ssytf2_rk.f

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*> \brief \b SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       REAL               A( LDA, * ), E ( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*> SSYTF2_RK computes the factorization of a real symmetric matrix A
*> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
*>
*>    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
*>
*> where U (or L) is unit upper (or lower) triangular matrix,
*> U**T (or L**T) is the transpose of U (or L), P is a permutation
*> matrix, P**T is the transpose of P, and D is symmetric and block
*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> For more information see Further Details section.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          On entry, the symmetric matrix A.
*>            If UPLO = 'U': the leading N-by-N upper triangular part
*>            of A contains the upper triangular part of the matrix A,
*>            and the strictly lower triangular part of A is not
*>            referenced.
*>
*>            If UPLO = 'L': the leading N-by-N lower triangular part
*>            of A contains the lower triangular part of the matrix A,
*>            and the strictly upper triangular part of A is not
*>            referenced.
*>
*>          On exit, contains:
*>            a) ONLY diagonal elements of the symmetric block diagonal
*>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
*>               (superdiagonal (or subdiagonal) elements of D
*>                are stored on exit in array E), and
*>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
*>               If UPLO = 'L': factor L in the subdiagonal part of A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is REAL array, dimension (N)
*>          On exit, contains the superdiagonal (or subdiagonal)
*>          elements of the symmetric block diagonal matrix D
*>          with 1-by-1 or 2-by-2 diagonal blocks, where
*>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
*>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
*>
*>          NOTE: For 1-by-1 diagonal block D(k), where
*>          1 <= k <= N, the element E(k) is set to 0 in both
*>          UPLO = 'U' or UPLO = 'L' cases.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          IPIV describes the permutation matrix P in the factorization
*>          of matrix A as follows. The absolute value of IPIV(k)
*>          represents the index of row and column that were
*>          interchanged with the k-th row and column. The value of UPLO
*>          describes the order in which the interchanges were applied.
*>          Also, the sign of IPIV represents the block structure of
*>          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
*>          diagonal blocks which correspond to 1 or 2 interchanges
*>          at each factorization step. For more info see Further
*>          Details section.
*>
*>          If UPLO = 'U',
*>          ( in factorization order, k decreases from N to 1 ):
*>            a) A single positive entry IPIV(k) > 0 means:
*>               D(k,k) is a 1-by-1 diagonal block.
*>               If IPIV(k) != k, rows and columns k and IPIV(k) were
*>               interchanged in the matrix A(1:N,1:N);
*>               If IPIV(k) = k, no interchange occurred.
*>
*>            b) A pair of consecutive negative entries
*>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
*>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
*>               (NOTE: negative entries in IPIV appear ONLY in pairs).
*>               1) If -IPIV(k) != k, rows and columns
*>                  k and -IPIV(k) were interchanged
*>                  in the matrix A(1:N,1:N).
*>                  If -IPIV(k) = k, no interchange occurred.
*>               2) If -IPIV(k-1) != k-1, rows and columns
*>                  k-1 and -IPIV(k-1) were interchanged
*>                  in the matrix A(1:N,1:N).
*>                  If -IPIV(k-1) = k-1, no interchange occurred.
*>
*>            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
*>
*>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
*>
*>          If UPLO = 'L',
*>          ( in factorization order, k increases from 1 to N ):
*>            a) A single positive entry IPIV(k) > 0 means:
*>               D(k,k) is a 1-by-1 diagonal block.
*>               If IPIV(k) != k, rows and columns k and IPIV(k) were
*>               interchanged in the matrix A(1:N,1:N).
*>               If IPIV(k) = k, no interchange occurred.
*>
*>            b) A pair of consecutive negative entries
*>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
*>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*>               (NOTE: negative entries in IPIV appear ONLY in pairs).
*>               1) If -IPIV(k) != k, rows and columns
*>                  k and -IPIV(k) were interchanged
*>                  in the matrix A(1:N,1:N).
