added standard precision (REAL) routines for 'rook' pivoting algorithm for symmetric indefinite matrices: slasyf_rook.f ssytri_rook.f ssytrs_rook.f ssytrf_rook.f ssysv_rook.f ssycon_rook.f ssytf2_rook.f

1 files changed, 815 insertions, 0 deletions

diff --git a/SRC/ssytf2_rook.f b/SRC/ssytf2_rook.f
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+*> \brief \b SSYTF2_ROOK
+*
+*  =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+*            http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download SSYTF2_ROOK + dependencies
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssytf2_rook.f">
+*> [TGZ]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssytf2_rook.f">
+*> [ZIP]</a>
+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssytf2_rook.f">
+*> [TXT]</a>
+*> \endhtmlonly
+*
+*  Definition:
+*  ===========
+*
+*       SUBROUTINE SSYTF2_ROOK( UPLO, N, A, LDA, IPIV, INFO )
+*
+*       .. Scalar Arguments ..
+*       CHARACTER          UPLO
+*       INTEGER            INFO, LDA, N
+*       ..
+*       .. Array Arguments ..
+*       INTEGER            IPIV( * )
+*       REAL               A( LDA, * )
+*       ..
+*
+*
+*> \par Purpose:
+*  =============
+*>
+*> \verbatim
+*>
+*> SSYTF2_ROOK computes the factorization of a real symmetric matrix A
+*> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:
+*>
+*>    A = U*D*U**T  or  A = L*D*L**T
+*>
+*> where U (or L) is a product of permutation and unit upper (lower)
+*> triangular matrices, U**T is the transpose of U, 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.
+*> \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, the block diagonal matrix D and the multipliers used
+*>          to obtain the factor U or L (see below for further details).
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*>          LDA is INTEGER
+*>          The leading dimension of the array A.  LDA >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] IPIV
+*> \verbatim
+*>          IPIV is INTEGER array, dimension (N)
+*>          Details of the interchanges and the block structure of D.
+*>
+*>          If UPLO = 'U':
+*>               If IPIV(k) > 0, then rows and columns k and IPIV(k)
+*>               were interchanged and D(k,k) is a 1-by-1 diagonal block.
+*>
+*>               If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
+*>               columns k and -IPIV(k) were interchanged and rows and
+*>               columns k-1 and -IPIV(k-1) were inerchaged,
+*>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
+*>
+*>          If UPLO = 'L':
+*>               If IPIV(k) > 0, then rows and columns k and IPIV(k)
+*>               were interchanged and D(k,k) is a 1-by-1 diagonal block.
+*>
+*>               If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
+*>               columns k and -IPIV(k) were interchanged and rows and
+*>               columns k+1 and -IPIV(k+1) were inerchaged,
+*>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
+*> \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, D(k,k) is exactly zero.  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.
+*> \endverbatim
+*
+*  Authors:
+*  ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup realSYcomputational
+*
+*> \par Further Details:
+*  =====================
+*>
+*> \verbatim
+*>
+*>  If UPLO = 'U', then A = U*D*U**T, where
+*>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
+*>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
+*>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
+*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
+*>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
+*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
+*>
+*>             (   I    v    0   )   k-s
+*>     U(k) =  (   0    I    0   )   s
+*>             (   0    0    I   )   n-k
+*>                k-s   s   n-k
+*>
+*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
+*>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
+*>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
+*>
+*>  If UPLO = 'L', then A = L*D*L**T, where
+*>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
+*>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
+*>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
+*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
+*>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
+*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
+*>
+*>             (   I    0     0   )  k-1
+*>     L(k) =  (   0    I     0   )  s
+*>             (   0    v     I   )  n-k-s+1
+*>                k-1   s  n-k-s+1
+*>
+*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
+*>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
+*>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
+*> \endverbatim
+*
+*> \par Contributors:
+*  ==================
+*>
+*> \verbatim
+*>
+*>   November 2011, 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, USA
+*>  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
+*>         Company
+*> \endverbatim
+*
+*  =====================================================================
+      SUBROUTINE SSYTF2_ROOK( UPLO, N, A, LDA, IPIV, INFO )
+*
+*  -- LAPACK computational routine (version 3.3.1) --
+*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
+*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
+*     November 2011
+*
+*     .. Scalar Arguments ..
+      CHARACTER          UPLO
+      INTEGER            INFO, LDA, N
+*     ..
+*     .. Array Arguments ..
+      INTEGER            IPIV( * )
+      REAL               A( LDA, * )
+*     ..
+*
+*  =====================================================================
+*
+*     .. 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, DTEMP, 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_ROOK', -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
+*
+*        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 70
+         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: set INFO and continue
+*
+            IF( INFO.EQ.0 )
+     $         INFO = K
+            KP = K
+         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 only ROWMAX.
+*
+                  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 )
+                     DTEMP = ABS( A( ITEMP, IMAX ) )
+                     IF( DTEMP.GT.ROWMAX ) THEN
+                        ROWMAX = DTEMP
+                        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
+            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
+            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
+               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
+*
+            END IF
+         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
+*
+      ELSE
+*
+*        Factorize A as L*D*L**T using the lower triangle of A
+*
+*        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 70
+         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: set INFO and continue
+*
+            IF( INFO.EQ.0 )
+     $         INFO = K
+            KP = K
+         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 )
+                     DTEMP = ABS( A( ITEMP, IMAX ) )
+                     IF( DTEMP.GT.ROWMAX ) THEN
+                        ROWMAX = DTEMP
+                        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
+            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
+            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
+               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
+*
+            END IF
+         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
+*
+      END IF
+*
+   70 CONTINUE
+*
+      RETURN
+*
+*     End of SSYTF2_ROOK
+*
+      END