![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief \b SPFTRI 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download SPFTRI + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spftri.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/spftri.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spftri.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SPFTRI( TRANSR, UPLO, N, A, INFO ) 00022 * 00023 * .. Scalar Arguments .. 00024 * CHARACTER TRANSR, UPLO 00025 * INTEGER INFO, N 00026 * .. Array Arguments .. 00027 * REAL A( 0: * ) 00028 * .. 00029 * 00030 * 00031 *> \par Purpose: 00032 * ============= 00033 *> 00034 *> \verbatim 00035 *> 00036 *> SPFTRI computes the inverse of a real (symmetric) positive definite 00037 *> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T 00038 *> computed by SPFTRF. 00039 *> \endverbatim 00040 * 00041 * Arguments: 00042 * ========== 00043 * 00044 *> \param[in] TRANSR 00045 *> \verbatim 00046 *> TRANSR is CHARACTER*1 00047 *> = 'N': The Normal TRANSR of RFP A is stored; 00048 *> = 'T': The Transpose TRANSR of RFP A is stored. 00049 *> \endverbatim 00050 *> 00051 *> \param[in] UPLO 00052 *> \verbatim 00053 *> UPLO is CHARACTER*1 00054 *> = 'U': Upper triangle of A is stored; 00055 *> = 'L': Lower triangle of A is stored. 00056 *> \endverbatim 00057 *> 00058 *> \param[in] N 00059 *> \verbatim 00060 *> N is INTEGER 00061 *> The order of the matrix A. N >= 0. 00062 *> \endverbatim 00063 *> 00064 *> \param[in,out] A 00065 *> \verbatim 00066 *> A is REAL array, dimension ( N*(N+1)/2 ) 00067 *> On entry, the symmetric matrix A in RFP format. RFP format is 00068 *> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' 00069 *> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is 00070 *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is 00071 *> the transpose of RFP A as defined when 00072 *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as 00073 *> follows: If UPLO = 'U' the RFP A contains the nt elements of 00074 *> upper packed A. If UPLO = 'L' the RFP A contains the elements 00075 *> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = 00076 *> 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N 00077 *> is odd. See the Note below for more details. 00078 *> 00079 *> On exit, the symmetric inverse of the original matrix, in the 00080 *> same storage format. 00081 *> \endverbatim 00082 *> 00083 *> \param[out] INFO 00084 *> \verbatim 00085 *> INFO is INTEGER 00086 *> = 0: successful exit 00087 *> < 0: if INFO = -i, the i-th argument had an illegal value 00088 *> > 0: if INFO = i, the (i,i) element of the factor U or L is 00089 *> zero, and the inverse could not be computed. 00090 *> \endverbatim 00091 * 00092 * Authors: 00093 * ======== 00094 * 00095 *> \author Univ. of Tennessee 00096 *> \author Univ. of California Berkeley 00097 *> \author Univ. of Colorado Denver 00098 *> \author NAG Ltd. 00099 * 00100 *> \date November 2011 00101 * 00102 *> \ingroup realOTHERcomputational 00103 * 00104 *> \par Further Details: 00105 * ===================== 00106 *> 00107 *> \verbatim 00108 *> 00109 *> We first consider Rectangular Full Packed (RFP) Format when N is 00110 *> even. We give an example where N = 6. 00111 *> 00112 *> AP is Upper AP is Lower 00113 *> 00114 *> 00 01 02 03 04 05 00 00115 *> 11 12 13 14 15 10 11 00116 *> 22 23 24 25 20 21 22 00117 *> 33 34 35 30 31 32 33 00118 *> 44 45 40 41 42 43 44 00119 *> 55 50 51 52 53 54 55 00120 *> 00121 *> 00122 *> Let TRANSR = 'N'. RFP holds AP as follows: 00123 *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last 00124 *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of 00125 *> the transpose of the first three columns of AP upper. 00126 *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first 00127 *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of 00128 *> the transpose of the last three columns of AP lower. 00129 *> This covers the case N even and TRANSR = 'N'. 