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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, 00002 $ TAU, WORK, 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 REAL AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ), 00014 $ U( LDU, * ), VP( * ), WORK( * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * SSPT21 generally checks a decomposition of the form 00021 * 00022 * A = U S U' 00023 * 00024 * where ' means transpose, A is symmetric (stored in packed format), U 00025 * is orthogonal, and S is diagonal (if KBAND=0) or symmetric 00026 * tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a 00027 * dense matrix, otherwise the U is expressed as a product of 00028 * Householder transformations, whose vectors are stored in the array 00029 * "V" and whose scaling constants are in "TAU"; we shall use the 00030 * letter "V" to refer to the product of Householder transformations 00031 * (which should be equal to U). 00032 * 00033 * Specifically, if ITYPE=1, then: 00034 * 00035 * RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and* 00036 * RESULT(2) = | I - UU' | / ( n ulp ) 00037 * 00038 * If ITYPE=2, then: 00039 * 00040 * RESULT(1) = | A - V S V' | / ( |A| n ulp ) 00041 * 00042 * If ITYPE=3, then: 00043 * 00044 * RESULT(1) = | I - VU' | / ( n ulp ) 00045 * 00046 * Packed storage means that, for example, if UPLO='U', then the columns 00047 * of the upper triangle of A are stored one after another, so that 00048 * A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if 00049 * UPLO='L', then the columns of the lower triangle of A are stored one 00050 * after another in AP, so that A(j+1,j+1) immediately follows A(n,j) 00051 * in the array AP. This means that A(i,j) is stored in: 00052 * 00053 * AP( i + j*(j-1)/2 ) if UPLO='U' 00054 * 00055 * AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L' 00056 * 00057 * The array VP bears the same relation to the matrix V that A does to 00058 * AP. 00059 * 00060 * For ITYPE > 1, the transformation U is expressed as a product 00061 * of Householder transformations: 00062 * 00063 * If UPLO='U', then V = H(n-1)...H(1), where 00064 * 00065 * H(j) = I - tau(j) v(j) v(j)' 00066 * 00067 * and the first j-1 elements of v(j) are stored in V(1:j-1,j+1), 00068 * (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ), 00069 * the j-th element is 1, and the last n-j elements are 0. 00070 * 00071 * If UPLO='L', then V = H(1)...H(n-1), where 00072 * 00073 * H(j) = I - tau(j) v(j) v(j)' 00074 * 00075 * and the first j elements of v(j) are 0, the (j+1)-st is 1, and the 00076 * (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e., 00077 * in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .) 00078 * 00079 * Arguments 00080 * ========= 00081 * 00082 * ITYPE (input) INTEGER 00083 * Specifies the type of tests to be performed. 00084 * 1: U expressed as a dense orthogonal matrix: 00085 * RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and* 00086 * RESULT(2) = | I - UU' | / ( n ulp ) 00087 * 00088 * 2: U expressed as a product V of Housholder transformations: 00089 * RESULT(1) = | A - V S V' | / ( |A| n ulp ) 00090 * 00091 * 3: U expressed both as a dense orthogonal matrix and 00092 * as a product of Housholder transformations: 00093 * RESULT(1) = | I - VU' | / ( n ulp ) 00094 * 00095 * UPLO (input) CHARACTER 00096 * If UPLO='U', AP and VP are considered to contain the upper 00097 * triangle of A and V. 00098 * If UPLO='L', AP and VP are considered to contain the lower 00099 * triangle of A and V. 00100 * 00101 * N (input) INTEGER 00102 * The size of the matrix. If it is zero, SSPT21 does nothing. 00103 * It must be at least zero. 00104 * 00105 * KBAND (input) INTEGER 00106 * The bandwidth of the matrix. It may only be zero or one. 00107 * If zero, then S is diagonal, and E is not referenced. If 00108 * one, then S is symmetric tri-diagonal. 