LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dsytf2.f
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00001 *> \brief \b DSYTF2
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DSYTF2 + 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/dsytf2.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DSYTF2( 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 *       DOUBLE PRECISION   A( LDA, * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *> DSYTF2 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 DOUBLE PRECISION 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 doubleSYcomputational
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. DISNAN(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 *> \endverbatim
00184 *
00185 *  =====================================================================
00186       SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
00187 *
00188 *  -- LAPACK computational routine (version 3.4.0) --
00189 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00190 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00191 *     November 2011
00192 *
00193 *     .. Scalar Arguments ..
00194       CHARACTER          UPLO
00195       INTEGER            INFO, LDA, N
00196 *     ..
00197 *     .. Array Arguments ..
00198       INTEGER            IPIV( * )
00199       DOUBLE PRECISION   A( LDA, * )
00200 *     ..
00201 *
00202 *  =====================================================================
00203 *
00204 *     .. Parameters ..
00205       DOUBLE PRECISION   ZERO, ONE
00206       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00207       DOUBLE PRECISION   EIGHT, SEVTEN
00208       PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
00209 *     ..
00210 *     .. Local Scalars ..
00211       LOGICAL            UPPER
00212       INTEGER            I, IMAX, J, JMAX, K, KK, KP, KSTEP
00213       DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
00214      $                   ROWMAX, T, WK, WKM1, WKP1
00215 *     ..
00216 *     .. External Functions ..
00217       LOGICAL            LSAME, DISNAN
00218       INTEGER            IDAMAX
00219       EXTERNAL           LSAME, IDAMAX, DISNAN
00220 *     ..
00221 *     .. External Subroutines ..
00222       EXTERNAL           DSCAL, DSWAP, DSYR, XERBLA
00223 *     ..
00224 *     .. Intrinsic Functions ..
00225       INTRINSIC          ABS, MAX, SQRT
00226 *     ..
00227 *     .. Executable Statements ..
00228 *
00229 *     Test the input parameters.
00230 *
00231       INFO = 0
00232       UPPER = LSAME( UPLO, 'U' )
00233       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00234          INFO = -1
00235       ELSE IF( N.LT.0 ) THEN
00236          INFO = -2
00237       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00238          INFO = -4
00239       END IF
00240       IF( INFO.NE.0 ) THEN
00241          CALL XERBLA( 'DSYTF2', -INFO )
00242          RETURN
00243       END IF
00244 *
00245 *     Initialize ALPHA for use in choosing pivot block size.
00246 *
00247       ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
00248 *
00249       IF( UPPER ) THEN
00250 *
00251 *        Factorize A as U*D*U**T using the upper triangle of A
00252 *
00253 *        K is the main loop index, decreasing from N to 1 in steps of
00254 *        1 or 2
00255 *
00256          K = N
00257    10    CONTINUE
00258 *
00259 *        If K < 1, exit from loop
00260 *
00261          IF( K.LT.1 )
00262      $      GO TO 70
00263          KSTEP = 1
00264 *
00265 *        Determine rows and columns to be interchanged and whether
00266 *        a 1-by-1 or 2-by-2 pivot block will be used
00267 *
00268          ABSAKK = ABS( A( K, K ) )
00269 *
00270 *        IMAX is the row-index of the largest off-diagonal element in
00271 *        column K, and COLMAX is its absolute value
00272 *
00273          IF( K.GT.1 ) THEN
00274             IMAX = IDAMAX( K-1, A( 1, K ), 1 )
00275             COLMAX = ABS( A( IMAX, K ) )
00276          ELSE
00277             COLMAX = ZERO
00278          END IF
00279 *
00280          IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
00281 *
00282 *           Column K is zero or contains a NaN: set INFO and continue
00283 *
00284             IF( INFO.EQ.0 )
00285      $         INFO = K
00286             KP = K
00287          ELSE
00288             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00289 *
00290 *              no interchange, use 1-by-1 pivot block
00291 *
00292                KP = K
00293             ELSE
00294 *
00295 *              JMAX is the column-index of the largest off-diagonal
00296 *              element in row IMAX, and ROWMAX is its absolute value
00297 *
00298                JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
00299                ROWMAX = ABS( A( IMAX, JMAX ) )
00300                IF( IMAX.GT.1 ) THEN
00301                   JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
00302                   ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
00303                END IF
00304 *