00130 *> 00131 *> RFP A RFP A 00132 *> 00133 *> 03 04 05 33 43 53 00134 *> 13 14 15 00 44 54 00135 *> 23 24 25 10 11 55 00136 *> 33 34 35 20 21 22 00137 *> 00 44 45 30 31 32 00138 *> 01 11 55 40 41 42 00139 *> 02 12 22 50 51 52 00140 *> 00141 *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the 00142 *> transpose of RFP A above. One therefore gets: 00143 *> 00144 *> 00145 *> RFP A RFP A 00146 *> 00147 *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50 00148 *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51 00149 *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52 00150 *> 00151 *> 00152 *> We then consider Rectangular Full Packed (RFP) Format when N is 00153 *> odd. We give an example where N = 5. 00154 *> 00155 *> AP is Upper AP is Lower 00156 *> 00157 *> 00 01 02 03 04 00 00158 *> 11 12 13 14 10 11 00159 *> 22 23 24 20 21 22 00160 *> 33 34 30 31 32 33 00161 *> 44 40 41 42 43 44 00162 *> 00163 *> 00164 *> Let TRANSR = 'N'. RFP holds AP as follows: 00165 *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last 00166 *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of 00167 *> the transpose of the first two columns of AP upper. 00168 *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first 00169 *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of 00170 *> the transpose of the last two columns of AP lower. 00171 *> This covers the case N odd and TRANSR = 'N'. 00172 *> 00173 *> RFP A RFP A 00174 *> 00175 *> 02 03 04 00 33 43 00176 *> 12 13 14 10 11 44 00177 *> 22 23 24 20 21 22 00178 *> 00 33 34 30 31 32 00179 *> 01 11 44 40 41 42 00180 *> 00181 *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the 00182 *> transpose of RFP A above. One therefore gets: 00183 *> 00184 *> RFP A RFP A 00185 *> 00186 *> 02 12 22 00 01 00 10 20 30 40 50 00187 *> 03 13 23 33 11 33 11 21 31 41 51 00188 *> 04 14 24 34 44 43 44 22 32 42 52 00189 *> \endverbatim 00190 *> 00191 * ===================================================================== 00192 SUBROUTINE SPFTRI( TRANSR, UPLO, N, A, INFO ) 00193 * 00194 * -- LAPACK computational routine (version 3.4.0) -- 00195 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00196 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00197 * November 2011 00198 * 00199 * .. Scalar Arguments .. 00200 CHARACTER TRANSR, UPLO 00201 INTEGER INFO, N 00202 * .. Array Arguments .. 00203 REAL A( 0: * ) 00204 * .. 00205 * 00206 * ===================================================================== 00207 * 00208 * .. Parameters .. 00209 REAL ONE 00210 PARAMETER ( ONE = 1.0E+0 ) 00211 * .. 00212 * .. Local Scalars .. 00213 LOGICAL LOWER, NISODD, NORMALTRANSR 00214 INTEGER N1, N2, K 00215 * .. 00216 * .. External Functions .. 00217 LOGICAL LSAME 00218 EXTERNAL LSAME 00219 * .. 00220 * .. External Subroutines .. 00221 EXTERNAL XERBLA, STFTRI, SLAUUM, STRMM, SSYRK 00222 * .. 00223 * .. Intrinsic Functions .. 00224 INTRINSIC MOD 00225 * .. 00226 * .. Executable Statements .. 00227 * 00228 * Test the input parameters. 00229 * 00230 INFO = 0 00231 NORMALTRANSR = LSAME( TRANSR, 'N' ) 00232 LOWER = LSAME( UPLO, 'L' ) 00233 IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN 00234 INFO = -1 00235 ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN 00236 INFO = -2 00237 ELSE IF( N.LT.0 ) THEN 00238 INFO = -3 00239 END IF 00240 IF( INFO.NE.0 ) THEN 00241 CALL XERBLA( 'SPFTRI', -INFO ) 00242 RETURN 00243 END IF 00244 * 00245 * Quick return if possible 00246 * 00247 IF( N.EQ.0 ) 00248 $ RETURN 00249 * 00250 * Invert the triangular Cholesky factor U or L. 00251 * 00252 CALL STFTRI( TRANSR, UPLO, 'N', N, A, INFO ) 00253 IF( INFO.GT.0 ) 00254 $ RETURN 00255 * 00256 * If N is odd, set NISODD = .TRUE. 00257 * If N is even, set K = N/2 and NISODD = .FALSE. 00258 * 00259 IF( MOD( N, 2 ).EQ.0 ) THEN 00260 K = N / 2 00261 NISODD = .FALSE. 00262 ELSE 00263 NISODD = .TRUE. 00264 END IF 00265 * 00266 * Set N1 and N2 depending on LOWER 00267 * 00268 IF( LOWER ) THEN 00269 N2 = N / 2 00270 N1 = N - N2 00271 ELSE 00272 N1 = N / 2 00273 N2 = N - N1 00274 END IF 00275 * 00276 * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or 00277 * inv(L)^C*inv(L). There are eight cases. 