*> \brief \b DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTF2 + dependencies
*>
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*>
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*
* Definition:
* ===========
*
* SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTF2 computes the factorization of a real symmetric matrix A using
*> the Bunch-Kaufman 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 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, 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) = IPIV(k-1) < 0, then rows and columns
*> k-1 and -IPIV(k) were interchanged and 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) = IPIV(k+1) < 0, then rows and columns
*> k+1 and -IPIV(k) were interchanged and 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.
*
*> \ingroup doubleSYcomputational
*
*> \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
*>
*> 09-29-06 - patch from
*> Bobby Cheng, MathWorks
*>
*> Replace l.204 and l.372
*> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*> by
*> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
*>
*> 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 DSYTF2( UPLO, N, A, LDA, 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, * )
* ..
*
* =====================================================================
*
* .. 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
INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
$ ROWMAX, T, WK, WKM1, WKP1
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
INTEGER IDAMAX
EXTERNAL LSAME, IDAMAX, DISNAN
* ..
* .. 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', -INFO )
RETURN
END IF
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
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
*
* 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) .OR. DISNAN(ABSAKK) ) THEN
*
* Column K is zero or underflow, or contains a NaN:
* set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
ROWMAX = ABS( A( IMAX, JMAX ) )
IF( IMAX.GT.1 ) THEN
JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K-1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
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)
*
CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 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
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
*
* 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
*
R1 = ONE / A( K, K )
CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
*
* Store U(k) in column k
*
CALL DSCAL( K-1, R1, A( 1, K ), 1 )
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 - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
*
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 )
D12 = T / D12
*
DO 30 J = K - 2, 1, -1
WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
WK = D12*( D22*A( J, K )-A( J, K-1 ) )
DO 20 I = J, 1, -1
A( I, J ) = A( I, J ) - A( I, K )*WK -
$ A( I, K-1 )*WKM1
20 CONTINUE
A( J, K ) = WK
A( J, K-1 ) = WKM1
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 ) = -KP
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
*
* 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) .OR. DISNAN(ABSAKK) ) THEN
*
* Column K is zero or underflow, or contains a NaN:
* set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
ROWMAX = ABS( A( IMAX, JMAX ) )
IF( IMAX.LT.N ) THEN
JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K+1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
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 )
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
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) 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 )
END IF
ELSE
*
* 2-by-2 pivot block D(k)
*
IF( K.LT.N-1 ) THEN
*
* Perform a rank-2 update of A(k+2:n,k+2:n) as
*
* A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))**T
*
* where L(k) and L(k+1) are the k-th and (k+1)-th
* columns of L
*
D21 = A( K+1, K )
D11 = A( K+1, K+1 ) / D21
D22 = A( K, K ) / D21
T = ONE / ( D11*D22-ONE )
D21 = T / D21
*
DO 60 J = K + 2, N
*
WK = D21*( D11*A( J, K )-A( J, K+1 ) )
WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
*
DO 50 I = J, N
A( I, J ) = A( I, J ) - A( I, K )*WK -
$ A( I, K+1 )*WKP1
50 CONTINUE
*
A( J, K ) = WK
A( J, K+1 ) = WKP1
*
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 ) = -KP
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 DSYTF2
*
END