LAPACK 3.3.0

spftri.f

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