*>                  If -IPIV(k) = k, no interchange occurred.
*>               2) If -IPIV(k+1) != k+1, rows and columns
*>                  k-1 and -IPIV(k-1) were interchanged
*>                  in the matrix A(1:N,1:N).
*>                  If -IPIV(k+1) = k+1, no interchange occurred.
*>
*>            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
*>
*>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>
*>          < 0: If INFO = -k, the k-th argument had an illegal value
*>
*>          > 0: If INFO = k, the matrix A is singular, because:
*>                 If UPLO = 'U': column k in the upper
*>                 triangular part of A contains all zeros.
*>                 If UPLO = 'L': column k in the lower
*>                 triangular part of A contains all zeros.
*>
*>               Therefore D(k,k) is exactly zero, and superdiagonal
*>               elements of column k of U (or subdiagonal elements of
*>               column k of L ) are all zeros. The factorization has
*>               been completed, but the block diagonal matrix D is
*>               exactly singular, and division by zero will occur if
*>               it is used to solve a system of equations.
*>
*>               NOTE: INFO only stores the first occurrence of
*>               a singularity, any subsequent occurrence of singularity
*>               is not stored in INFO even though the factorization
*>               always completes.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetf2_rk
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*> TODO: put further details
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  December 2016,  Igor Kozachenko,
*>                  Computer Science Division,
*>                  University of California, Berkeley
*>
*>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
*>                  School of Mathematics,
*>                  University of Manchester
*>
*>  01-01-96 - Based on modifications by
*>    J. Lewis, Boeing Computer Services Company
*>    A. Petitet, Computer Science Dept.,
*>                Univ. of Tenn., Knoxville abd , USA
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE ssytf2_rk( UPLO, N, A, LDA, E, IPIV, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      REAL               A( LDA, * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
      REAL               EIGHT, SEVTEN
      parameter( eight = 8.0e+0, sevten = 17.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER, DONE
      INTEGER            I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
     $                   P, II
      REAL               ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
     $                   ROWMAX, STEMP, T, WK, WKM1, WKP1, SFMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SLAMCH
      EXTERNAL           lsame, isamax, slamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           sscal, sswap, ssyr, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      upper = lsame( uplo, 'U' )
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -4
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SSYTF2_RK', -info )
         RETURN
      END IF
*
*     Initialize ALPHA for use in choosing pivot block size.
*
      alpha = ( one+sqrt( sevten ) ) / eight
*
*     Compute machine safe minimum
*
      sfmin = slamch( 'S' )
*
      IF( upper ) THEN
*
*        Factorize A as U*D*U**T using the upper triangle of A
*
*        Initialize the first entry of array E, where superdiagonal
*        elements of D are stored
*
         e( 1 ) = zero
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2
*
         k = n
   10    CONTINUE
*
*        If K < 1, exit from loop
*
         IF( k.LT.1 )
     $      GO TO 34
         kstep = 1
         p = k
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         absakk = abs( a( k, k ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value.
*        Determine both COLMAX and IMAX.
*
         IF( k.GT.1 ) THEN
            imax = isamax( k-1, a( 1, k ), 1 )
            colmax = abs( a( imax, k ) )
         ELSE
            colmax = zero
         END IF
*
         IF( (max( absakk, colmax ).EQ.zero) ) THEN
*
*           Column K is zero or underflow: set INFO and continue
*
            IF( info.EQ.0 )
     $         info = k
            kp = k
*
*           Set E( K ) to zero
*
            IF( k.GT.1 )
     $         e( k ) = zero
*
         ELSE
*
*           Test for interchange
*
*           Equivalent to testing for (used to handle NaN and Inf)
*           ABSAKK.GE.ALPHA*COLMAX
*
            IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
*
*              no interchange,
*              use 1-by-1 pivot block
*
               kp = k
            ELSE
*
               done = .false.
