LAPACK 3.3.0

zpftrf.f

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00001       SUBROUTINE ZPFTRF( TRANSR, UPLO, N, A, INFO )
00002 *
00003 *  -- LAPACK routine (version 3.3.0)                                    --
00004 *
00005 *  -- Contributed by Fred Gustavson of the IBM Watson Research Center --
00006 *     November 2010                                                   --
00007 *
00008 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00009 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00010 *
00011 *     ..
00012 *     .. Scalar Arguments ..
00013       CHARACTER          TRANSR, UPLO
00014       INTEGER            N, INFO
00015 *     ..
00016 *     .. Array Arguments ..
00017       COMPLEX*16         A( 0: * )
00018 *
00019 *  Purpose
00020 *  =======
00021 *
00022 *  ZPFTRF computes the Cholesky factorization of a complex Hermitian
00023 *  positive definite matrix A.
00024 *
00025 *  The factorization has the form
00026 *     A = U**H * U,  if UPLO = 'U', or
00027 *     A = L  * L**H,  if UPLO = 'L',
00028 *  where U is an upper triangular matrix and L is lower triangular.
00029 *
00030 *  This is the block version of the algorithm, calling Level 3 BLAS.
00031 *
00032 *  Arguments
00033 *  =========
00034 *
00035 *  TRANSR    (input) CHARACTER*1
00036 *          = 'N':  The Normal TRANSR of RFP A is stored;
00037 *          = 'C':  The Conjugate-transpose TRANSR of RFP A is stored.
00038 *
00039 *  UPLO    (input) CHARACTER*1
00040 *          = 'U':  Upper triangle of RFP A is stored;
00041 *          = 'L':  Lower triangle of RFP A is stored.
00042 *
00043 *  N       (input) INTEGER
00044 *          The order of the matrix A.  N >= 0.
00045 *
00046 *  A       (input/output) COMPLEX array, dimension ( N*(N+1)/2 );
00047 *          On entry, the Hermitian matrix A in RFP format. RFP format is
00048 *          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
00049 *          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
00050 *          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
00051 *          the Conjugate-transpose of RFP A as defined when
00052 *          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
00053 *          follows: If UPLO = 'U' the RFP A contains the nt elements of
00054 *          upper packed A. If UPLO = 'L' the RFP A contains the elements
00055 *          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
00056 *          'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
00057 *          is odd. See the Note below for more details.
00058 *
00059 *          On exit, if INFO = 0, the factor U or L from the Cholesky
00060 *          factorization RFP A = U**H*U or RFP A = L*L**H.
00061 *
00062 *  INFO    (output) INTEGER
00063 *          = 0:  successful exit
00064 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00065 *          > 0:  if INFO = i, the leading minor of order i is not
00066 *                positive definite, and the factorization could not be
00067 *                completed.
00068 *
00069 *  Further Notes on RFP Format:
00070 *  ============================
00071 *
00072 *  We first consider Standard Packed Format when N is even.
00073 *  We give an example where N = 6.
00074 *
00075 *     AP is Upper             AP is Lower
00076 *
00077 *   00 01 02 03 04 05       00
00078 *      11 12 13 14 15       10 11
00079 *         22 23 24 25       20 21 22
00080 *            33 34 35       30 31 32 33
00081 *               44 45       40 41 42 43 44
00082 *                  55       50 51 52 53 54 55
00083 *
00084 *
00085 *  Let TRANSR = 'N'. RFP holds AP as follows:
00086 *  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
00087 *  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
00088 *  conjugate-transpose of the first three columns of AP upper.
00089 *  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
00090 *  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
00091 *  conjugate-transpose of the last three columns of AP lower.
00092 *  To denote conjugate we place -- above the element. This covers the
00093 *  case N even and TRANSR = 'N'.