00109 * 00110 * AP (input) REAL array, dimension (N*(N+1)/2) 00111 * The original (unfactored) matrix. It is assumed to be 00112 * symmetric, and contains the columns of just the upper 00113 * triangle (UPLO='U') or only the lower triangle (UPLO='L'), 00114 * packed one after another. 00115 * 00116 * D (input) REAL array, dimension (N) 00117 * The diagonal of the (symmetric tri-) diagonal matrix. 00118 * 00119 * E (input) REAL array, dimension (N-1) 00120 * The off-diagonal of the (symmetric tri-) diagonal matrix. 00121 * E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and 00122 * (3,2) element, etc. 00123 * Not referenced if KBAND=0. 00124 * 00125 * U (input) REAL array, dimension (LDU, N) 00126 * If ITYPE=1 or 3, this contains the orthogonal matrix in 00127 * the decomposition, expressed as a dense matrix. If ITYPE=2, 00128 * then it is not referenced. 00129 * 00130 * LDU (input) INTEGER 00131 * The leading dimension of U. LDU must be at least N and 00132 * at least 1. 00133 * 00134 * VP (input) REAL array, dimension (N*(N+1)/2) 00135 * If ITYPE=2 or 3, the columns of this array contain the 00136 * Householder vectors used to describe the orthogonal matrix 00137 * in the decomposition, as described in purpose. 00138 * *NOTE* If ITYPE=2 or 3, V is modified and restored. The 00139 * subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') 00140 * is set to one, and later reset to its original value, during 00141 * the course of the calculation. 00142 * If ITYPE=1, then it is neither referenced nor modified. 00143 * 00144 * TAU (input) REAL array, dimension (N) 00145 * If ITYPE >= 2, then TAU(j) is the scalar factor of 00146 * v(j) v(j)' in the Householder transformation H(j) of 00147 * the product U = H(1)...H(n-2) 00148 * If ITYPE < 2, then TAU is not referenced. 00149 * 00150 * WORK (workspace) REAL array, dimension (N**2+N) 00151 * Workspace. 00152 * 00153 * RESULT (output) REAL array, dimension (2) 00154 * The values computed by the two tests described above. The 00155 * values are currently limited to 1/ulp, to avoid overflow. 00156 * RESULT(1) is always modified. RESULT(2) is modified only 00157 * if ITYPE=1. 00158 * 00159 * ===================================================================== 00160 * 00161 * .. Parameters .. 00162 REAL ZERO, ONE, TEN 00163 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TEN = 10.0E0 ) 00164 REAL HALF 00165 PARAMETER ( HALF = 1.0E+0 / 2.0E+0 ) 00166 * .. 00167 * .. Local Scalars .. 00168 LOGICAL LOWER 00169 CHARACTER CUPLO 00170 INTEGER IINFO, J, JP, JP1, JR, LAP 00171 REAL ANORM, TEMP, ULP, UNFL, VSAVE, WNORM 00172 * .. 00173 * .. External Functions .. 00174 LOGICAL LSAME 00175 REAL SDOT, SLAMCH, SLANGE, SLANSP 00176 EXTERNAL LSAME, SDOT, SLAMCH, SLANGE, SLANSP 00177 * .. 00178 * .. External Subroutines .. 00179 EXTERNAL SAXPY, SCOPY, SGEMM, SLACPY, SLASET, SOPMTR, 00180 $ SSPMV, SSPR, SSPR2 00181 * .. 00182 * .. Intrinsic Functions .. 00183 INTRINSIC MAX, MIN, REAL 00184 * .. 00185 * .. Executable Statements .. 00186 * 00187 * 1) Constants 00188 * 00189 RESULT( 1 ) = ZERO 00190 IF( ITYPE.EQ.1 ) 00191 $ RESULT( 2 ) = ZERO 00192 IF( N.LE.0 ) 00193 $ RETURN 00194 * 00195 LAP = ( N*( N+1 ) ) / 2 00196 * 00197 IF( LSAME( UPLO, 'U' ) ) THEN 00198 LOWER = .FALSE. 00199 CUPLO = 'U' 00200 ELSE 00201 LOWER = .TRUE. 00202 CUPLO = 'L' 00203 END IF 00204 * 00205 UNFL = SLAMCH( 'Safe minimum' ) 00206 ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' ) 00207 * 00208 * Some Error Checks 00209 * 00210 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN 00211 RESULT( 1 ) = TEN / ULP 00212 RETURN 00213 END IF 00214 * 00215 * Do Test 1 00216 * 00217 * Norm of A: 00218 * 00219 IF( ITYPE.EQ.3 ) THEN 00220 ANORM = ONE 00221 ELSE 00222 ANORM = MAX( SLANSP( '1', CUPLO, N, AP, WORK ), UNFL ) 00223 END IF 00224 * 00225 * Compute error matrix: 00226 * 00227 IF( ITYPE.EQ.1 ) THEN 00228 * 00229 * ITYPE=1: error = A - U S U' 00230 * 00231 CALL SLASET( 'Full', N, N, ZERO, ZERO, WORK, N ) 00232 CALL SCOPY( LAP, AP, 1, WORK, 1 ) 00233 * 00234 DO 10 J = 1, N 00235 CALL SSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK ) 00236 10 CONTINUE 00237 * 00238 IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN 00239 DO 20 J = 1, N - 1 00240 CALL SSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), 00241 $ 1, WORK ) 00242 20 CONTINUE 00243 END IF 00244 WNORM = SLANSP( '1', CUPLO, N, WORK, WORK( N**2+1 ) ) 00245 * 00246 ELSE IF( ITYPE.EQ.2 ) THEN 00247 * 00248 * ITYPE=2: error = V S V' - A 00249 * 00250 CALL SLASET( 'Full', N, N, ZERO, ZERO, WORK, N ) 00251 * 00252 IF( LOWER ) THEN 00253 WORK( LAP ) = D( N ) 00254 DO 40 J = N - 1, 1, -1 00255 JP = ( ( 2*N-J )*( J-1 ) ) / 2 00256 JP1 = JP + N - J 00257 IF( KBAND.EQ.1 ) THEN 00258 WORK( JP+J+1 ) = ( ONE-TAU( J ) )*E( J ) 00259 DO 30 JR = J + 2, N 00260 WORK( JP+JR ) = -TAU( J )*E( J )*VP( JP+JR ) 00261 30 CONTINUE 00262 END IF 00263 * 00264 IF( TAU( J ).NE.ZERO ) THEN 00265 VSAVE = VP( JP+J+1 ) 00266 VP( JP+J+1 ) = ONE 00267 CALL SSPMV( 'L', N-J, ONE, WORK( JP1+J+1 ), 00268 $ VP( JP+J+1 ), 1, ZERO, WORK( LAP+1 ), 1 ) 00269 TEMP = -HALF*TAU( J )*SDOT( N-J, WORK( LAP+1 ), 1, 00270 $ VP( JP+J+1 ), 1 ) 00271 CALL SAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ), 00272 $ 1 ) 00273 CALL SSPR2( 'L', N-J, -TAU( J ), VP( JP+J+1 ), 1, 00274 $ WORK( LAP+1 ), 1, WORK( JP1+J+1 ) ) 00275 VP( JP+J+1 ) = VSAVE 00276 END IF 00277 WORK( JP+J ) = D( J ) 00278 40 CONTINUE 00279 ELSE 00280 WORK( 1 ) = D( 1 ) 00281 DO 60 J = 1, N - 1 00282 JP = ( J*( J-1 ) ) / 2 00283 JP1 = JP + J 00284 IF( KBAND.EQ.1 ) THEN 00285 WORK( JP1+J ) = ( ONE-TAU( J ) )*E( J ) 00286 DO 50 JR = 1, J - 1 00287 WORK( JP1+JR ) = -TAU( J )*E( J )*VP( JP1+JR ) 00288 50 CONTINUE 00289 END IF 00290 * 00291 IF( TAU( J ).NE.ZERO ) THEN 00292 VSAVE = VP( JP1+J ) 00293 VP( JP1+J ) = ONE 00294 CALL SSPMV( 'U', J, ONE, WORK, VP( JP1+1 ), 1, ZERO, 00295 $ WORK( LAP+1 ), 1 ) 00296 TEMP = -HALF*TAU( J )*SDOT( J, WORK( LAP+1 ), 1, 00297 $ VP( JP1+1 ), 1 ) 00298 CALL SAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ), 00299 $ 1 ) 00300 CALL SSPR2( 'U', J, -TAU( J ), VP( JP1+1 ), 1, 00301 $ WORK( LAP+1 ), 1, WORK ) 00302 VP( JP1+J ) = VSAVE 00303 END IF 00304 WORK( JP1+J+1 ) = D( J+1 ) 00305 60 CONTINUE 00306 END IF 00307 * 00308 DO 70 J = 1, LAP 00309 WORK( J ) = WORK( J ) - AP( J ) 00310 70 CONTINUE 00311 WNORM = SLANSP( '1', CUPLO, N, WORK, WORK( LAP+1 ) ) 00312 * 00313 ELSE IF( ITYPE.EQ.3 ) THEN 00314 * 00315 * ITYPE=3: error = U V' - I 00316 * 00317 IF( N.LT.2 ) 00318 $ RETURN 00319 CALL SLACPY( ' ', N, N, U, LDU, WORK, N ) 00320 CALL SOPMTR( 'R', CUPLO, 'T', N, N, VP, TAU, WORK, N, 00321 $ WORK( N**2+1 ), IINFO ) 00322 IF( IINFO.NE.0 ) THEN 00323 RESULT( 1 ) = TEN / ULP 00324 RETURN 00325 END IF 00326 * 00327 DO 80 J = 1, N 00328 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 00329 80 CONTINUE 00330 * 00331 WNORM = SLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ) 00332 END IF 00333 * 00334 IF( ANORM.GT.WNORM ) THEN 00335 RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP ) 00336 ELSE 00337 IF( ANORM.LT.ONE ) THEN 00338 RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP ) 00339 ELSE 00340 RESULT( 1 ) = MIN( WNORM / ANORM, REAL( N ) ) / ( N*ULP ) 00341 END IF 00342 END IF 00343 * 00344 * Do Test 2 00345 * 00346 * Compute UU' - I 00347 * 00348 IF( ITYPE.EQ.1 ) THEN 00349 CALL SGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK, 00350 $ N ) 00351 * 00352 DO 90 J = 1, N 00353 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 00354 90 CONTINUE 00355 * 00356 RESULT( 2 ) = MIN( SLANGE( '1', N, N, WORK, N, 00357 $ WORK( N**2+1 ) ), REAL( N ) ) / ( N*ULP ) 00358 END IF 00359 * 00360 RETURN 00361 * 00362 * End of SSPT21 00363 * 00364 END
1.7.3 