*
*              Loop until pivot found
*
   12          CONTINUE
*
*                 Begin pivot search loop body
*
*                 JMAX is the column-index of the largest off-diagonal
*                 element in row IMAX, and ROWMAX is its absolute value.
*                 Determine both ROWMAX and JMAX.
*
                  IF( imax.NE.k ) THEN
                     jmax = imax + isamax( k-imax, a( imax, imax+1 ),
     $                                    lda )
                     rowmax = abs( a( imax, jmax ) )
                  ELSE
                     rowmax = zero
                  END IF
*
                  IF( imax.GT.1 ) THEN
                     itemp = isamax( imax-1, a( 1, imax ), 1 )
                     stemp = abs( a( itemp, imax ) )
                     IF( stemp.GT.rowmax ) THEN
                        rowmax = stemp
                        jmax = itemp
                     END IF
                  END IF
*
*                 Equivalent to testing for (used to handle NaN and Inf)
*                 ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
*
                  IF( .NOT.( abs( a( imax, imax ) ).LT.alpha*rowmax ) )
     $            THEN
*
*                    interchange rows and columns K and IMAX,
*                    use 1-by-1 pivot block
*
                     kp = imax
                     done = .true.
*
*                 Equivalent to testing for ROWMAX .EQ. COLMAX,
*                 used to handle NaN and Inf
*
                  ELSE IF( ( p.EQ.jmax ).OR.( rowmax.LE.colmax ) ) THEN
*
*                    interchange rows and columns K+1 and IMAX,
*                    use 2-by-2 pivot block
*
                     kp = imax
                     kstep = 2
                     done = .true.
                  ELSE
*
*                    Pivot NOT found, set variables and repeat
*
                     p = imax
                     colmax = rowmax
                     imax = jmax
                  END IF
*
*                 End pivot search loop body
*
               IF( .NOT. done ) GOTO 12
*
            END IF
*
*           Swap TWO rows and TWO columns
*
*           First swap
*
            IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
*
*              Interchange rows and column K and P in the leading
*              submatrix A(1:k,1:k) if we have a 2-by-2 pivot
*
               IF( p.GT.1 )
     $            CALL sswap( p-1, a( 1, k ), 1, a( 1, p ), 1 )
               IF( p.LT.(k-1) )
     $            CALL sswap( k-p-1, a( p+1, k ), 1, a( p, p+1 ),
     $                     lda )
               t = a( k, k )
               a( k, k ) = a( p, p )
               a( p, p ) = t
*
*              Convert upper triangle of A into U form by applying
*              the interchanges in columns k+1:N.
*
               IF( k.LT.n )
     $            CALL sswap( n-k, a( k, k+1 ), lda, a( p, k+1 ), lda )
*
            END IF
*
*           Second swap
*
            kk = k - kstep + 1
            IF( kp.NE.kk ) THEN
*
*              Interchange rows and columns KK and KP in the leading
*              submatrix A(1:k,1:k)
*
               IF( kp.GT.1 )
     $            CALL sswap( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
               IF( ( kk.GT.1 ) .AND. ( kp.LT.(kk-1) ) )
     $            CALL sswap( kk-kp-1, a( kp+1, kk ), 1, a( kp, kp+1 ),
     $                     lda )
               t = a( kk, kk )
               a( kk, kk ) = a( kp, kp )
               a( kp, kp ) = t
               IF( kstep.EQ.2 ) THEN
                  t = a( k-1, k )
                  a( k-1, k ) = a( kp, k )
                  a( kp, k ) = t
               END IF
*
*              Convert upper triangle of A into U form by applying
*              the interchanges in columns k+1:N.
