SUBROUTINE SLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KB, LDA, LDW, N, NB * .. * .. Array Arguments .. INTEGER IPIV( * ) REAL A( LDA, * ), W( LDW, * ) * .. * * Purpose * ======= * * SLASYF computes a partial factorization of a real symmetric matrix A * using the Bunch-Kaufman diagonal pivoting method. The partial * factorization has the form: * * A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: * ( 0 U22 ) ( 0 D ) ( U12' U22' ) * * A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L' * ( L21 I ) ( 0 A22 ) ( 0 I ) * * where the order of D is at most NB. The actual order is returned in * the argument KB, and is either NB or NB-1, or N if N <= NB. * * SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code * (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or * A22 (if UPLO = 'L'). * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * symmetric matrix A is stored: * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NB (input) INTEGER * The maximum number of columns of the matrix A that should be * factored. NB should be at least 2 to allow for 2-by-2 pivot * blocks. * * KB (output) INTEGER * The number of columns of A that were actually factored. * KB is either NB-1 or NB, or N if N <= NB. * * A (input/output) 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, A contains details of the partial factorization. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * IPIV (output) INTEGER array, dimension (N) * Details of the interchanges and the block structure of D. * If UPLO = 'U', only the last KB elements of IPIV are set; * if UPLO = 'L', only the first KB elements are set. * * 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 UPLO = 'U' and 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' and 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. * * W (workspace) REAL array, dimension (LDW,NB) * * LDW (input) INTEGER * The leading dimension of the array W. LDW >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * > 0: if INFO = k, D(k,k) is exactly zero. The factorization * has been completed, but the block diagonal matrix D is * exactly singular. * * ===================================================================== * * .. 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 .. INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP, $ KSTEP, KW REAL ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1, $ ROWMAX, T * .. * .. External Functions .. LOGICAL LSAME INTEGER ISAMAX EXTERNAL LSAME, ISAMAX * .. * .. External Subroutines .. EXTERNAL SCOPY, SGEMM, SGEMV, SSCAL, SSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT * .. * .. Executable Statements .. * INFO = 0 * * Initialize ALPHA for use in choosing pivot block size. * ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT * IF( LSAME( UPLO, 'U' ) ) THEN * * Factorize the trailing columns of A using the upper triangle * of A and working backwards, and compute the matrix W = U12*D * for use in updating A11 * * K is the main loop index, decreasing from N in steps of 1 or 2 * * KW is the column of W which corresponds to column K of A * K = N 10 CONTINUE KW = NB + K - N * * Exit from loop * IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 ) $ GO TO 30 * * Copy column K of A to column KW of W and update it * CALL SCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 ) IF( K.LT.N ) $ CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), LDA, $ W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 ) * 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( W( K, KW ) ) * * IMAX is the row-index of the largest off-diagonal element in * column K, and COLMAX is its absolute value * IF( K.GT.1 ) THEN IMAX = ISAMAX( K-1, W( 1, KW ), 1 ) COLMAX = ABS( W( IMAX, KW ) ) 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 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN * * no interchange, use 1-by-1 pivot block * KP = K ELSE * * Copy column IMAX to column KW-1 of W and update it * CALL SCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) CALL SCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, $ W( IMAX+1, KW-1 ), 1 ) IF( K.LT.N ) $ CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), $ LDA, W( IMAX, KW+1 ), LDW, ONE, $ W( 1, KW-1 ), 1 ) * * JMAX is the column-index of the largest off-diagonal * element in row IMAX, and ROWMAX is its absolute value * JMAX = IMAX + ISAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) ROWMAX = ABS( W( JMAX, KW-1 ) ) IF( IMAX.GT.1 ) THEN JMAX = ISAMAX( IMAX-1, W( 1, KW-1 ), 1 ) ROWMAX = MAX( ROWMAX, ABS( W( JMAX, KW-1 ) ) ) END IF * IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN * * no interchange, use 1-by-1 pivot block * KP = K ELSE IF( ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN * * interchange rows and columns K and IMAX, use 1-by-1 * pivot block * KP = IMAX * * copy column KW-1 of W to column KW * CALL SCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) 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 KKW = NB + KK - N * * Updated column KP is already stored in column KKW of W * IF( KP.NE.KK ) THEN * * Copy non-updated column KK to column KP * A( KP, K ) = A( KK, K ) CALL SCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), $ LDA ) CALL SCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 ) * * Interchange rows KK and KP in last KK columns of A and W * CALL SSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA ) CALL SSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), $ LDW ) END IF * IF( KSTEP.EQ.1 ) THEN * * 1-by-1 pivot block D(k): column KW of W now holds * * W(k) = U(k)*D(k) * * where U(k) is the k-th column of U * * Store U(k) in column k of A * CALL SCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) R1 = ONE / A( K, K ) CALL SSCAL( K-1, R1, A( 1, K ), 1 ) ELSE * * 2-by-2 pivot block D(k): columns KW and KW-1 of W 