LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
ssytf2.f
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00001 *> \brief \b SSYTF2
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SSYTF2 + 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/ssytf2.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SSYTF2( UPLO, N, A, LDA, IPIV, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          UPLO
00025 *       INTEGER            INFO, LDA, N
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       INTEGER            IPIV( * )
00029 *       REAL               A( LDA, * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *> SSYTF2 computes the factorization of a real symmetric matrix A using
00039 *> the Bunch-Kaufman diagonal pivoting method:
00040 *>
00041 *>    A = U*D*U**T  or  A = L*D*L**T
00042 *>
00043 *> where U (or L) is a product of permutation and unit upper (lower)
00044 *> triangular matrices, U**T is the transpose of U, and D is symmetric and
00045 *> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
00046 *>
00047 *> This is the unblocked version of the algorithm, calling Level 2 BLAS.
00048 *> \endverbatim
00049 *
00050 *  Arguments:
00051 *  ==========
00052 *
00053 *> \param[in] UPLO
00054 *> \verbatim
00055 *>          UPLO is CHARACTER*1
00056 *>          Specifies whether the upper or lower triangular part of the
00057 *>          symmetric matrix A is stored:
00058 *>          = 'U':  Upper triangular
00059 *>          = 'L':  Lower triangular
00060 *> \endverbatim
00061 *>
00062 *> \param[in] N
00063 *> \verbatim
00064 *>          N is INTEGER
00065 *>          The order of the matrix A.  N >= 0.
00066 *> \endverbatim
00067 *>
00068 *> \param[in,out] A
00069 *> \verbatim
00070 *>          A is REAL array, dimension (LDA,N)
00071 *>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
00072 *>          n-by-n upper triangular part of A contains the upper
00073 *>          triangular part of the matrix A, and the strictly lower
00074 *>          triangular part of A is not referenced.  If UPLO = 'L', the
00075 *>          leading n-by-n lower triangular part of A contains the lower
00076 *>          triangular part of the matrix A, and the strictly upper
00077 *>          triangular part of A is not referenced.
00078 *>
00079 *>          On exit, the block diagonal matrix D and the multipliers used
00080 *>          to obtain the factor U or L (see below for further details).
00081 *> \endverbatim
00082 *>
00083 *> \param[in] LDA
00084 *> \verbatim
00085 *>          LDA is INTEGER
00086 *>          The leading dimension of the array A.  LDA >= max(1,N).
00087 *> \endverbatim
00088 *>
00089 *> \param[out] IPIV
00090 *> \verbatim
00091 *>          IPIV is INTEGER array, dimension (N)
00092 *>          Details of the interchanges and the block structure of D.
00093 *>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
00094 *>          interchanged and D(k,k) is a 1-by-1 diagonal block.
00095 *>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
00096 *>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
00097 *>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
00098 *>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
00099 *>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
00100 *> \endverbatim
00101 *>
00102 *> \param[out] INFO
00103 *> \verbatim
00104 *>          INFO is INTEGER
00105 *>          = 0: successful exit
00106 *>          < 0: if INFO = -k, the k-th argument had an illegal value
00107 *>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
00108 *>               has been completed, but the block diagonal matrix D is
00109 *>               exactly singular, and division by zero will occur if it
00110 *>               is used to solve a system of equations.
00111 *> \endverbatim
00112 *
00113 *  Authors:
00114 *  ========
00115 *
00116 *> \author Univ. of Tennessee 
00117 *> \author Univ. of California Berkeley 
00118 *> \author Univ. of Colorado Denver 
00119 *> \author NAG Ltd. 
00120 *
00121 *> \date November 2011
00122 *
00123 *> \ingroup realSYcomputational
00124 *
00125 *> \par Further Details:
00126 *  =====================
00127 *>
00128 *> \verbatim
00129 *>
00130 *>  If UPLO = 'U', then A = U*D*U**T, where
00131 *>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
00132 *>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
00133 *>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00134 *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00135 *>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
00136 *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00137 *>
00138 *>             (   I    v    0   )   k-s
00139 *>     U(k) =  (   0    I    0   )   s
00140 *>             (   0    0    I   )   n-k
00141 *>                k-s   s   n-k
00142 *>
00143 *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
00144 *>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
00145 *>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
00146 *>
00147 *>  If UPLO = 'L', then A = L*D*L**T, where
00148 *>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
00149 *>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
00150 *>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00151 *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00152 *>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
00153 *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00154 *>
00155 *>             (   I    0     0   )  k-1
00156 *>     L(k) =  (   0    I     0   )  s
00157 *>             (   0    v     I   )  n-k-s+1
00158 *>                k-1   s  n-k-s+1
00159 *>
00160 *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
00161 *>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
00162 *>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
00163 *> \endverbatim
00164 *
00165 *> \par Contributors:
00166 *  ==================
00167 *>
00168 *> \verbatim
00169 *>
00170 *>  09-29-06 - patch from
00171 *>    Bobby Cheng, MathWorks
00172 *>
00173 *>    Replace l.204 and l.372
00174 *>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
00175 *>    by
00176 *>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
00177 *>
00178 *>  01-01-96 - Based on modifications by
00179 *>    J. Lewis, Boeing Computer Services Company
00180 *>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
00181 *>  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
00182 *>         Company
00183 *>
00184 *> \endverbatim
00185 *
00186 *  =====================================================================
00187       SUBROUTINE SSYTF2( UPLO, N, A, LDA, IPIV, INFO )
00188 *
00189 *  -- LAPACK computational routine (version 3.4.0) --
00190 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00191 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00192 *     November 2011
00193 *
00194 *     .. Scalar Arguments ..
