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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE ZHPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, 00002 $ TAU, WORK, RWORK, RESULT ) 00003 * 00004 * -- LAPACK test routine (version 3.1) -- 00005 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER ITYPE, KBAND, LDU, N 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION D( * ), E( * ), RESULT( 2 ), RWORK( * ) 00014 COMPLEX*16 AP( * ), TAU( * ), U( LDU, * ), VP( * ), 00015 $ WORK( * ) 00016 * .. 00017 * 00018 * Purpose 00019 * ======= 00020 * 00021 * ZHPT21 generally checks a decomposition of the form 00022 * 00023 * A = U S U* 00024 * 00025 * where * means conjugate transpose, A is hermitian, U is 00026 * unitary, and S is diagonal (if KBAND=0) or (real) symmetric 00027 * tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as 00028 * a dense matrix, otherwise the U is expressed as a product of 00029 * Householder transformations, whose vectors are stored in the 00030 * array "V" and whose scaling constants are in "TAU"; we shall 00031 * use the letter "V" to refer to the product of Householder 00032 * transformations (which should be equal to U). 00033 * 00034 * Specifically, if ITYPE=1, then: 00035 * 00036 * RESULT(1) = | A - U S U* | / ( |A| n ulp ) *and* 00037 * RESULT(2) = | I - UU* | / ( n ulp ) 00038 * 00039 * If ITYPE=2, then: 00040 * 00041 * RESULT(1) = | A - V S V* | / ( |A| n ulp ) 00042 * 00043 * If ITYPE=3, then: 00044 * 00045 * RESULT(1) = | I - UV* | / ( n ulp ) 00046 * 00047 * Packed storage means that, for example, if UPLO='U', then the columns 00048 * of the upper triangle of A are stored one after another, so that 00049 * A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if 00050 * UPLO='L', then the columns of the lower triangle of A are stored one 00051 * after another in AP, so that A(j+1,j+1) immediately follows A(n,j) 00052 * in the array AP. This means that A(i,j) is stored in: 00053 * 00054 * AP( i + j*(j-1)/2 ) if UPLO='U' 00055 * 00056 * AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L' 00057 * 00058 * The array VP bears the same relation to the matrix V that A does to 00059 * AP. 00060 * 00061 * For ITYPE > 1, the transformation U is expressed as a product 00062 * of Householder transformations: 00063 * 00064 * If UPLO='U', then V = H(n-1)...H(1), where 00065 * 00066 * H(j) = I - tau(j) v(j) v(j)* 00067 * 00068 * and the first j-1 elements of v(j) are stored in V(1:j-1,j+1), 00069 * (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ), 00070 * the j-th element is 1, and the last n-j elements are 0. 00071 * 00072 * If UPLO='L', then V = H(1)...H(n-1), where 00073 * 00074 * H(j) = I - tau(j) v(j) v(j)* 00075 * 00076 * and the first j elements of v(j) are 0, the (j+1)-st is 1, and the 00077 * (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e., 00078 * in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .) 00079 * 00080 * Arguments 00081 * ========= 00082 * 00083 * ITYPE (input) INTEGER 00084 * Specifies the type of tests to be performed. 