*
               IF( k.LT.n )
     $            CALL sswap( n-k, a( kk, k+1 ), lda, a( kp, k+1 ),
     $                        lda )
*
            END IF
*
*           Update the leading submatrix
*
            IF( kstep.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = U(k)*D(k)
*
*              where U(k) is the k-th column of U
*
               IF( k.GT.1 ) THEN
*
*                 Perform a rank-1 update of A(1:k-1,1:k-1) and
*                 store U(k) in column k
*
                  IF( abs( a( k, k ) ).GE.sfmin ) THEN
*
*                    Perform a rank-1 update of A(1:k-1,1:k-1) as
*                    A := A - U(k)*D(k)*U(k)**T
*                       = A - W(k)*1/D(k)*W(k)**T
*
                     d11 = one / a( k, k )
                     CALL ssyr( uplo, k-1, -d11, a( 1, k ), 1, a, lda )
*
*                    Store U(k) in column k
*
                     CALL sscal( k-1, d11, a( 1, k ), 1 )
                  ELSE
*
*                    Store L(k) in column K
*
                     d11 = a( k, k )
                     DO 16 ii = 1, k - 1
                        a( ii, k ) = a( ii, k ) / d11
   16                CONTINUE
*
*                    Perform a rank-1 update of A(k+1:n,k+1:n) as
*                    A := A - U(k)*D(k)*U(k)**T
*                       = A - W(k)*(1/D(k))*W(k)**T
*                       = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
*
                     CALL ssyr( uplo, k-1, -d11, a( 1, k ), 1, a, lda )
                  END IF
*
*                 Store the superdiagonal element of D in array E
*
                  e( k ) = zero
*
               END IF
*
            ELSE
*
*              2-by-2 pivot block D(k): columns k and k-1 now hold
*
*              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
*              where U(k) and U(k-1) are the k-th and (k-1)-th columns
*              of U
*
*              Perform a rank-2 update of A(1:k-2,1:k-2) as
*
*              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
*                 = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
*
*              and store L(k) and L(k+1) in columns k and k+1
*
               IF( k.GT.2 ) THEN
*
                  d12 = a( k-1, k )
                  d22 = a( k-1, k-1 ) / d12
                  d11 = a( k, k ) / d12
                  t = one / ( d11*d22-one )
*
                  DO 30 j = k - 2, 1, -1
*
                     wkm1 = t*( d11*a( j, k-1 )-a( j, k ) )
                     wk = t*( d22*a( j, k )-a( j, k-1 ) )
*
                     DO 20 i = j, 1, -1
                        a( i, j ) = a( i, j ) - (a( i, k ) / d12 )*wk -
     $                              ( a( i, k-1 ) / d12 )*wkm1
   20                CONTINUE
*
*                    Store U(k) and U(k-1) in cols k and k-1 for row J
*
                     a( j, k ) = wk / d12
                     a( j, k-1 ) = wkm1 / d12
*
   30             CONTINUE
*
               END IF
*
*              Copy superdiagonal elements of D(K) to E(K) and
*              ZERO out superdiagonal entry of A
*
               e( k ) = a( k-1, k )
               e( k-1 ) = zero
               a( k-1, k ) = zero
*
            END IF
*
*           End column K is nonsingular
*
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( kstep.EQ.1 ) THEN
            ipiv( k ) = kp
         ELSE
            ipiv( k ) = -p
            ipiv( k-1 ) = -kp
         END IF
*
*        Decrease K and return to the start of the main loop
*
         k = k - kstep
         GO TO 10
*
   34    CONTINUE
*
      ELSE
*
*        Factorize A as L*D*L**T using the lower triangle of A
*
*        Initialize the unused last entry of the subdiagonal array E.
*
         e( n ) = zero
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2
*
         k = 1
   40    CONTINUE
*
*        If K > N, exit from loop
*
         IF( k.GT.n )
     $      GO TO 64
         kstep = 1
         p = k
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         absakk = abs( a( k, k ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value.
*        Determine both COLMAX and IMAX.