00195       CHARACTER          UPLO
00196       INTEGER            INFO, LDA, N
00197 *     ..
00198 *     .. Array Arguments ..
00199       INTEGER            IPIV( * )
00200       REAL               A( LDA, * )
00201 *     ..
00202 *
00203 *  =====================================================================
00204 *
00205 *     .. Parameters ..
00206       REAL               ZERO, ONE
00207       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00208       REAL               EIGHT, SEVTEN
00209       PARAMETER          ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 )
00210 *     ..
00211 *     .. Local Scalars ..
00212       LOGICAL            UPPER
00213       INTEGER            I, IMAX, J, JMAX, K, KK, KP, KSTEP
00214       REAL               ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
00215      $                   ROWMAX, T, WK, WKM1, WKP1
00216 *     ..
00217 *     .. External Functions ..
00218       LOGICAL            LSAME, SISNAN
00219       INTEGER            ISAMAX
00220       EXTERNAL           LSAME, ISAMAX, SISNAN
00221 *     ..
00222 *     .. External Subroutines ..
00223       EXTERNAL           SSCAL, SSWAP, SSYR, XERBLA
00224 *     ..
00225 *     .. Intrinsic Functions ..
00226       INTRINSIC          ABS, MAX, SQRT
00227 *     ..
00228 *     .. Executable Statements ..
00229 *
00230 *     Test the input parameters.
00231 *
00232       INFO = 0
00233       UPPER = LSAME( UPLO, 'U' )
00234       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00235          INFO = -1
00236       ELSE IF( N.LT.0 ) THEN
00237          INFO = -2
00238       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00239          INFO = -4
00240       END IF
00241       IF( INFO.NE.0 ) THEN
00242          CALL XERBLA( 'SSYTF2', -INFO )
00243          RETURN
00244       END IF
00245 *
00246 *     Initialize ALPHA for use in choosing pivot block size.
00247 *
00248       ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
00249 *
00250       IF( UPPER ) THEN
00251 *
00252 *        Factorize A as U*D*U**T using the upper triangle of A
00253 *
00254 *        K is the main loop index, decreasing from N to 1 in steps of
00255 *        1 or 2
00256 *
00257          K = N
00258    10    CONTINUE
00259 *
00260 *        If K < 1, exit from loop
00261 *
00262          IF( K.LT.1 )
00263      $      GO TO 70
00264          KSTEP = 1
00265 *
00266 *        Determine rows and columns to be interchanged and whether
00267 *        a 1-by-1 or 2-by-2 pivot block will be used
00268 *
00269          ABSAKK = ABS( A( K, K ) )
00270 *
00271 *        IMAX is the row-index of the largest off-diagonal element in
00272 *        column K, and COLMAX is its absolute value
00273 *
00274          IF( K.GT.1 ) THEN
00275             IMAX = ISAMAX( K-1, A( 1, K ), 1 )
00276             COLMAX = ABS( A( IMAX, K ) )
00277          ELSE
00278             COLMAX = ZERO
00279          END IF
00280 *
00281          IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
00282 *
00283 *           Column K is zero or contains a NaN: set INFO and continue
00284 *
00285             IF( INFO.EQ.0 )
00286      $         INFO = K
00287             KP = K
00288          ELSE
00289             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00290 *
00291 *              no interchange, use 1-by-1 pivot block
00292 *
00293                KP = K
00294             ELSE
00295 *
00296 *              JMAX is the column-index of the largest off-diagonal
00297 *              element in row IMAX, and ROWMAX is its absolute value
00298 *
00299                JMAX = IMAX + ISAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
00300                ROWMAX = ABS( A( IMAX, JMAX ) )
00301                IF( IMAX.GT.1 ) THEN
00302                   JMAX = ISAMAX( IMAX-1, A( 1, IMAX ), 1 )
00303                   ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
00304                END IF
00305 *
00306                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00307 *
00308 *                 no interchange, use 1-by-1 pivot block
00309 *
00310                   KP = K
00311                ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
00312 *
00313 *                 interchange rows and columns K and IMAX, use 1-by-1
00314 *                 pivot block
00315 *
00316                   KP = IMAX
00317                ELSE
00318 *
00319 *                 interchange rows and columns K-1 and IMAX, use 2-by-2
00320 *                 pivot block
00321 *
00322                   KP = IMAX
00323                   KSTEP = 2
00324                END IF
00325             END IF
00326 *
00327             KK = K - KSTEP + 1
