LAPACK 3.3.1 Linear Algebra PACKage

dpftrf.f

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```00001       SUBROUTINE DPFTRF( TRANSR, UPLO, N, A, INFO )
00002 *
00003 *  -- LAPACK routine (version 3.3.1)                                    --
00004 *
00005 *  -- Contributed by Fred Gustavson of the IBM Watson Research Center --
00006 *  -- April 2011                                                      --
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       DOUBLE PRECISION   A( 0: * )
00018 *
00019 *  Purpose
00020 *  =======
00021 *
00022 *  DPFTRF computes the Cholesky factorization of a real symmetric
00023 *  positive definite matrix A.
00024 *
00025 *  The factorization has the form
00026 *     A = U**T * U,  if UPLO = 'U', or
00027 *     A = L  * L**T,  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 *          = 'T':  The 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) DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
00047 *          On entry, the symmetric 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 = 'T' then RFP is
00051 *          the 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 *          'T'. 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**T*U or RFP A = L*L**T.
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 Details
00070 *  ===============
00071 *
00072 *  We first consider Rectangular Full Packed (RFP) Format when N is
00073 *  even. 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 *  the 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 *  the transpose of the last three columns of AP lower.
00092 *  This covers the case N even and TRANSR = 'N'.
00093 *
00094 *         RFP A                   RFP A
00095 *
00096 *        03 04 05                33 43 53
00097 *        13 14 15                00 44 54
00098 *        23 24 25                10 11 55
00099 *        33 34 35                20 21 22
00100 *        00 44 45                30 31 32
00101 *        01 11 55                40 41 42
00102 *        02 12 22                50 51 52
00103 *
00104 *  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
00105 *  transpose of RFP A above. One therefore gets:
00106 *
00107 *
00108 *           RFP A                   RFP A
00109 *
00110 *     03 13 23 33 00 01 02    33 00 10 20 30 40 50
00111 *     04 14 24 34 44 11 12    43 44 11 21 31 41 51
00112 *     05 15 25 35 45 55 22    53 54 55 22 32 42 52
00113 *
00114 *
00115 *  We then consider Rectangular Full Packed (RFP) Format when N is
00116 *  odd. We give an example where N = 5.
00117 *
00118 *     AP is Upper                 AP is Lower
00119 *
00120 *   00 01 02 03 04              00
00121 *      11 12 13 14              10 11
00122 *         22 23 24              20 21 22
00123 *            33 34              30 31 32 33
00124 *               44              40 41 42 43 44
00125 *
00126 *
00127 *  Let TRANSR = 'N'. RFP holds AP as follows:
00128 *  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
00129 *  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
00130 *  the transpose of the first two columns of AP upper.
00131 *  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
00132 *  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
00133 *  the transpose of the last two columns of AP lower.
00134 *  This covers the case N odd and TRANSR = 'N'.
00135 *
00136 *         RFP A                   RFP A
00137 *
00138 *        02 03 04                00 33 43
00139 *        12 13 14                10 11 44
00140 *        22 23 24                20 21 22
00141 *        00 33 34                30 31 32
00142 *        01 11 44                40 41 42
00143 *
00144 *  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
00145 *  transpose of RFP A above. One therefore gets:
00146 *
00147 *           RFP A                   RFP A
00148 *
00149 *     02 12 22 00 01             00 10 20 30 40 50
00150 *     03 13 23 33 11             33 11 21 31 41 51
00151 *     04 14 24 34 44             43 44 22 32 42 52
00152 *
00153 *  =====================================================================
00154 *
00155 *     .. Parameters ..
00156       DOUBLE PRECISION   ONE
00157       PARAMETER          ( ONE = 1.0D+0 )
00158 *     ..
00159 *     .. Local Scalars ..
00160       LOGICAL            LOWER, NISODD, NORMALTRANSR
00161       INTEGER            N1, N2, K
00162 *     ..
00163 *     .. External Functions ..
00164       LOGICAL            LSAME
00165       EXTERNAL           LSAME
00166 *     ..
00167 *     .. External Subroutines ..
00168       EXTERNAL           XERBLA, DSYRK, DPOTRF, DTRSM
00169 *     ..
00170 *     .. Intrinsic Functions ..
00171       INTRINSIC          MOD
00172 *     ..
00173 *     .. Executable Statements ..
00174 *
00175 *     Test the input parameters.