00305                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00306 *
00307 *                 no interchange, use 1-by-1 pivot block
00308 *
00309                   KP = K
00310                ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
00311 *
00312 *                 interchange rows and columns K and IMAX, use 1-by-1
00313 *                 pivot block
00314 *
00315                   KP = IMAX
00316                ELSE
00317 *
00318 *                 interchange rows and columns K-1 and IMAX, use 2-by-2
00319 *                 pivot block
00320 *
00321                   KP = IMAX
00322                   KSTEP = 2
00323                END IF
00324             END IF
00325 *
00326             KK = K - KSTEP + 1
00327             IF( KP.NE.KK ) THEN
00328 *
00329 *              Interchange rows and columns KK and KP in the leading
00330 *              submatrix A(1:k,1:k)
00331 *
00332                CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
00333                CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
00334      $                     LDA )
00335                T = A( KK, KK )
00336                A( KK, KK ) = A( KP, KP )
00337                A( KP, KP ) = T
00338                IF( KSTEP.EQ.2 ) THEN
00339                   T = A( K-1, K )
00340                   A( K-1, K ) = A( KP, K )
00341                   A( KP, K ) = T
00342                END IF
00343             END IF
00344 *
00345 *           Update the leading submatrix
00346 *
00347             IF( KSTEP.EQ.1 ) THEN
00348 *
00349 *              1-by-1 pivot block D(k): column k now holds
00350 *
00351 *              W(k) = U(k)*D(k)
00352 *
00353 *              where U(k) is the k-th column of U
00354 *
00355 *              Perform a rank-1 update of A(1:k-1,1:k-1) as
00356 *
00357 *              A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
00358 *
00359                R1 = ONE / A( K, K )
00360                CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
00361 *
00362 *              Store U(k) in column k
00363 *
00364                CALL DSCAL( K-1, R1, A( 1, K ), 1 )
00365             ELSE
00366 *
00367 *              2-by-2 pivot block D(k): columns k and k-1 now hold
00368 *
00369 *              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
00370 *
00371 *              where U(k) and U(k-1) are the k-th and (k-1)-th columns
00372 *              of U
00373 *
00374 *              Perform a rank-2 update of A(1:k-2,1:k-2) as
00375 *
00376 *              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
00377 *                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
00378 *
00379                IF( K.GT.2 ) THEN
00380 *
00381                   D12 = A( K-1, K )
00382                   D22 = A( K-1, K-1 ) / D12
00383                   D11 = A( K, K ) / D12
00384                   T = ONE / ( D11*D22-ONE )
00385                   D12 = T / D12
00386 *
00387                   DO 30 J = K - 2, 1, -1
00388                      WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
00389                      WK = D12*( D22*A( J, K )-A( J, K-1 ) )
00390                      DO 20 I = J, 1, -1
00391                         A( I, J ) = A( I, J ) - A( I, K )*WK -
00392      $                              A( I, K-1 )*WKM1
00393    20                CONTINUE
00394                      A( J, K ) = WK
00395                      A( J, K-1 ) = WKM1
00396    30             CONTINUE
00397 *
00398                END IF
00399 *
00400             END IF
00401          END IF
00402 *
00403 *        Store details of the interchanges in IPIV
00404 *
00405          IF( KSTEP.EQ.1 ) THEN
00406             IPIV( K ) = KP
00407          ELSE
00408             IPIV( K ) = -KP
00409             IPIV( K-1 ) = -KP
00410          END IF
00411 *
00412 *        Decrease K and return to the start of the main loop
00413 *
00414          K = K - KSTEP
00415          GO TO 10
00416 *
00417       ELSE
00418 *
00419 *        Factorize A as L*D*L**T using the lower triangle of A
00420 *
00421 *        K is the main loop index, increasing from 1 to N in steps of
00422 *        1 or 2
00423 *
00424          K = 1
00425    40    CONTINUE
00426 *
00427 *        If K > N, exit from loop
00428 *
00429          IF( K.GT.N )
00430      $      GO TO 70
00431          KSTEP = 1
00432 *
00433 *        Determine rows and columns to be interchanged and whether
00434 *        a 1-by-1 or 2-by-2 pivot block will be used
00435 *
00436          ABSAKK = ABS( A( K, K ) )
00437 *
00438 *        IMAX is the row-index of the largest off-diagonal element in
00439 *        column K, and COLMAX is its absolute value
00440 *
00441          IF( K.LT.N ) THEN
00442             IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
00443             COLMAX = ABS( A( IMAX, K ) )
00444          ELSE
00445             COLMAX = ZERO
00446          END IF
00447 *
00448          IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
00449 *
00450 *           Column K is zero or contains a NaN: set INFO and continue
