SUBROUTINE DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DSYTRI computes the inverse of a real symmetric indefinite matrix
* A using the factorization A = U*D*U**T or A = L*D*L**T computed by
* DSYTRF.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the details of the factorization are stored
* as an upper or lower triangular matrix.
* = 'U': Upper triangular, form is A = U*D*U**T;
* = 'L': Lower triangular, form is A = L*D*L**T.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the block diagonal matrix D and the multipliers
* used to obtain the factor U or L as computed by DSYTRF.
*
* On exit, if INFO = 0, the (symmetric) inverse of the original
* matrix. If UPLO = 'U', the upper triangular part of the
* inverse is formed and the part of A below the diagonal is not
* referenced; if UPLO = 'L' the lower triangular part of the
* inverse is formed and the part of A above the diagonal is
* not referenced.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D
* as determined by DSYTRF.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
* inverse could not be computed.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER K, KP, KSTEP
DOUBLE PRECISION AK, AKKP1, AKP1, D, T, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DSWAP, DSYMV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. 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( 'DSYTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
DO 10 INFO = N, 1, -1
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
$ RETURN
10 CONTINUE
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
DO 20 INFO = 1, N
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO )
$ RETURN
20 CONTINUE
END IF
INFO = 0
*
IF( UPPER ) THEN
*
* Compute inv(A) from the factorization A = U*D*U'.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
30 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 40
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
A( K, K ) = ONE / A( K, K )
*
* Compute column K of the inverse.
*
IF( K.GT.1 ) THEN
CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
$ A( 1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ),
$ 1 )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( A( K, K+1 ) )
AK = A( K, K ) / T
AKP1 = A( K+1, K+1 ) / T
AKKP1 = A( K, K+1 ) / T
D = T*( AK*AKP1-ONE )
A( K, K ) = AKP1 / D
A( K+1, K+1 ) = AK / D
A( K, K+1 ) = -AKKP1 / D
*
* Compute columns K and K+1 of the inverse.
*
IF( K.GT.1 ) THEN
CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
$ A( 1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ),
$ 1 )
A( K, K+1 ) = A( K, K+1 ) -
$ DDOT( K-1, A( 1, K ), 1, A( 1, K+1 ), 1 )
CALL DCOPY( K-1, A( 1, K+1 ), 1, WORK, 1 )
CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO,
$ A( 1, K+1 ), 1 )
A( K+1, K+1 ) = A( K+1, K+1 ) -
$ DDOT( K-1, WORK, 1, A( 1, K+1 ), 1 )
END IF
KSTEP = 2
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the leading
* submatrix A(1:k+1,1:k+1)
*
CALL DSWAP( KP-1, A( 1, K ), 1, A( 1, KP ), 1 )
CALL DSWAP( K-KP-1, A( KP+1, K ), 1, A( KP, KP+1 ), LDA )
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = A( K, K+1 )
A( K, K+1 ) = A( KP, K+1 )
A( KP, K+1 ) = TEMP
END IF
END IF
*
K = K + KSTEP
GO TO 30
40 CONTINUE
*
ELSE
*
* Compute inv(A) from the factorization A = L*D*L'.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
50 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 60
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
A( K, K ) = ONE / A( K, K )
*
* Compute column K of the inverse.
*
IF( K.LT.N ) THEN
CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK, 1,
$ ZERO, A( K+1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1, K ),
$ 1 )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( A( K, K-1 ) )
AK = A( K-1, K-1 ) / T
AKP1 = A( K, K ) / T
AKKP1 = A( K, K-1 ) / T
D = T*( AK*AKP1-ONE )
A( K-1, K-1 ) = AKP1 / D
A( K, K ) = AK / D
A( K, K-1 ) = -AKKP1 / D
*
* Compute columns K-1 and K of the inverse.
*
IF( K.LT.N ) THEN
CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK, 1,
$ ZERO, A( K+1, K ), 1 )
A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1, K ),
$ 1 )
A( K, K-1 ) = A( K, K-1 ) -
$ DDOT( N-K, A( K+1, K ), 1, A( K+1, K-1 ),
$ 1 )
CALL DCOPY( N-K, A( K+1, K-1 ), 1, WORK, 1 )
CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK, 1,
$ ZERO, A( K+1, K-1 ), 1 )
A( K-1, K-1 ) = A( K-1, K-1 ) -
$ DDOT( N-K, WORK, 1, A( K+1, K-1 ), 1 )
END IF
KSTEP = 2
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the trailing
* submatrix A(k-1:n,k-1:n)
*
IF( KP.LT.N )
$ CALL DSWAP( N-KP, A( KP+1, K ), 1, A( KP+1, KP ), 1 )
CALL DSWAP( KP-K-1, A( K+1, K ), 1, A( KP, K+1 ), LDA )
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = A( K, K-1 )
A( K, K-1 ) = A( KP, K-1 )
A( KP, K-1 ) = TEMP
END IF
END IF
*
K = K - KSTEP
GO TO 50
60 CONTINUE
END IF
*
RETURN
*
* End of DSYTRI
*
END