00176 *
00177       INFO = 0
00178       NORMALTRANSR = LSAME( TRANSR, 'N' )
00179       LOWER = LSAME( UPLO, 'L' )
00180       IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
00181          INFO = -1
00182       ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
00183          INFO = -2
00184       ELSE IF( N.LT.0 ) THEN
00185          INFO = -3
00186       END IF
00187       IF( INFO.NE.0 ) THEN
00188          CALL XERBLA( 'DPFTRF', -INFO )
00189          RETURN
00190       END IF
00191 *
00192 *     Quick return if possible
00193 *
00194       IF( N.EQ.0 )
00195      \$   RETURN
00196 *
00197 *     If N is odd, set NISODD = .TRUE.
00198 *     If N is even, set K = N/2 and NISODD = .FALSE.
00199 *
00200       IF( MOD( N, 2 ).EQ.0 ) THEN
00201          K = N / 2
00202          NISODD = .FALSE.
00203       ELSE
00204          NISODD = .TRUE.
00205       END IF
00206 *
00207 *     Set N1 and N2 depending on LOWER
00208 *
00209       IF( LOWER ) THEN
00210          N2 = N / 2
00211          N1 = N - N2
00212       ELSE
00213          N1 = N / 2
00214          N2 = N - N1
00215       END IF
00216 *
00217 *     start execution: there are eight cases
00218 *
00219       IF( NISODD ) THEN
00220 *
00221 *        N is odd
00222 *
00223          IF( NORMALTRANSR ) THEN
00224 *
00225 *           N is odd and TRANSR = 'N'
00226 *
00227             IF( LOWER ) THEN
00228 *
00229 *             SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
00230 *             T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
00231 *             T1 -> a(0), T2 -> a(n), S -> a(n1)
00232 *
00233                CALL DPOTRF( 'L', N1, A( 0 ), N, INFO )
00234                IF( INFO.GT.0 )
00235      \$            RETURN
00236                CALL DTRSM( 'R', 'L', 'T', 'N', N2, N1, ONE, A( 0 ), N,
00237      \$                     A( N1 ), N )
00238                CALL DSYRK( 'U', 'N', N2, N1, -ONE, A( N1 ), N, ONE,
00239      \$                     A( N ), N )
00240                CALL DPOTRF( 'U', N2, A( N ), N, INFO )
00241                IF( INFO.GT.0 )
00242      \$            INFO = INFO + N1
00243 *
00244             ELSE
00245 *
00246 *             SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
00247 *             T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
00248 *             T1 -> a(n2), T2 -> a(n1), S -> a(0)
00249 *
00250                CALL DPOTRF( 'L', N1, A( N2 ), N, INFO )
00251                IF( INFO.GT.0 )
00252      \$            RETURN
00253                CALL DTRSM( 'L', 'L', 'N', 'N', N1, N2, ONE, A( N2 ), N,
00254      \$                     A( 0 ), N )
00255                CALL DSYRK( 'U', 'T', N2, N1, -ONE, A( 0 ), N, ONE,
00256      \$                     A( N1 ), N )
00257                CALL DPOTRF( 'U', N2, A( N1 ), N, INFO )
00258                IF( INFO.GT.0 )
00259      \$            INFO = INFO + N1
00260 *
00261             END IF
00262 *
00263          ELSE
00264 *
00265 *           N is odd and TRANSR = 'T'
00266 *
00267             IF( LOWER ) THEN
00268 *
00269 *              SRPA for LOWER, TRANSPOSE and N is odd
00270 *              T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
00271 *              T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
00272 *
00273                CALL DPOTRF( 'U', N1, A( 0 ), N1, INFO )
00274                IF( INFO.GT.0 )
00275      \$            RETURN
00276                CALL DTRSM( 'L', 'U', 'T', 'N', N1, N2, ONE, A( 0 ), N1,
00277      \$                     A( N1*N1 ), N1 )
00278                CALL DSYRK( 'L', 'T', N2, N1, -ONE, A( N1*N1 ), N1, ONE,
00279      \$                     A( 1 ), N1 )
00280                CALL DPOTRF( 'L', N2, A( 1 ), N1, INFO )
00281                IF( INFO.GT.0 )
00282      \$            INFO = INFO + N1
00283 *
00284             ELSE
00285 *
00286 *              SRPA for UPPER, TRANSPOSE and N is odd
00287 *              T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
00288 *              T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
00289 *
00290                CALL DPOTRF( 'U', N1, A( N2*N2 ), N2, INFO )