00085 * 1: U expressed as a dense unitary matrix: 00086 * RESULT(1) = | A - U S U* | / ( |A| n ulp ) *and* 00087 * RESULT(2) = | I - UU* | / ( n ulp ) 00088 * 00089 * 2: U expressed as a product V of Housholder transformations: 00090 * RESULT(1) = | A - V S V* | / ( |A| n ulp ) 00091 * 00092 * 3: U expressed both as a dense unitary matrix and 00093 * as a product of Housholder transformations: 00094 * RESULT(1) = | I - UV* | / ( n ulp ) 00095 * 00096 * UPLO (input) CHARACTER 00097 * If UPLO='U', the upper triangle of A and V will be used and 00098 * the (strictly) lower triangle will not be referenced. 00099 * If UPLO='L', the lower triangle of A and V will be used and 00100 * the (strictly) upper triangle will not be referenced. 00101 * 00102 * N (input) INTEGER 00103 * The size of the matrix. If it is zero, ZHPT21 does nothing. 00104 * It must be at least zero. 00105 * 00106 * KBAND (input) INTEGER 00107 * The bandwidth of the matrix. It may only be zero or one. 00108 * If zero, then S is diagonal, and E is not referenced. If 00109 * one, then S is symmetric tri-diagonal. 00110 * 00111 * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2) 00112 * The original (unfactored) matrix. It is assumed to be 00113 * hermitian, and contains the columns of just the upper 00114 * triangle (UPLO='U') or only the lower triangle (UPLO='L'), 00115 * packed one after another. 00116 * 00117 * D (input) DOUBLE PRECISION array, dimension (N) 00118 * The diagonal of the (symmetric tri-) diagonal matrix. 00119 * 00120 * E (input) DOUBLE PRECISION array, dimension (N) 00121 * The off-diagonal of the (symmetric tri-) diagonal matrix. 00122 * E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and 00123 * (3,2) element, etc. 00124 * Not referenced if KBAND=0. 00125 * 00126 * U (input) COMPLEX*16 array, dimension (LDU, N) 00127 * If ITYPE=1 or 3, this contains the unitary matrix in 00128 * the decomposition, expressed as a dense matrix. If ITYPE=2, 00129 * then it is not referenced. 00130 * 00131 * LDU (input) INTEGER 00132 * The leading dimension of U. LDU must be at least N and 00133 * at least 1. 00134 * 00135 * VP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) 00136 * If ITYPE=2 or 3, the columns of this array contain the 00137 * Householder vectors used to describe the unitary matrix 00138 * in the decomposition, as described in purpose. 00139 * *NOTE* If ITYPE=2 or 3, V is modified and restored. The 00140 * subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') 00141 * is set to one, and later reset to its original value, during 00142 * the course of the calculation. 00143 * If ITYPE=1, then it is neither referenced nor modified. 00144 * 00145 * TAU (input) COMPLEX*16 array, dimension (N) 00146 * If ITYPE >= 2, then TAU(j) is the scalar factor of 00147 * v(j) v(j)* in the Householder transformation H(j) of 00148 * the product U = H(1)...H(n-2) 00149 * If ITYPE < 2, then TAU is not referenced. 00150 * 00151 * WORK (workspace) COMPLEX*16 array, dimension (N**2) 00152 * Workspace. 00153 * 00154 * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 00155 * Workspace. 