00094 *
00095 *         RFP A                   RFP A
00096 *
00097 *                                -- -- --
00098 *        03 04 05                33 43 53
00099 *                                   -- --
00100 *        13 14 15                00 44 54
00101 *                                      --
00102 *        23 24 25                10 11 55
00103 *
00104 *        33 34 35                20 21 22
00105 *        --
00106 *        00 44 45                30 31 32
00107 *        -- --
00108 *        01 11 55                40 41 42
00109 *        -- -- --
00110 *        02 12 22                50 51 52
00111 *
00112 *  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
00113 *  transpose of RFP A above. One therefore gets:
00114 *
00115 *
00116 *           RFP A                   RFP A
00117 *
00118 *     -- -- -- --                -- -- -- -- -- --
00119 *     03 13 23 33 00 01 02    33 00 10 20 30 40 50
00120 *     -- -- -- -- --                -- -- -- -- --
00121 *     04 14 24 34 44 11 12    43 44 11 21 31 41 51
00122 *     -- -- -- -- -- --                -- -- -- --
00123 *     05 15 25 35 45 55 22    53 54 55 22 32 42 52
00124 *
00125 *
00126 *  We next  consider Standard Packed Format when N is odd.
00127 *  We give an example where N = 5.
00128 *
00129 *     AP is Upper                 AP is Lower
00130 *
00131 *   00 01 02 03 04              00
00132 *      11 12 13 14              10 11
00133 *         22 23 24              20 21 22
00134 *            33 34              30 31 32 33
00135 *               44              40 41 42 43 44
00136 *
00137 *
00138 *  Let TRANSR = 'N'. RFP holds AP as follows:
00139 *  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
00140 *  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
00141 *  conjugate-transpose of the first two   columns of AP upper.
00142 *  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
00143 *  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
00144 *  conjugate-transpose of the last two   columns of AP lower.
00145 *  To denote conjugate we place -- above the element. This covers the
00146 *  case N odd  and TRANSR = 'N'.
00147 *
00148 *         RFP A                   RFP A
00149 *
00150 *                                   -- --
00151 *        02 03 04                00 33 43
00152 *                                      --
00153 *        12 13 14                10 11 44
00154 *
00155 *        22 23 24                20 21 22
00156 *        --
00157 *        00 33 34                30 31 32
00158 *        -- --
00159 *        01 11 44                40 41 42
00160 *
00161 *  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
00162 *  transpose of RFP A above. One therefore gets:
00163 *
00164 *
00165 *           RFP A                   RFP A
00166 *
00167 *     -- -- --                   -- -- -- -- -- --
00168 *     02 12 22 00 01             00 10 20 30 40 50
00169 *     -- -- -- --                   -- -- -- -- --
00170 *     03 13 23 33 11             33 11 21 31 41 51
00171 *     -- -- -- -- --                   -- -- -- --
00172 *     04 14 24 34 44             43 44 22 32 42 52
00173 *
00174 *  =====================================================================
00175 *
00176 *     .. Parameters ..
00177       DOUBLE PRECISION   ONE
00178       COMPLEX*16         CONE
00179       PARAMETER          ( ONE = 1.0D+0, CONE = ( 1.0D+0, 0.0D+0 ) )
00180 *     ..
00181 *     .. Local Scalars ..
00182       LOGICAL            LOWER, NISODD, NORMALTRANSR
00183       INTEGER            N1, N2, K
00184 *     ..
00185 *     .. External Functions ..
00186       LOGICAL            LSAME
00187       EXTERNAL           LSAME
00188 *     ..
00189 *     .. External Subroutines ..
00190       EXTERNAL           XERBLA, ZHERK, ZPOTRF, ZTRSM
00191 *     ..
00192 *     .. Intrinsic Functions ..
00193       INTRINSIC          MOD
00194 *     ..
00195 *     .. Executable Statements ..
00196 *
00197 *     Test the input parameters.
00198 *
00199       INFO = 0
00200       NORMALTRANSR = LSAME( TRANSR, 'N' )
00201       LOWER = LSAME( UPLO, 'L' )
00202       IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
00203          INFO = -1
00204       ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
00205          INFO = -2
00206       ELSE IF( N.LT.0 ) THEN
00207          INFO = -3
00208       END IF
00209       IF( INFO.NE.0 ) THEN
00210          CALL XERBLA( 'ZPFTRF', -INFO )
00211          RETURN
00212       END IF
00213 *
00214 *     Quick return if possible
00215 *
00216       IF( N.EQ.0 )
00217      +   RETURN
00218 *
00219 *     If N is odd, set NISODD = .TRUE.