00328             IF( KP.NE.KK ) THEN
00329 *
00330 *              Interchange rows and columns KK and KP in the leading
00331 *              submatrix A(1:k,1:k)
00332 *
00333                CALL SSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
00334                CALL SSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
00335      $                     LDA )
00336                T = A( KK, KK )
00337                A( KK, KK ) = A( KP, KP )
00338                A( KP, KP ) = T
00339                IF( KSTEP.EQ.2 ) THEN
00340                   T = A( K-1, K )
00341                   A( K-1, K ) = A( KP, K )
00342                   A( KP, K ) = T
00343                END IF
00344             END IF
00345 *
00346 *           Update the leading submatrix
00347 *
00348             IF( KSTEP.EQ.1 ) THEN
00349 *
00350 *              1-by-1 pivot block D(k): column k now holds
00351 *
00352 *              W(k) = U(k)*D(k)
00353 *
00354 *              where U(k) is the k-th column of U
00355 *
00356 *              Perform a rank-1 update of A(1:k-1,1:k-1) as
00357 *
00358 *              A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
00359 *
00360                R1 = ONE / A( K, K )
00361                CALL SSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
00362 *
00363 *              Store U(k) in column k
00364 *
00365                CALL SSCAL( K-1, R1, A( 1, K ), 1 )
00366             ELSE
00367 *
00368 *              2-by-2 pivot block D(k): columns k and k-1 now hold
00369 *
00370 *              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
00371 *
00372 *              where U(k) and U(k-1) are the k-th and (k-1)-th columns
00373 *              of U
00374 *
00375 *              Perform a rank-2 update of A(1:k-2,1:k-2) as
00376 *
00377 *              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
00378 *                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
00379 *
00380                IF( K.GT.2 ) THEN
00381 *
00382                   D12 = A( K-1, K )
00383                   D22 = A( K-1, K-1 ) / D12
00384                   D11 = A( K, K ) / D12
00385                   T = ONE / ( D11*D22-ONE )
00386                   D12 = T / D12
00387 *
00388                   DO 30 J = K - 2, 1, -1
00389                      WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
00390                      WK = D12*( D22*A( J, K )-A( J, K-1 ) )
00391                      DO 20 I = J, 1, -1
00392                         A( I, J ) = A( I, J ) - A( I, K )*WK -
00393      $                              A( I, K-1 )*WKM1
00394    20                CONTINUE
00395                      A( J, K ) = WK
00396                      A( J, K-1 ) = WKM1
00397    30             CONTINUE
00398 *
00399                END IF
00400 *
00401             END IF
00402          END IF
00403 *
00404 *        Store details of the interchanges in IPIV
00405 *
00406          IF( KSTEP.EQ.1 ) THEN
00407             IPIV( K ) = KP
00408          ELSE
00409             IPIV( K ) = -KP
00410             IPIV( K-1 ) = -KP
00411          END IF
00412 *
00413 *        Decrease K and return to the start of the main loop
00414 *
00415          K = K - KSTEP
00416          GO TO 10
00417 *
00418       ELSE
00419 *
00420 *        Factorize A as L*D*L**T using the lower triangle of A
00421 *
00422 *        K is the main loop index, increasing from 1 to N in steps of
00423 *        1 or 2
00424 *
00425          K = 1
00426    40    CONTINUE
00427 *
00428 *        If K > N, exit from loop
00429 *
00430          IF( K.GT.N )
00431      $      GO TO 70
00432          KSTEP = 1
00433 *
00434 *        Determine rows and columns to be interchanged and whether
00435 *        a 1-by-1 or 2-by-2 pivot block will be used
00436 *
00437          ABSAKK = ABS( A( K, K ) )
00438 *
00439 *        IMAX is the row-index of the largest off-diagonal element in
00440 *        column K, and COLMAX is its absolute value
00441 *
00442          IF( K.LT.N ) THEN
00443             IMAX = K + ISAMAX( N-K, A( K+1, K ), 1 )
00444             COLMAX = ABS( A( IMAX, K ) )
00445          ELSE
00446             COLMAX = ZERO
00447          END IF
00448 *
00449          IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
00450 *
00451 *           Column K is zero or contains a NaN: set INFO and continue
00452 *
00453             IF( INFO.EQ.0 )
00454      $         INFO = K
00455             KP = K
00456          ELSE
00457             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00458 *
00459 *              no interchange, use 1-by-1 pivot block
00460 *
00461                KP = K
00462             ELSE
00463 *
00464 *              JMAX is the column-index of the largest off-diagonal