00291                IF( INFO.GT.0 )
00292      \$            RETURN
00293                CALL DTRSM( 'R', 'U', 'N', 'N', N2, N1, ONE, A( N2*N2 ),
00294      \$                     N2, A( 0 ), N2 )
00295                CALL DSYRK( 'L', 'N', N2, N1, -ONE, A( 0 ), N2, ONE,
00296      \$                     A( N1*N2 ), N2 )
00297                CALL DPOTRF( 'L', N2, A( N1*N2 ), N2, INFO )
00298                IF( INFO.GT.0 )
00299      \$            INFO = INFO + N1
00300 *
00301             END IF
00302 *
00303          END IF
00304 *
00305       ELSE
00306 *
00307 *        N is even
00308 *
00309          IF( NORMALTRANSR ) THEN
00310 *
00311 *           N is even and TRANSR = 'N'
00312 *
00313             IF( LOWER ) THEN
00314 *
00315 *              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
00316 *              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
00317 *              T1 -> a(1), T2 -> a(0), S -> a(k+1)
00318 *
00319                CALL DPOTRF( 'L', K, A( 1 ), N+1, INFO )
00320                IF( INFO.GT.0 )
00321      \$            RETURN
00322                CALL DTRSM( 'R', 'L', 'T', 'N', K, K, ONE, A( 1 ), N+1,
00323      \$                     A( K+1 ), N+1 )
00324                CALL DSYRK( 'U', 'N', K, K, -ONE, A( K+1 ), N+1, ONE,
00325      \$                     A( 0 ), N+1 )
00326                CALL DPOTRF( 'U', K, A( 0 ), N+1, INFO )
00327                IF( INFO.GT.0 )
00328      \$            INFO = INFO + K
00329 *
00330             ELSE
00331 *
00332 *              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
00333 *              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
00334 *              T1 -> a(k+1), T2 -> a(k), S -> a(0)
00335 *
00336                CALL DPOTRF( 'L', K, A( K+1 ), N+1, INFO )
00337                IF( INFO.GT.0 )
00338      \$            RETURN
00339                CALL DTRSM( 'L', 'L', 'N', 'N', K, K, ONE, A( K+1 ),
00340      \$                     N+1, A( 0 ), N+1 )
00341                CALL DSYRK( 'U', 'T', K, K, -ONE, A( 0 ), N+1, ONE,
00342      \$                     A( K ), N+1 )
00343                CALL DPOTRF( 'U', K, A( K ), N+1, INFO )
00344                IF( INFO.GT.0 )
00345      \$            INFO = INFO + K
00346 *
00347             END IF
00348 *
00349          ELSE
00350 *
00351 *           N is even and TRANSR = 'T'
00352 *
00353             IF( LOWER ) THEN
00354 *
00355 *              SRPA for LOWER, TRANSPOSE and N is even (see paper)
00356 *              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
00357 *              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
00358 *
00359                CALL DPOTRF( 'U', K, A( 0+K ), K, INFO )
00360                IF( INFO.GT.0 )
00361      \$            RETURN
00362                CALL DTRSM( 'L', 'U', 'T', 'N', K, K, ONE, A( K ), N1,
00363      \$                     A( K*( K+1 ) ), K )
00364                CALL DSYRK( 'L', 'T', K, K, -ONE, A( K*( K+1 ) ), K, ONE,
00365      \$                     A( 0 ), K )
00366                CALL DPOTRF( 'L', K, A( 0 ), K, INFO )
00367                IF( INFO.GT.0 )
00368      \$            INFO = INFO + K
00369 *
00370             ELSE
00371 *
00372 *              SRPA for UPPER, TRANSPOSE and N is even (see paper)
00373 *              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0)
00374 *              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
00375 *
00376                CALL DPOTRF( 'U', K, A( K*( K+1 ) ), K, INFO )
00377                IF( INFO.GT.0 )
00378      \$            RETURN
00379                CALL DTRSM( 'R', 'U', 'N', 'N', K, K, ONE,
00380      \$                     A( K*( K+1 ) ), K, A( 0 ), K )
00381                CALL DSYRK( 'L', 'N', K, K, -ONE, A( 0 ), K, ONE,
00382      \$                     A( K*K ), K )
00383                CALL DPOTRF( 'L', K, A( K*K ), K, INFO )
00384                IF( INFO.GT.0 )
00385      \$            INFO = INFO + K
00386 *
00387             END IF
00388 *
00389          END IF
00390 *
00391       END IF
00392 *
00393       RETURN
00394 *
00395 *     End of DPFTRF
00396 *
00397       END
```