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 * IF( K.GT.2 ) THEN * * Store U(k) and U(k-1) in columns k and k-1 of A * D21 = W( K-1, KW ) D11 = W( K, KW ) / D21 D22 = W( K-1, KW-1 ) / D21 T = ONE / ( D11*D22-ONE ) D21 = T / D21 DO 20 J = 1, K - 2 A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) ) 20 CONTINUE END IF * * Copy D(k) to A * A( K-1, K-1 ) = W( K-1, KW-1 ) A( K-1, K ) = W( K-1, KW ) A( K, K ) = W( K, KW ) 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 * 30 CONTINUE * * Update the upper triangle of A11 (= A(1:k,1:k)) as * * A11 := A11 - U12*D*U12' = A11 - U12*W' * * computing blocks of NB columns at a time * DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB JB = MIN( NB, K-J+1 ) * * Update the upper triangle of the diagonal block * DO 40 JJ = J, J + JB - 1 CALL SGEMV( 'No transpose', JJ-J+1, N-K, -ONE, $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE, $ A( J, JJ ), 1 ) 40 CONTINUE * * Update the rectangular superdiagonal block * CALL SGEMM( 'No transpose', 'Transpose', J-1, JB, N-K, -ONE, $ A( 1, K+1 ), LDA, W( J, KW+1 ), LDW, ONE, $ A( 1, J ), LDA ) 50 CONTINUE * * Put U12 in standard form by partially undoing the interchanges * in columns k+1:n * J = K + 1 60 CONTINUE JJ = J JP = IPIV( J ) IF( JP.LT.0 ) THEN JP = -JP J = J + 1 END IF J = J + 1 IF( JP.NE.JJ .AND. J.LE.N ) $ CALL SSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) IF( J.LE.N ) $ GO TO 60 * * Set KB to the number of columns factorized * KB = N - K * ELSE * * Factorize the leading columns of A using the lower triangle * of A and working forwards, and compute the matrix W = L21*D * for use in updating A22 * * K is the main loop index, increasing from 1 in steps of 1 or 2 * K = 1 70 CONTINUE * * Exit from loop * IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N ) $ GO TO 90 * * Copy column K of A to column K of W and update it * CALL SCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 ) CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), LDA, $ W( K, 1 ), LDW, ONE, W( K, K ), 1 ) * 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( W( K, K ) ) * * IMAX is the row-index of the largest off-diagonal element in * column K, and COLMAX is its absolute value * IF( K.LT.N ) THEN IMAX = K + ISAMAX( N-K, W( K+1, K ), 1 ) COLMAX = ABS( W( 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 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN * * no interchange, use 1-by-1 pivot block * KP = K ELSE * * Copy column IMAX to column K+1 of W and update it * CALL SCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) CALL SCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ), $ 1 ) CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), $ LDA, W( IMAX, 1 ), LDW, ONE, W( K, K+1 ), 1 ) * * JMAX is the column-index of the largest off-diagonal * element in row IMAX, and ROWMAX is its absolute value * JMAX = K - 1 + ISAMAX( IMAX-K, W( K, K+1 ), 1 ) ROWMAX = ABS( W( JMAX, K+1 ) ) IF( IMAX.LT.N ) THEN JMAX = IMAX + ISAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 ) ROWMAX = MAX( ROWMAX, ABS( W( JMAX, K+1 ) ) ) END IF * IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN * * no interchange, use 1-by-1 pivot block * KP = K ELSE IF( ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN * * interchange rows and columns K and IMAX, use 1-by-1 * pivot block * KP = IMAX * * copy column K+1 of W to column K * CALL SCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) 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 * * Updated column KP is already stored in column KK of W * IF( KP.NE.KK ) THEN * * Copy non-updated column KK to column KP * A( KP, K ) = A( KK, K ) CALL SCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA ) CALL SCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 ) * * Interchange rows KK and KP in first KK columns of A and W * CALL SSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) CALL SSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) END IF * IF( KSTEP.EQ.1 ) THEN * * 1-by-1 pivot block D(k): column k of W now holds * * W(k) = L(k)*D(k) * * where L(k) is the k-th column of L * * Store L(k) in column k of A * CALL SCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) IF( K.LT.N ) THEN R1 = ONE / A( K, K ) CALL SSCAL( N-K, R1, A( K+1, K ), 1 ) END IF ELSE * * 2-by-2 pivot block D(k): columns k and k+1 of W 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 * IF( K.LT.N-1 ) THEN * * Store L(k) and L(k+1) in columns k and k+1 of A * D21 = W( K+1, K ) D11 = W( K+1, K+1 ) / D21 D22 = W( K, K ) / D21 T = ONE / ( D11*D22-ONE ) D21 = T / D21 DO 80 J = K + 2, N A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) ) A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) ) 80 CONTINUE END IF * * Copy D(k) to A * A( K, K ) = W( K, K ) A( K+1, K ) = W( K+1, K ) A( K+1, K+1 ) = W( K+1, K+1 ) 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 70 * 90 CONTINUE * * Update the lower triangle of A22 (= A(k:n,k:n)) as * * A22 := A22 - L21*D*L21' = A22 - L21*W' * * computing blocks of NB columns at a time * DO 110 J = K, N, NB JB = MIN( NB, N-J+1 ) * * Update the lower triangle of the diagonal block * DO 100 JJ = J, J + JB - 1 CALL SGEMV( 'No transpose', J+JB-JJ, K-1, -ONE, $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE, $ A( JJ, JJ ), 1 ) 100 CONTINUE * * Update the rectangular subdiagonal block * IF( J+JB.LE.N ) $ CALL SGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW, $ ONE, A( J+JB, J ), LDA ) 110 CONTINUE * * Put L21 in standard form by partially undoing the interchanges * in columns 1:k-1 * J = K - 1 120 CONTINUE JJ = J JP = IPIV( J ) IF( JP.LT.0 ) THEN JP = -JP J = J - 1 END IF J = J - 1 IF( JP.NE.JJ .AND. J.GE.1 ) $ CALL SSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) IF( J.GE.1 ) $ GO TO 120 * * Set KB to the number of columns factorized * KB = K - 1 * END IF RETURN * * End of SLASYF * END