*> \brief \b DSYTF2_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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*> [ZIP]</a>
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*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), E ( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*> DSYTF2_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSYTF2_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( * )
DOUBLE PRECISION A( LDA, * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION EIGHT, SEVTEN
PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER, DONE
INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
$ P, II
DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
$ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, DSYR, 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( 'DSYTF2_RK', -INFO )
RETURN
END IF
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
* Compute machine safe minimum
*
SFMIN = DLAMCH( '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 = IDAMAX( 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 + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
$ LDA )
ROWMAX = ABS( A( IMAX, JMAX ) )
ELSE
ROWMAX = ZERO
END IF
*
IF( IMAX.GT.1 ) THEN
ITEMP = IDAMAX( 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 DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
IF( P.LT.(K-1) )
$ CALL DSWAP( 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 DSWAP( 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 DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
$ CALL DSWAP( 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 DSWAP( 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 DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A,
$ LDA )
*
* Store U(k) in column k
*
CALL DSCAL( 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 DSYR( 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 + IDAMAX( 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 + IDAMAX( IMAX-K, A( IMAX, K ),
$ LDA )
ROWMAX = ABS( A( IMAX, JMAX ) )
ELSE
ROWMAX = ZERO
END IF
*
IF( IMAX.LT.N ) THEN
ITEMP = IMAX + IDAMAX( 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 DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
IF( P.GT.(K+1) )
$ CALL DSWAP( 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 DSWAP( 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 DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ),
$ 1 )
IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
$ CALL DSWAP( 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 DSWAP( 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 DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
$ A( K+1, K+1 ), LDA )
*
* Store L(k) in column k
*
CALL DSCAL( 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 DSYR( 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 DSYTF2_RK
*
END