00220 *     If N is even, set K = N/2 and NISODD = .FALSE.
00221 *
00222       IF( MOD( N, 2 ).EQ.0 ) THEN
00223          K = N / 2
00224          NISODD = .FALSE.
00225       ELSE
00226          NISODD = .TRUE.
00227       END IF
00228 *
00229 *     Set N1 and N2 depending on LOWER
00230 *
00231       IF( LOWER ) THEN
00232          N2 = N / 2
00233          N1 = N - N2
00234       ELSE
00235          N1 = N / 2
00236          N2 = N - N1
00237       END IF
00238 *
00239 *     start execution: there are eight cases
00240 *
00241       IF( NISODD ) THEN
00242 *
00243 *        N is odd
00244 *
00245          IF( NORMALTRANSR ) THEN
00246 *
00247 *           N is odd and TRANSR = 'N'
00248 *
00249             IF( LOWER ) THEN
00250 *
00251 *             SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
00252 *             T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
00253 *             T1 -> a(0), T2 -> a(n), S -> a(n1)
00254 *
00255                CALL ZPOTRF( 'L', N1, A( 0 ), N, INFO )
00256                IF( INFO.GT.0 )
00257      +            RETURN
00258                CALL ZTRSM( 'R', 'L', 'C', 'N', N2, N1, CONE, A( 0 ), N,
00259      +                     A( N1 ), N )
00260                CALL ZHERK( 'U', 'N', N2, N1, -ONE, A( N1 ), N, ONE,
00261      +                     A( N ), N )
00262                CALL ZPOTRF( 'U', N2, A( N ), N, INFO )
00263                IF( INFO.GT.0 )
00264      +            INFO = INFO + N1
00265 *
00266             ELSE
00267 *
00268 *             SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
00269 *             T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
00270 *             T1 -> a(n2), T2 -> a(n1), S -> a(0)
00271 *
00272                CALL ZPOTRF( 'L', N1, A( N2 ), N, INFO )
00273                IF( INFO.GT.0 )
00274      +            RETURN
00275                CALL ZTRSM( 'L', 'L', 'N', 'N', N1, N2, CONE, A( N2 ), N,
00276      +                     A( 0 ), N )
00277                CALL ZHERK( 'U', 'C', N2, N1, -ONE, A( 0 ), N, ONE,
00278      +                     A( N1 ), N )
00279                CALL ZPOTRF( 'U', N2, A( N1 ), N, INFO )
00280                IF( INFO.GT.0 )
00281      +            INFO = INFO + N1
00282 *
00283             END IF
00284 *
00285          ELSE
00286 *
00287 *           N is odd and TRANSR = 'C'
00288 *
00289             IF( LOWER ) THEN
00290 *
00291 *              SRPA for LOWER, TRANSPOSE and N is odd
00292 *              T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
00293 *              T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
00294 *
00295                CALL ZPOTRF( 'U', N1, A( 0 ), N1, INFO )
00296                IF( INFO.GT.0 )
00297      +            RETURN
00298                CALL ZTRSM( 'L', 'U', 'C', 'N', N1, N2, CONE, A( 0 ), N1,
00299      +                     A( N1*N1 ), N1 )
00300                CALL ZHERK( 'L', 'C', N2, N1, -ONE, A( N1*N1 ), N1, ONE,
00301      +                     A( 1 ), N1 )
00302                CALL ZPOTRF( 'L', N2, A( 1 ), N1, INFO )
00303                IF( INFO.GT.0 )
00304      +            INFO = INFO + N1
00305 *
00306             ELSE
00307 *
00308 *              SRPA for UPPER, TRANSPOSE and N is odd
00309 *              T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
00310 *              T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
00311 *
00312                CALL ZPOTRF( 'U', N1, A( N2*N2 ), N2, INFO )
00313                IF( INFO.GT.0 )
00314      +            RETURN
00315                CALL ZTRSM( 'R', 'U', 'N', 'N', N2, N1, CONE, A( N2*N2 ),
00316      +                     N2, A( 0 ), N2 )
00317                CALL ZHERK( 'L', 'N', N2, N1, -ONE, A( 0 ), N2, ONE,
00318      +                     A( N1*N2 ), N2 )
00319                CALL ZPOTRF( 'L', N2, A( N1*N2 ), N2, INFO )
