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Diffstat (limited to '2.3-1/src/fortran/lapack/dlasyf.f')
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diff --git a/2.3-1/src/fortran/lapack/dlasyf.f b/2.3-1/src/fortran/lapack/dlasyf.f new file mode 100644 index 00000000..67b9c147 --- /dev/null +++ b/2.3-1/src/fortran/lapack/dlasyf.f @@ -0,0 +1,587 @@ + SUBROUTINE DLASYF( 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( * ) + DOUBLE PRECISION A( LDA, * ), W( LDW, * ) +* .. +* +* Purpose +* ======= +* +* DLASYF 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. +* +* DLASYF is an auxiliary routine called by DSYTRF. 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) 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, 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) DOUBLE PRECISION 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 .. + 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 .. + INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP, + $ KSTEP, KW + DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1, + $ ROWMAX, T +* .. +* .. External Functions .. + LOGICAL LSAME + INTEGER IDAMAX + EXTERNAL LSAME, IDAMAX +* .. +* .. External Subroutines .. + EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP +* .. +* .. 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 DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 ) + IF( K.LT.N ) + $ CALL DGEMV( '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 = IDAMAX( 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 DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) + CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, + $ W( IMAX+1, KW-1 ), 1 ) + IF( K.LT.N ) + $ CALL DGEMV( '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 + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) + ROWMAX = ABS( W( JMAX, KW-1 ) ) + IF( IMAX.GT.1 ) THEN + JMAX = IDAMAX( 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 DCOPY( 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 DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), + $ LDA ) + CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 ) +* +* Interchange rows KK and KP in last KK columns of A and W +* + CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA ) + CALL DSWAP( 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 DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) + R1 = ONE / A( K, K ) + CALL DSCAL( 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 DGEMV( '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 DGEMM( '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 DSWAP( 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 DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 ) + CALL DGEMV( '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 + IDAMAX( 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 DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) + CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ), + $ 1 ) + CALL DGEMV( '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 + IDAMAX( IMAX-K, W( K, K+1 ), 1 ) + ROWMAX = ABS( W( JMAX, K+1 ) ) + IF( IMAX.LT.N ) THEN + JMAX = IMAX + IDAMAX( 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 DCOPY( 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 DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA ) + CALL DCOPY( 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 DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) + CALL DSWAP( 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 DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) + IF( K.LT.N ) THEN + R1 = ONE / A( K, K ) + CALL DSCAL( 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 DGEMV( '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 DGEMM( '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 DSWAP( 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 DLASYF +* + END |