00278 * 00279 IF( NISODD ) THEN 00280 * 00281 * N is odd 00282 * 00283 IF( NORMALTRANSR ) THEN 00284 * 00285 * N is odd and TRANSR = 'N' 00286 * 00287 IF( LOWER ) THEN 00288 * 00289 * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) ) 00290 * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0) 00291 * T1 -> a(0), T2 -> a(n), S -> a(N1) 00292 * 00293 CALL SLAUUM( 'L', N1, A( 0 ), N, INFO ) 00294 CALL SSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE, 00295 $ A( 0 ), N ) 00296 CALL STRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N, 00297 $ A( N1 ), N ) 00298 CALL SLAUUM( 'U', N2, A( N ), N, INFO ) 00299 * 00300 ELSE 00301 * 00302 * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1) 00303 * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0) 00304 * T1 -> a(N2), T2 -> a(N1), S -> a(0) 00305 * 00306 CALL SLAUUM( 'L', N1, A( N2 ), N, INFO ) 00307 CALL SSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE, 00308 $ A( N2 ), N ) 00309 CALL STRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N, 00310 $ A( 0 ), N ) 00311 CALL SLAUUM( 'U', N2, A( N1 ), N, INFO ) 00312 * 00313 END IF 00314 * 00315 ELSE 00316 * 00317 * N is odd and TRANSR = 'T' 00318 * 00319 IF( LOWER ) THEN 00320 * 00321 * SRPA for LOWER, TRANSPOSE, and N is odd 00322 * T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1) 00323 * 00324 CALL SLAUUM( 'U', N1, A( 0 ), N1, INFO ) 00325 CALL SSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE, 00326 $ A( 0 ), N1 ) 00327 CALL STRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1, 00328 $ A( N1*N1 ), N1 ) 00329 CALL SLAUUM( 'L', N2, A( 1 ), N1, INFO ) 00330 * 00331 ELSE 00332 * 00333 * SRPA for UPPER, TRANSPOSE, and N is odd 00334 * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0) 00335 * 00336 CALL SLAUUM( 'U', N1, A( N2*N2 ), N2, INFO ) 00337 CALL SSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE, 00338 $ A( N2*N2 ), N2 ) 00339 CALL STRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ), 00340 $ N2, A( 0 ), N2 ) 00341 CALL SLAUUM( 'L', N2, A( N1*N2 ), N2, INFO ) 00342 * 00343 END IF 00344 * 00345 END IF 00346 * 00347 ELSE 00348 * 00349 * N is even 00350 * 00351 IF( NORMALTRANSR ) THEN 00352 * 00353 * N is even and TRANSR = 'N' 00354 * 00355 IF( LOWER ) THEN 00356 * 00357 * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) ) 00358 * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0) 00359 * T1 -> a(1), T2 -> a(0), S -> a(k+1) 00360 * 00361 CALL SLAUUM( 'L', K, A( 1 ), N+1, INFO ) 00362 CALL SSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE, 00363 $ A( 1 ), N+1 ) 00364 CALL STRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1, 00365 $ A( K+1 ), N+1 ) 00366 CALL SLAUUM( 'U', K, A( 0 ), N+1, INFO ) 00367 * 00368 ELSE 00369 * 00370 * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) ) 00371 * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0) 00372 * T1 -> a(k+1), T2 -> a(k), S -> a(0) 00373 * 00374 CALL SLAUUM( 'L', K, A( K+1 ), N+1, INFO ) 00375 CALL SSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE, 00376 $ A( K+1 ), N+1 ) 00377 CALL STRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1, 00378 $ A( 0 ), N+1 ) 00379 CALL SLAUUM( 'U', K, A( K ), N+1, INFO ) 00380 * 00381 END IF 00382 * 00383 ELSE 00384 * 00385 * N is even and TRANSR = 'T' 00386 * 00387 IF( LOWER ) THEN 00388 * 00389 * SRPA for LOWER, TRANSPOSE, and N is even (see paper) 00390 * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1), 00391 * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k 00392 * 00393 CALL SLAUUM( 'U', K, A( K ), K, INFO ) 00394 CALL SSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE, 00395 $ A( K ), K ) 00396 CALL STRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K, 00397 $ A( K*( K+1 ) ), K ) 00398 CALL SLAUUM( 'L', K, A( 0 ), K, INFO ) 00399 * 00400 ELSE 00401 * 00402 * SRPA for UPPER, TRANSPOSE, and N is even (see paper) 00403 * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0), 00404 * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k 00405 * 00406 CALL SLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO ) 00407 CALL SSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE, 00408 $ A( K*( K+1 ) ), K ) 00409 CALL STRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K, 00410 $ A( 0 ), K ) 00411 CALL SLAUUM( 'L', K, A( K*K ), K, INFO ) 00412 * 00413 END IF 00414 * 00415 END IF 00416 * 00417 END IF 00418 * 00419 RETURN 00420 * 00421 * End of SPFTRI 00422 * 00423 END