00451 *
00452             IF( INFO.EQ.0 )
00453      $         INFO = K
00454             KP = K
00455          ELSE
00456             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00457 *
00458 *              no interchange, use 1-by-1 pivot block
00459 *
00460                KP = K
00461             ELSE
00462 *
00463 *              JMAX is the column-index of the largest off-diagonal
00464 *              element in row IMAX, and ROWMAX is its absolute value
00465 *
00466                JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
00467                ROWMAX = ABS( A( IMAX, JMAX ) )
00468                IF( IMAX.LT.N ) THEN
00469                   JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
00470                   ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
00471                END IF
00472 *
00473                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00474 *
00475 *                 no interchange, use 1-by-1 pivot block
00476 *
00477                   KP = K
00478                ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
00479 *
00480 *                 interchange rows and columns K and IMAX, use 1-by-1
00481 *                 pivot block
00482 *
00483                   KP = IMAX
00484                ELSE
00485 *
00486 *                 interchange rows and columns K+1 and IMAX, use 2-by-2
00487 *                 pivot block
00488 *
00489                   KP = IMAX
00490                   KSTEP = 2
00491                END IF
00492             END IF
00493 *
00494             KK = K + KSTEP - 1
00495             IF( KP.NE.KK ) THEN
00496 *
00497 *              Interchange rows and columns KK and KP in the trailing
00498 *              submatrix A(k:n,k:n)
00499 *
00500                IF( KP.LT.N )
00501      $            CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
00502                CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
00503      $                     LDA )
00504                T = A( KK, KK )
00505                A( KK, KK ) = A( KP, KP )
00506                A( KP, KP ) = T
00507                IF( KSTEP.EQ.2 ) THEN
00508                   T = A( K+1, K )
00509                   A( K+1, K ) = A( KP, K )
00510                   A( KP, K ) = T
00511                END IF
00512             END IF
00513 *
00514 *           Update the trailing submatrix
00515 *
00516             IF( KSTEP.EQ.1 ) THEN
00517 *
00518 *              1-by-1 pivot block D(k): column k now holds
00519 *
00520 *              W(k) = L(k)*D(k)
00521 *
00522 *              where L(k) is the k-th column of L
00523 *
00524                IF( K.LT.N ) THEN
00525 *
00526 *                 Perform a rank-1 update of A(k+1:n,k+1:n) as
00527 *
00528 *                 A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
00529 *
00530                   D11 = ONE / A( K, K )
00531                   CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
00532      $                       A( K+1, K+1 ), LDA )
00533 *
00534 *                 Store L(k) in column K
00535 *
00536                   CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
00537                END IF
00538             ELSE
00539 *
00540 *              2-by-2 pivot block D(k)
00541 *
00542                IF( K.LT.N-1 ) THEN
00543 *
00544 *                 Perform a rank-2 update of A(k+2:n,k+2:n) as
00545 *
00546 *                 A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))**T
00547 *
00548 *                 where L(k) and L(k+1) are the k-th and (k+1)-th
00549 *                 columns of L
00550 *
00551                   D21 = A( K+1, K )
00552                   D11 = A( K+1, K+1 ) / D21
00553                   D22 = A( K, K ) / D21
00554                   T = ONE / ( D11*D22-ONE )
00555                   D21 = T / D21
00556 *
00557                   DO 60 J = K + 2, N
00558 *
00559                      WK = D21*( D11*A( J, K )-A( J, K+1 ) )
00560                      WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
00561 *
00562                      DO 50 I = J, N
00563                         A( I, J ) = A( I, J ) - A( I, K )*WK -
00564      $                              A( I, K+1 )*WKP1
00565    50                CONTINUE
00566 *
00567                      A( J, K ) = WK
00568                      A( J, K+1 ) = WKP1
00569 *
00570    60             CONTINUE
00571                END IF
00572             END IF
00573          END IF
00574 *
00575 *        Store details of the interchanges in IPIV
00576 *
00577          IF( KSTEP.EQ.1 ) THEN
00578             IPIV( K ) = KP
00579          ELSE
00580             IPIV( K ) = -KP
00581             IPIV( K+1 ) = -KP
00582          END IF
00583 *
00584 *        Increase K and return to the start of the main loop
00585 *
00586          K = K + KSTEP
00587          GO TO 40
00588 *
00589       END IF
00590 *
00591    70 CONTINUE
00592 *
00593       RETURN
00594 *
00595 *     End of DSYTF2
00596 *
00597       END
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