00320                IF( INFO.GT.0 )
00321      +            INFO = INFO + N1
00322 *
00323             END IF
00324 *
00325          END IF
00326 *
00327       ELSE
00328 *
00329 *        N is even
00330 *
00331          IF( NORMALTRANSR ) THEN
00332 *
00333 *           N is even and TRANSR = 'N'
00334 *
00335             IF( LOWER ) THEN
00336 *
00337 *              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
00338 *              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
00339 *              T1 -> a(1), T2 -> a(0), S -> a(k+1)
00340 *
00341                CALL ZPOTRF( 'L', K, A( 1 ), N+1, INFO )
00342                IF( INFO.GT.0 )
00343      +            RETURN
00344                CALL ZTRSM( 'R', 'L', 'C', 'N', K, K, CONE, A( 1 ), N+1,
00345      +                     A( K+1 ), N+1 )
00346                CALL ZHERK( 'U', 'N', K, K, -ONE, A( K+1 ), N+1, ONE,
00347      +                     A( 0 ), N+1 )
00348                CALL ZPOTRF( 'U', K, A( 0 ), N+1, INFO )
00349                IF( INFO.GT.0 )
00350      +            INFO = INFO + K
00351 *
00352             ELSE
00353 *
00354 *              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
00355 *              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
00356 *              T1 -> a(k+1), T2 -> a(k), S -> a(0)
00357 *
00358                CALL ZPOTRF( 'L', K, A( K+1 ), N+1, INFO )
00359                IF( INFO.GT.0 )
00360      +            RETURN
00361                CALL ZTRSM( 'L', 'L', 'N', 'N', K, K, CONE, A( K+1 ),
00362      +                     N+1, A( 0 ), N+1 )
00363                CALL ZHERK( 'U', 'C', K, K, -ONE, A( 0 ), N+1, ONE,
00364      +                     A( K ), N+1 )
00365                CALL ZPOTRF( 'U', K, A( K ), N+1, INFO )
00366                IF( INFO.GT.0 )
00367      +            INFO = INFO + K
00368 *
00369             END IF
00370 *
00371          ELSE
00372 *
00373 *           N is even and TRANSR = 'C'
00374 *
00375             IF( LOWER ) THEN
00376 *
00377 *              SRPA for LOWER, TRANSPOSE and N is even (see paper)
00378 *              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
00379 *              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
00380 *
00381                CALL ZPOTRF( 'U', K, A( 0+K ), K, INFO )
00382                IF( INFO.GT.0 )
00383      +            RETURN
00384                CALL ZTRSM( 'L', 'U', 'C', 'N', K, K, CONE, A( K ), N1,
00385      +                     A( K*( K+1 ) ), K )
00386                CALL ZHERK( 'L', 'C', K, K, -ONE, A( K*( K+1 ) ), K, ONE,
00387      +                     A( 0 ), K )
00388                CALL ZPOTRF( 'L', K, A( 0 ), K, INFO )
00389                IF( INFO.GT.0 )
00390      +            INFO = INFO + K
00391 *
00392             ELSE
00393 *
00394 *              SRPA for UPPER, TRANSPOSE and N is even (see paper)
00395 *              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0)
00396 *              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
00397 *
00398                CALL ZPOTRF( 'U', K, A( K*( K+1 ) ), K, INFO )
00399                IF( INFO.GT.0 )
00400      +            RETURN
00401                CALL ZTRSM( 'R', 'U', 'N', 'N', K, K, CONE,
00402      +                     A( K*( K+1 ) ), K, A( 0 ), K )
00403                CALL ZHERK( 'L', 'N', K, K, -ONE, A( 0 ), K, ONE,
00404      +                     A( K*K ), K )
00405                CALL ZPOTRF( 'L', K, A( K*K ), K, INFO )
00406                IF( INFO.GT.0 )
00407      +            INFO = INFO + K
00408 *
00409             END IF
00410 *
00411          END IF
00412 *
00413       END IF
00414 *
00415       RETURN
00416 *
00417 *     End of ZPFTRF
00418 *
00419       END
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