LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zpftri.f
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00001 *> \brief \b ZPFTRI
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download ZPFTRI + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpftri.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          TRANSR, UPLO
00025 *       INTEGER            INFO, N
00026 *       .. Array Arguments ..
00027 *       COMPLEX*16         A( 0: * )
00028 *       ..
00029 *  
00030 *
00031 *> \par Purpose:
00032 *  =============
00033 *>
00034 *> \verbatim
00035 *>
00036 *> ZPFTRI computes the inverse of a complex Hermitian positive definite
00037 *> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
00038 *> computed by ZPFTRF.
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 *>          = 'C':  The Conjugate-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 COMPLEX*16 array, dimension ( N*(N+1)/2 );
00067 *>          On entry, the Hermitian 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 = 'C' then RFP is
00071 *>          the Conjugate-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 *>          'C'. 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 Hermitian 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 complex16OTHERcomputational
00103 *
00104 *> \par Further Details:
00105 *  =====================
00106 *>
00107 *> \verbatim
00108 *>
00109 *>  We first consider Standard Packed Format when N is even.
00110 *>  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 *>  conjugate-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 *>  conjugate-transpose of the last three columns of AP lower.
00129 *>  To denote conjugate we place -- above the element. This covers the
00130 *>  case N even and TRANSR = 'N'.
00131 *>
00132 *>         RFP A                   RFP A
00133 *>
00134 *>                                -- -- --
00135 *>        03 04 05                33 43 53
00136 *>                                   -- --
00137 *>        13 14 15                00 44 54
00138 *>                                      --
00139 *>        23 24 25                10 11 55
00140 *>
00141 *>        33 34 35                20 21 22
00142 *>        --
00143 *>        00 44 45                30 31 32
00144 *>        -- --
00145 *>        01 11 55                40 41 42
00146 *>        -- -- --
00147 *>        02 12 22                50 51 52
00148 *>
00149 *>  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
00150 *>  transpose of RFP A above. One therefore gets:
00151 *>
00152 *>
00153 *>           RFP A                   RFP A
00154 *>
00155 *>     -- -- -- --                -- -- -- -- -- --
00156 *>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
00157 *>     -- -- -- -- --                -- -- -- -- --
00158 *>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
00159 *>     -- -- -- -- -- --                -- -- -- --
00160 *>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
00161 *>
00162 *>
00163 *>  We next  consider Standard Packed Format when N is odd.
00164 *>  We give an example where N = 5.
00165 *>
00166 *>     AP is Upper                 AP is Lower
00167 *>
00168 *>   00 01 02 03 04              00
00169 *>      11 12 13 14              10 11
00170 *>         22 23 24              20 21 22
00171 *>            33 34              30 31 32 33
00172 *>               44              40 41 42 43 44
00173 *>
00174 *>
00175 *>  Let TRANSR = 'N'. RFP holds AP as follows:
00176 *>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
00177 *>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
00178 *>  conjugate-transpose of the first two   columns of AP upper.
00179 *>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
00180 *>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
00181 *>  conjugate-transpose of the last two   columns of AP lower.
00182 *>  To denote conjugate we place -- above the element. This covers the
00183 *>  case N odd  and TRANSR = 'N'.
00184 *>
00185 *>         RFP A                   RFP A
00186 *>
00187 *>                                   -- --
00188 *>        02 03 04                00 33 43
00189 *>                                      --
00190 *>        12 13 14                10 11 44
00191 *>
00192 *>        22 23 24                20 21 22
00193 *>        --
00194 *>        00 33 34                30 31 32
00195 *>        -- --
00196 *>        01 11 44                40 41 42
00197 *>
00198 *>  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
00199 *>  transpose of RFP A above. One therefore gets:
00200 *>
00201 *>
00202 *>           RFP A                   RFP A
00203 *>
00204 *>     -- -- --                   -- -- -- -- -- --
00205 *>     02 12 22 00 01             00 10 20 30 40 50
00206 *>     -- -- -- --                   -- -- -- -- --
00207 *>     03 13 23 33 11             33 11 21 31 41 51
00208 *>     -- -- -- -- --                   -- -- -- --
00209 *>     04 14 24 34 44             43 44 22 32 42 52
00210 *> \endverbatim
00211 *>
00212 *  =====================================================================
00213       SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
00214 *
00215 *  -- LAPACK computational routine (version 3.4.0) --
00216 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00217 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00218 *     November 2011
00219 *
00220 *     .. Scalar Arguments ..