00156 * 00157 * RESULT (output) DOUBLE PRECISION array, dimension (2) 00158 * The values computed by the two tests described above. The 00159 * values are currently limited to 1/ulp, to avoid overflow. 00160 * RESULT(1) is always modified. RESULT(2) is modified only 00161 * if ITYPE=1. 00162 * 00163 * ===================================================================== 00164 * 00165 * .. Parameters .. 00166 DOUBLE PRECISION ZERO, ONE, TEN 00167 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 10.0D+0 ) 00168 DOUBLE PRECISION HALF 00169 PARAMETER ( HALF = 1.0D+0 / 2.0D+0 ) 00170 COMPLEX*16 CZERO, CONE 00171 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), 00172 $ CONE = ( 1.0D+0, 0.0D+0 ) ) 00173 * .. 00174 * .. Local Scalars .. 00175 LOGICAL LOWER 00176 CHARACTER CUPLO 00177 INTEGER IINFO, J, JP, JP1, JR, LAP 00178 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM 00179 COMPLEX*16 TEMP, VSAVE 00180 * .. 00181 * .. External Functions .. 00182 LOGICAL LSAME 00183 DOUBLE PRECISION DLAMCH, ZLANGE, ZLANHP 00184 COMPLEX*16 ZDOTC 00185 EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANHP, ZDOTC 00186 * .. 00187 * .. External Subroutines .. 00188 EXTERNAL ZAXPY, ZCOPY, ZGEMM, ZHPMV, ZHPR, ZHPR2, 00189 $ ZLACPY, ZLASET, ZUPMTR 00190 * .. 00191 * .. Intrinsic Functions .. 00192 INTRINSIC DBLE, DCMPLX, MAX, MIN 00193 * .. 00194 * .. Executable Statements .. 00195 * 00196 * Constants 00197 * 00198 RESULT( 1 ) = ZERO 00199 IF( ITYPE.EQ.1 ) 00200 $ RESULT( 2 ) = ZERO 00201 IF( N.LE.0 ) 00202 $ RETURN 00203 * 00204 LAP = ( N*( N+1 ) ) / 2 00205 * 00206 IF( LSAME( UPLO, 'U' ) ) THEN 00207 LOWER = .FALSE. 00208 CUPLO = 'U' 00209 ELSE 00210 LOWER = .TRUE. 00211 CUPLO = 'L' 00212 END IF 00213 * 00214 UNFL = DLAMCH( 'Safe minimum' ) 00215 ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) 00216 * 00217 * Some Error Checks 00218 * 00219 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN 00220 RESULT( 1 ) = TEN / ULP 00221 RETURN 00222 END IF 00223 * 00224 * Do Test 1 00225 * 00226 * Norm of A: 00227 * 00228 IF( ITYPE.EQ.3 ) THEN 00229 ANORM = ONE 00230 ELSE 00231 ANORM = MAX( ZLANHP( '1', CUPLO, N, AP, RWORK ), UNFL ) 00232 END IF 00233 * 00234 * Compute error matrix: 00235 * 00236 IF( ITYPE.EQ.1 ) THEN 00237 * 00238 * ITYPE=1: error = A - U S U* 00239 * 00240 CALL ZLASET( 'Full', N, N, CZERO, CZERO, WORK, N ) 00241 CALL ZCOPY( LAP, AP, 1, WORK, 1 ) 00242 * 00243 DO 10 J = 1, N 00244 CALL ZHPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK ) 00245 10 CONTINUE 00246 * 00247 IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN 00248 DO 20 J = 1, N - 1 00249 CALL ZHPR2( CUPLO, N, -DCMPLX( E( J ) ), U( 1, J ), 1, 00250 $ U( 1, J-1 ), 1, WORK ) 00251 20 CONTINUE 00252 END IF 00253 WNORM = ZLANHP( '1', CUPLO, N, WORK, RWORK ) 00254 * 00255 ELSE IF( ITYPE.EQ.2 ) THEN 00256 * 00257 * ITYPE=2: error = V S V* - A 00258 * 00259 CALL ZLASET( 'Full', N, N, CZERO, CZERO, WORK, N ) 00260 * 00261 IF( LOWER ) THEN 00262 WORK( LAP ) = D( N ) 00263 DO 40 J = N - 1, 1, -1 00264 JP = ( ( 