE 00221 IF( KBAND.EQ.1 .AND. N.GT.1 ) THEN 00222 DO 20 J = 2, M 00223 JJ1 = NN + ( J-1 )*N + J - 1 00224 JJ2 = NN + ( J-2 )*N + J 00225 WORK( JJ1 ) = WORK( JJ1 ) - E( J-1 ) 00226 WORK( JJ2 ) = WORK( JJ2 ) - E( J-1 ) 00227 20 CONTINUE 00228 END IF 00229 WNORM = DLANSY( '1', UPLO, M, WORK( NNP1 ), N, WORK( 1 ) ) 00230 * 00231 IF( ANORM.GT.WNORM ) THEN 00232 RESULT( 1 ) = ( WNORM / ANORM ) / ( M*ULP ) 00233 ELSE 00234 IF( ANORM.LT.ONE ) THEN 00235 RESULT( 1 ) = ( MIN( WNORM, M*ANORM ) / ANORM ) / ( M*ULP ) 00236 ELSE 00237 RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( M ) ) / ( M*ULP ) 00238 END IF 00239 END IF 00240 * 00241 * Do Test 2 00242 * 00243 * Compute U'U - I 00244 * 00245 IF( ITYPE.EQ.1 ) 00246 $ CALL DORT01( 'Columns', N, M, U, LDU, WORK, 2*N*N, 00247 $ RESULT( 2 ) ) 00248 * 00249 RETURN 00250 * 00251 * End of DSYT22 00252 * 00253 END