*> \brief \b DSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DSYTF2_ROOK + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DSYTF2_ROOK( 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_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 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) < 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 2013 * *> \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 *> *> November 2013, 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_ROOK( UPLO, N, A, LDA, IPIV, INFO ) * * -- LAPACK computational routine (version 3.5.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2013 * * .. 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, 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_ROOK', -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 * * 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 = 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 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 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 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 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 + 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 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 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 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 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 DSYTF2_ROOK * END