2*N-J )*( J-1 ) ) / 2 00265 JP1 = JP + N - J 00266 IF( KBAND.EQ.1 ) THEN 00267 WORK( JP+J+1 ) = ( CONE-TAU( J ) )*E( J ) 00268 DO 30 JR = J + 2, N 00269 WORK( JP+JR ) = -TAU( J )*E( J )*VP( JP+JR ) 00270 30 CONTINUE 00271 END IF 00272 * 00273 IF( TAU( J ).NE.CZERO ) THEN 00274 VSAVE = VP( JP+J+1 ) 00275 VP( JP+J+1 ) = CONE 00276 CALL ZHPMV( 'L', N-J, CONE, WORK( JP1+J+1 ), 00277 $ VP( JP+J+1 ), 1, CZERO, WORK( LAP+1 ), 1 ) 00278 TEMP = -HALF*TAU( J )*ZDOTC( N-J, WORK( LAP+1 ), 1, 00279 $ VP( JP+J+1 ), 1 ) 00280 CALL ZAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ), 00281 $ 1 ) 00282 CALL ZHPR2( 'L', N-J, -TAU( J ), VP( JP+J+1 ), 1, 00283 $ WORK( LAP+1 ), 1, WORK( JP1+J+1 ) ) 00284 * 00285 VP( JP+J+1 ) = VSAVE 00286 END IF 00287 WORK( JP+J ) = D( J ) 00288 40 CONTINUE 00289 ELSE 00290 WORK( 1 ) = D( 1 ) 00291 DO 60 J = 1, N - 1 00292 JP = ( J*( J-1 ) ) / 2 00293 JP1 = JP + J 00294 IF( KBAND.EQ.1 ) THEN 00295 WORK( JP1+J ) = ( CONE-TAU( J ) )*E( J ) 00296 DO 50 JR = 1, J - 1 00297 WORK( JP1+JR ) = -TAU( J )*E( J )*VP( JP1+JR ) 00298 50 CONTINUE 00299 END IF 00300 * 00301 IF( TAU( J ).NE.CZERO ) THEN 00302 VSAVE = VP( JP1+J ) 00303 VP( JP1+J ) = CONE 00304 CALL ZHPMV( 'U', J, CONE, WORK, VP( JP1+1 ), 1, CZERO, 00305 $ WORK( LAP+1 ), 1 ) 00306 TEMP = -HALF*TAU( J )*ZDOTC( J, WORK( LAP+1 ), 1, 00307 $ VP( JP1+1 ), 1 ) 00308 CALL ZAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ), 00309 $ 1 ) 00310 CALL ZHPR2( 'U', J, -TAU( J ), VP( JP1+1 ), 1, 00311 $ WORK( LAP+1 ), 1, WORK ) 00312 VP( JP1+J ) = VSAVE 00313 END IF 00314 WORK( JP1+J+1 ) = D( J+1 ) 00315 60 CONTINUE 00316 END IF 00317 * 00318 DO 70 J = 1, LAP 00319 WORK( J ) = WORK( J ) - AP( J ) 00320 70 CONTINUE 00321 WNORM = ZLANHP( '1', CUPLO, N, WORK, RWORK ) 00322 * 00323 ELSE IF( ITYPE.EQ.3 ) THEN 00324 * 00325 * ITYPE=3: error = U V* - I 00326 * 00327 IF( N.LT.2 ) 00328 $ RETURN 00329 CALL ZLACPY( ' ', N, N, U, LDU, WORK, N ) 00330 CALL ZUPMTR( 'R', CUPLO, 'C', N, N, VP, TAU, WORK, N, 00331 $ WORK( N**2+1 ), IINFO ) 00332 IF( IINFO.NE.0 ) THEN 00333 RESULT( 1 ) = TEN / ULP 00334 RETURN 00335 END IF 00336 * 00337 DO 80 J = 1, N 00338 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - CONE 00339 80 CONTINUE 00340 * 00341 WNORM = ZLANGE( '1', N, N, WORK, N, RWORK ) 00342 END IF 00343 * 00344 IF( ANORM.GT.WNORM ) THEN 00345 RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP ) 00346 ELSE 00347 IF( ANORM.LT.ONE ) THEN 00348 RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP ) 00349 ELSE 00350 RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP ) 00351 END IF 00352 END IF 00353 * 00354 * Do Test 2 00355 * 00356 * Compute UU* - I 00357 * 00358 IF( ITYPE.EQ.1 ) THEN 00359 CALL ZGEMM( 'N', 'C', N, N, N, CONE, U, LDU, U, LDU, CZERO, 00360 $ WORK, N ) 00361 * 00362 DO 90 J = 1, N 00363 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - CONE 00364 90 CONTINUE 00365 * 00366 RESULT( 2 ) = MIN( ZLANGE( '1', N, N, WORK, N, RWORK ), 00367 $ DBLE( N ) ) / ( N*ULP ) 00368 END IF 00369 * 00370 RETURN 00371 * 00372 * End of ZHPT21 00373 * 00374 END
1.7.3 