00465 *              element in row IMAX, and ROWMAX is its absolute value
00466 *
00467                JMAX = K - 1 + ISAMAX( IMAX-K, A( IMAX, K ), LDA )
00468                ROWMAX = ABS( A( IMAX, JMAX ) )
00469                IF( IMAX.LT.N ) THEN
00470                   JMAX = IMAX + ISAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
00471                   ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
00472                END IF
00473 *
00474                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00475 *
00476 *                 no interchange, use 1-by-1 pivot block
00477 *
00478                   KP = K
00479                ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
00480 *
00481 *                 interchange rows and columns K and IMAX, use 1-by-1
00482 *                 pivot block
00483 *
00484                   KP = IMAX
00485                ELSE
00486 *
00487 *                 interchange rows and columns K+1 and IMAX, use 2-by-2
00488 *                 pivot block
00489 *
00490                   KP = IMAX
00491                   KSTEP = 2
00492                END IF
00493             END IF
00494 *
00495             KK = K + KSTEP - 1
00496             IF( KP.NE.KK ) THEN
00497 *
00498 *              Interchange rows and columns KK and KP in the trailing
00499 *              submatrix A(k:n,k:n)
00500 *
00501                IF( KP.LT.N )
00502      $            CALL SSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
00503                CALL SSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
00504      $                     LDA )
00505                T = A( KK, KK )
00506                A( KK, KK ) = A( KP, KP )
00507                A( KP, KP ) = T
00508                IF( KSTEP.EQ.2 ) THEN
00509                   T = A( K+1, K )
00510                   A( K+1, K ) = A( KP, K )
00511                   A( KP, K ) = T
00512                END IF
00513             END IF
00514 *
00515 *           Update the trailing submatrix
00516 *
00517             IF( KSTEP.EQ.1 ) THEN
00518 *
00519 *              1-by-1 pivot block D(k): column k now holds
00520 *
00521 *              W(k) = L(k)*D(k)
00522 *
00523 *              where L(k) is the k-th column of L
00524 *
00525                IF( K.LT.N ) THEN
00526 *
00527 *                 Perform a rank-1 update of A(k+1:n,k+1:n) as
00528 *
00529 *                 A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
00530 *
00531                   D11 = ONE / A( K, K )
00532                   CALL SSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
00533      $                       A( K+1, K+1 ), LDA )
00534 *
00535 *                 Store L(k) in column K
00536 *
00537                   CALL SSCAL( N-K, D11, A( K+1, K ), 1 )
00538                END IF
00539             ELSE
00540 *
00541 *              2-by-2 pivot block D(k)
00542 *
00543                IF( K.LT.N-1 ) THEN
00544 *
00545 *                 Perform a rank-2 update of A(k+2:n,k+2:n) as
00546 *
00547 *                 A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))**T
00548 *
00549 *                 where L(k) and L(k+1) are the k-th and (k+1)-th
00550 *                 columns of L
00551 *
00552                   D21 = A( K+1, K )
00553                   D11 = A( K+1, K+1 ) / D21
00554                   D22 = A( K, K ) / D21
00555                   T = ONE / ( D11*D22-ONE )
00556                   D21 = T / D21
00557 *
00558                   DO 60 J = K + 2, N
00559 *
00560                      WK = D21*( D11*A( J, K )-A( J, K+1 ) )
00561                      WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
00562 *
00563                      DO 50 I = J, N
00564                         A( I, J ) = A( I, J ) - A( I, K )*WK -
00565      $                              A( I, K+1 )*WKP1
00566    50                CONTINUE
00567 *
00568                      A( J, K ) = WK
00569                      A( J, K+1 ) = WKP1
00570 *
00571    60             CONTINUE
00572                END IF
00573             END IF
00574          END IF
00575 *
00576 *        Store details of the interchanges in IPIV
00577 *
00578          IF( KSTEP.EQ.1 ) THEN
00579             IPIV( K ) = KP
00580          ELSE
00581             IPIV( K ) = -KP
00582             IPIV( K+1 ) = -KP
00583          END IF
00584 *
00585 *        Increase K and return to the start of the main loop
00586 *
00587          K = K + KSTEP
00588          GO TO 40
00589 *
00590       END IF
00591 *
00592    70 CONTINUE
00593 *
00594       RETURN
00595 *
00596 *     End of SSYTF2
00597 *
00598       END
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