*
         IF( k.LT.n ) THEN
            imax = k + isamax( n-k, a( k+1, k ), 1 )
            colmax = abs( a( imax, k ) )
         ELSE
            colmax = zero
         END IF
*
         IF( ( max( absakk, colmax ).EQ.zero ) ) THEN
*
*           Column K is zero or underflow: set INFO and continue
*
            IF( info.EQ.0 )
     $         info = k
            kp = k
*
*           Set E( K ) to zero
*
            IF( k.LT.n )
     $         e( k ) = zero
*
         ELSE
*
*           Test for interchange
*
*           Equivalent to testing for (used to handle NaN and Inf)
*           ABSAKK.GE.ALPHA*COLMAX
*
            IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               kp = k
*
            ELSE
*
               done = .false.
*
*              Loop until pivot found
*
   42          CONTINUE
*
*                 Begin pivot search loop body
*
*                 JMAX is the column-index of the largest off-diagonal
*                 element in row IMAX, and ROWMAX is its absolute value.
*                 Determine both ROWMAX and JMAX.
*
                  IF( imax.NE.k ) THEN
                     jmax = k - 1 + isamax( imax-k, a( imax, k ), lda )
                     rowmax = abs( a( imax, jmax ) )
                  ELSE
                     rowmax = zero
                  END IF
*
                  IF( imax.LT.n ) THEN
                     itemp = imax + isamax( n-imax, a( imax+1, imax ),
     $                                     1 )
                     stemp = abs( a( itemp, imax ) )
                     IF( stemp.GT.rowmax ) THEN
                        rowmax = stemp
                        jmax = itemp
                     END IF
                  END IF
*
*                 Equivalent to testing for (used to handle NaN and Inf)
*                 ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
*
                  IF( .NOT.( abs( a( imax, imax ) ).LT.alpha*rowmax ) )
     $            THEN
*
*                    interchange rows and columns K and IMAX,
*                    use 1-by-1 pivot block
*
                     kp = imax
                     done = .true.
*
*                 Equivalent to testing for ROWMAX .EQ. COLMAX,
*                 used to handle NaN and Inf
*
                  ELSE IF( ( p.EQ.jmax ).OR.( rowmax.LE.colmax ) ) THEN
*
*                    interchange rows and columns K+1 and IMAX,
*                    use 2-by-2 pivot block
*
                     kp = imax
                     kstep = 2
                     done = .true.
                  ELSE
*
*                    Pivot NOT found, set variables and repeat
*
                     p = imax
                     colmax = rowmax
                     imax = jmax
                  END IF
*
*                 End pivot search loop body
*
               IF( .NOT. done ) GOTO 42
*
            END IF
*
*           Swap TWO rows and TWO columns
*
*           First swap
*
            IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
*
*              Interchange rows and column K and P in the trailing
*              submatrix A(k:n,k:n) if we have a 2-by-2 pivot
*
               IF( p.LT.n )
     $            CALL sswap( n-p, a( p+1, k ), 1, a( p+1, p ), 1 )
               IF( p.GT.(k+1) )
     $            CALL sswap( p-k-1, a( k+1, k ), 1, a( p, k+1 ), lda )
               t = a( k, k )
               a( k, k ) = a( p, p )
               a( p, p ) = t
*
*              Convert lower triangle of A into L form by applying
*              the interchanges in columns 1:k-1.
*
               IF ( k.GT.1 )
     $            CALL sswap( k-1, a( k, 1 ), lda, a( p, 1 ), lda )
*
            END IF
*
*           Second swap
*
            kk = k + kstep - 1
            IF( kp.NE.kk ) THEN
*
*              Interchange rows and columns KK and KP in the trailing
*              submatrix A(k:n,k:n)
*
               IF( kp.LT.n )
     $            CALL sswap( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
               IF( ( kk.LT.n ) .AND. ( kp.GT.(kk+1) ) )
     $            CALL sswap( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
     $                     lda )
               t = a( kk, kk )
               a( kk, kk ) = a( kp, kp )
               a( kp, kp ) = t
               IF( kstep.EQ.2 ) THEN
                  t = a( k+1, k )
                  a( k+1, k ) = a( kp, k )
                  a( kp, k ) = t
               END IF
*
*              Convert lower triangle of A into L form by applying
*              the interchanges in columns 1:k-1.