00221       CHARACTER          TRANSR, UPLO
00222       INTEGER            INFO, N
00223 *     .. Array Arguments ..
00224       COMPLEX*16         A( 0: * )
00225 *     ..
00226 *
00227 *  =====================================================================
00228 *
00229 *     .. Parameters ..
00230       DOUBLE PRECISION   ONE
00231       COMPLEX*16         CONE
00232       PARAMETER          ( ONE = 1.D0, CONE = ( 1.D0, 0.D0 ) )
00233 *     ..
00234 *     .. Local Scalars ..
00235       LOGICAL            LOWER, NISODD, NORMALTRANSR
00236       INTEGER            N1, N2, K
00237 *     ..
00238 *     .. External Functions ..
00239       LOGICAL            LSAME
00240       EXTERNAL           LSAME
00241 *     ..
00242 *     .. External Subroutines ..
00243       EXTERNAL           XERBLA, ZTFTRI, ZLAUUM, ZTRMM, ZHERK
00244 *     ..
00245 *     .. Intrinsic Functions ..
00246       INTRINSIC          MOD
00247 *     ..
00248 *     .. Executable Statements ..
00249 *
00250 *     Test the input parameters.
00251 *
00252       INFO = 0
00253       NORMALTRANSR = LSAME( TRANSR, 'N' )
00254       LOWER = LSAME( UPLO, 'L' )
00255       IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
00256          INFO = -1
00257       ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
00258          INFO = -2
00259       ELSE IF( N.LT.0 ) THEN
00260          INFO = -3
00261       END IF
00262       IF( INFO.NE.0 ) THEN
00263          CALL XERBLA( 'ZPFTRI', -INFO )
00264          RETURN
00265       END IF
00266 *
00267 *     Quick return if possible
00268 *
00269       IF( N.EQ.0 )
00270      $   RETURN
00271 *
00272 *     Invert the triangular Cholesky factor U or L.
00273 *
00274       CALL ZTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
00275       IF( INFO.GT.0 )
00276      $   RETURN
00277 *
00278 *     If N is odd, set NISODD = .TRUE.
00279 *     If N is even, set K = N/2 and NISODD = .FALSE.
00280 *
00281       IF( MOD( N, 2 ).EQ.0 ) THEN
00282          K = N / 2
00283          NISODD = .FALSE.
00284       ELSE
00285          NISODD = .TRUE.
00286       END IF
00287 *
00288 *     Set N1 and N2 depending on LOWER
00289 *
00290       IF( LOWER ) THEN
00291          N2 = N / 2
00292          N1 = N - N2
00293       ELSE
00294          N1 = N / 2
00295          N2 = N - N1
00296       END IF
00297 *
00298 *     Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
00299 *     inv(L)^C*inv(L). There are eight cases.
00300 *
00301       IF( NISODD ) THEN
00302 *
00303 *        N is odd
00304 *
00305          IF( NORMALTRANSR ) THEN
00306 *
00307 *           N is odd and TRANSR = 'N'
00308 *
00309             IF( LOWER ) THEN
00310 *
00311 *              SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
00312 *              T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
00313 *              T1 -> a(0), T2 -> a(n), S -> a(N1)
00314 *
00315                CALL ZLAUUM( 'L', N1, A( 0 ), N, INFO )
00316                CALL ZHERK( 'L', 'C', N1, N2, ONE, A( N1 ), N, ONE,
00317      $                     A( 0 ), N )
00318                CALL ZTRMM( 'L', 'U', 'N', 'N', N2, N1, CONE, A( N ), N,
00319      $                     A( N1 ), N )
00320                CALL ZLAUUM( 'U', N2, A( N ), N, INFO )
00321 *
00322             ELSE
00323 *
00324 *              SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
00325 *              T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
00326 *              T1 -> a(N2), T2 -> a(N1), S -> a(0)
00327 *
00328                CALL ZLAUUM( 'L', N1, A( N2 ), N, INFO )
00329                CALL ZHERK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
00330      $                     A( N2 ), N )
00331                CALL ZTRMM( 'R', 'U', 'C', 'N', N1, N2, CONE, A( N1 ), N,
00332      $                     A( 0 ), N )
00333                CALL ZLAUUM( 'U', N2, A( N1 ), N, INFO )
00334 *
00335             END IF
00336 *
00337          ELSE
00338 *
00339 *           N is odd and TRANSR = 'C'
00340 *
00341             IF( LOWER ) THEN
00342 *
00343 *              SRPA for LOWER, TRANSPOSE, and N is odd
00344 *              T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