*
               IF ( k.GT.1 )
     $            CALL sswap( k-1, a( kk, 1 ), lda, a( kp, 1 ), lda )
*
            END IF
*
*           Update the trailing submatrix
*
            IF( kstep.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = L(k)*D(k)
*
*              where L(k) is the k-th column of L
*
               IF( k.LT.n ) THEN
*
*              Perform a rank-1 update of A(k+1:n,k+1:n) and
*              store L(k) in column k
*
                  IF( abs( a( k, k ) ).GE.sfmin ) THEN
*
*                    Perform a rank-1 update of A(k+1:n,k+1:n) as
*                    A := A - L(k)*D(k)*L(k)**T
*                       = A - W(k)*(1/D(k))*W(k)**T
*
                     d11 = one / a( k, k )
                     CALL ssyr( uplo, n-k, -d11, a( k+1, k ), 1,
     $                          a( k+1, k+1 ), lda )
*
*                    Store L(k) in column k
*
                     CALL sscal( n-k, d11, a( k+1, k ), 1 )
                  ELSE
*
*                    Store L(k) in column k
*
                     d11 = a( k, k )
                     DO 46 ii = k + 1, n
                        a( ii, k ) = a( ii, k ) / d11
   46                CONTINUE
*
*                    Perform a rank-1 update of A(k+1:n,k+1:n) as
*                    A := A - L(k)*D(k)*L(k)**T
*                       = A - W(k)*(1/D(k))*W(k)**T
*                       = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
*
                     CALL ssyr( uplo, n-k, -d11, a( k+1, k ), 1,
     $                          a( k+1, k+1 ), lda )
                  END IF
*
*                 Store the subdiagonal element of D in array E
*
                  e( k ) = zero
*
               END IF
*
            ELSE
*
*              2-by-2 pivot block D(k): columns k and k+1 now hold
*
*              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
*
*              where L(k) and L(k+1) are the k-th and (k+1)-th columns
*              of L
*
*
*              Perform a rank-2 update of A(k+2:n,k+2:n) as
*
*              A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
*                 = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
*
*              and store L(k) and L(k+1) in columns k and k+1
*
               IF( k.LT.n-1 ) THEN
*
                  d21 = a( k+1, k )
                  d11 = a( k+1, k+1 ) / d21
                  d22 = a( k, k ) / d21
                  t = one / ( d11*d22-one )
*
                  DO 60 j = k + 2, n
*
*                    Compute  D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
*
                     wk = t*( d11*a( j, k )-a( j, k+1 ) )
                     wkp1 = t*( d22*a( j, k+1 )-a( j, k ) )
*
*                    Perform a rank-2 update of A(k+2:n,k+2:n)
*
                     DO 50 i = j, n
                        a( i, j ) = a( i, j ) - ( a( i, k ) / d21 )*wk -
     $                              ( a( i, k+1 ) / d21 )*wkp1
   50                CONTINUE
*
*                    Store L(k) and L(k+1) in cols k and k+1 for row J
*
                     a( j, k ) = wk / d21
                     a( j, k+1 ) = wkp1 / d21
*
   60             CONTINUE
*
               END IF
*
*              Copy subdiagonal elements of D(K) to E(K) and
*              ZERO out subdiagonal entry of A
*
               e( k ) = a( k+1, k )
               e( k+1 ) = zero
               a( k+1, k ) = zero
*
            END IF
*
*           End column K is nonsingular
*
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( kstep.EQ.1 ) THEN
            ipiv( k ) = kp
         ELSE
            ipiv( k ) = -p
            ipiv( k+1 ) = -kp
         END IF
*
*        Increase K and return to the start of the main loop
*
         k = k + kstep
         GO TO 40
*
   64    CONTINUE
*
      END IF
*
      RETURN
*
*     End of SSYTF2_RK
*
      END