00345 *
00346                CALL ZLAUUM( 'U', N1, A( 0 ), N1, INFO )
00347                CALL ZHERK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
00348      $                     A( 0 ), N1 )
00349                CALL ZTRMM( 'R', 'L', 'N', 'N', N1, N2, CONE, A( 1 ), N1,
00350      $                     A( N1*N1 ), N1 )
00351                CALL ZLAUUM( 'L', N2, A( 1 ), N1, INFO )
00352 *
00353             ELSE
00354 *
00355 *              SRPA for UPPER, TRANSPOSE, and N is odd
00356 *              T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
00357 *
00358                CALL ZLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
00359                CALL ZHERK( 'U', 'C', N1, N2, ONE, A( 0 ), N2, ONE,
00360      $                     A( N2*N2 ), N2 )
00361                CALL ZTRMM( 'L', 'L', 'C', 'N', N2, N1, CONE, A( N1*N2 ),
00362      $                     N2, A( 0 ), N2 )
00363                CALL ZLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
00364 *
00365             END IF
00366 *
00367          END IF
00368 *
00369       ELSE
00370 *
00371 *        N is even
00372 *
00373          IF( NORMALTRANSR ) THEN
00374 *
00375 *           N is even and TRANSR = 'N'
00376 *
00377             IF( LOWER ) THEN
00378 *
00379 *              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
00380 *              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
00381 *              T1 -> a(1), T2 -> a(0), S -> a(k+1)
00382 *
00383                CALL ZLAUUM( 'L', K, A( 1 ), N+1, INFO )
00384                CALL ZHERK( 'L', 'C', K, K, ONE, A( K+1 ), N+1, ONE,
00385      $                     A( 1 ), N+1 )
00386                CALL ZTRMM( 'L', 'U', 'N', 'N', K, K, CONE, A( 0 ), N+1,
00387      $                     A( K+1 ), N+1 )
00388                CALL ZLAUUM( 'U', K, A( 0 ), N+1, INFO )
00389 *
00390             ELSE
00391 *
00392 *              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
00393 *              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
00394 *              T1 -> a(k+1), T2 -> a(k), S -> a(0)
00395 *
00396                CALL ZLAUUM( 'L', K, A( K+1 ), N+1, INFO )
00397                CALL ZHERK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
00398      $                     A( K+1 ), N+1 )
00399                CALL ZTRMM( 'R', 'U', 'C', 'N', K, K, CONE, A( K ), N+1,
00400      $                     A( 0 ), N+1 )
00401                CALL ZLAUUM( 'U', K, A( K ), N+1, INFO )
00402 *
00403             END IF
00404 *
00405          ELSE
00406 *
00407 *           N is even and TRANSR = 'C'
00408 *
00409             IF( LOWER ) THEN
00410 *
00411 *              SRPA for LOWER, TRANSPOSE, and N is even (see paper)
00412 *              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
00413 *              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
00414 *
00415                CALL ZLAUUM( 'U', K, A( K ), K, INFO )
00416                CALL ZHERK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
00417      $                     A( K ), K )
00418                CALL ZTRMM( 'R', 'L', 'N', 'N', K, K, CONE, A( 0 ), K,
00419      $                     A( K*( K+1 ) ), K )
00420                CALL ZLAUUM( 'L', K, A( 0 ), K, INFO )
00421 *
00422             ELSE
00423 *
00424 *              SRPA for UPPER, TRANSPOSE, and N is even (see paper)
00425 *              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0),
00426 *              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
00427 *
00428                CALL ZLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
00429                CALL ZHERK( 'U', 'C', K, K, ONE, A( 0 ), K, ONE,
00430      $                     A( K*( K+1 ) ), K )
00431                CALL ZTRMM( 'L', 'L', 'C', 'N', K, K, CONE, A( K*K ), K,
00432      $                     A( 0 ), K )
00433                CALL ZLAUUM( 'L', K, A( K*K ), K, INFO )
00434 *
00435             END IF
00436 *
00437          END IF
00438 *
00439       END IF
00440 *
00441       RETURN
00442 *
00443 *     End of ZPFTRI
00444 *
00445       END
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