SUBROUTINE ZHETF2( UPLO, N, A, LDA, IPIV, 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( * ) COMPLEX*16 A( LDA, * ) * .. * * Purpose * ======= * * ZHETF2 computes the factorization of a complex Hermitian matrix A * using the Bunch-Kaufman diagonal pivoting method: * * A = U*D*U' or A = L*D*L' * * where U (or L) is a product of permutation and unit upper (lower) * triangular matrices, U' is the conjugate transpose of U, and D is * Hermitian 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. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * Hermitian matrix A is stored: * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) COMPLEX*16 array, dimension (LDA,N) * On entry, the Hermitian 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). * * 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 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. * * INFO (output) 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. * * Further Details * =============== * * 09-29-06 - patch from * Bobby Cheng, MathWorks * * Replace l.210 and l.393 * 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 * * If UPLO = 'U', then A = U*D*U', 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', 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). * * ===================================================================== * * .. 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, D, D11, D22, R1, ROWMAX, $ TT COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, ZDUM * .. * .. External Functions .. LOGICAL LSAME, DISNAN INTEGER IZAMAX DOUBLE PRECISION DLAPY2 EXTERNAL LSAME, IZAMAX, DLAPY2, DISNAN * .. * .. External Subroutines .. EXTERNAL XERBLA, ZDSCAL, ZHER, ZSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) * .. * .. 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( 'ZHETF2', -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' 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 90 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( DBLE( A( K, K ) ) ) * * 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 = IZAMAX( K-1, A( 1, K ), 1 ) COLMAX = CABS1( A( IMAX, K ) ) ELSE COLMAX = ZERO END IF * IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN * * Column K is zero or contains a NaN: set INFO and continue * IF( INFO.EQ.0 ) $ INFO = K KP = K A( K, K ) = DBLE( A( K, 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 + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA ) ROWMAX = CABS1( A( IMAX, JMAX ) ) IF( IMAX.GT.1 ) THEN JMAX = IZAMAX( IMAX-1, A( 1, IMAX ), 1 ) ROWMAX = MAX( ROWMAX, CABS1( 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( DBLE( 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 ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) DO 20 J = KP + 1, KK - 1 T = DCONJG( A( J, KK ) ) A( J, KK ) = DCONJG( A( KP, J ) ) A( KP, J ) = T 20 CONTINUE A( KP, KK ) = DCONJG( A( KP, KK ) ) R1 = DBLE( A( KK, KK ) ) A( KK, KK ) = DBLE( A( KP, KP ) ) A( KP, KP ) = R1 IF( KSTEP.EQ.2 ) THEN A( K, K ) = DBLE( A( K, K ) ) T = A( K-1, K ) A( K-1, K ) = A( KP, K ) A( KP, K ) = T END IF ELSE A( K, K ) = DBLE( A( K, K ) ) IF( KSTEP.EQ.2 ) $ A( K-1, K-1 ) = DBLE( A( K-1, K-1 ) ) 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)' = A - W(k)*1/D(k)*W(k)' * R1 = ONE / DBLE( A( K, K ) ) CALL ZHER( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA ) * * Store U(k) in column k * CALL ZDSCAL( 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) )' * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' * IF( K.GT.2 ) THEN * D = DLAPY2( DBLE( A( K-1, K ) ), $ DIMAG( A( K-1, K ) ) ) D22 = DBLE( A( K-1, K-1 ) ) / D D11 = DBLE( A( K, K ) ) / D TT = ONE / ( D11*D22-ONE ) D12 = A( K-1, K ) / D D = TT / D * DO 40 J = K - 2, 1, -1 WKM1 = D*( D11*A( J, K-1 )-DCONJG( D12 )* $ A( J, K ) ) WK = D*( D22*A( J, K )-D12*A( J, K-1 ) ) DO 30 I = J, 1, -1 A( I, J ) = A( I, J ) - A( I, K )*DCONJG( WK ) - $ A( I, K-1 )*DCONJG( WKM1 ) 30 CONTINUE A( J, K ) = WK A( J, K-1 ) = WKM1 A( J, J ) = DCMPLX( DBLE( A( J, J ) ), 0.0D+0 ) 40 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' 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 50 CONTINUE * * If K > N, exit from loop * IF( K.GT.N ) $ GO TO 90 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( DBLE( A( 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 + IZAMAX( N-K, A( K+1, K ), 1 ) COLMAX = CABS1( A( IMAX, K ) ) ELSE COLMAX = ZERO END IF * IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN * * Column K is zero or contains a NaN: set INFO and continue * IF( INFO.EQ.0 ) $ INFO = K KP = K A( K, K ) = DBLE( A( K, 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 + IZAMAX( IMAX-K, A( IMAX, K ), LDA ) ROWMAX = CABS1( A( IMAX, JMAX ) ) IF( IMAX.LT.N ) THEN JMAX = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 ) ROWMAX = MAX( ROWMAX, CABS1( 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( DBLE( 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 ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) DO 60 J = KK + 1, KP - 1 T = DCONJG( A( J, KK ) ) A( J, KK ) = DCONJG( A( KP, J ) ) A( KP, J ) = T 60 CONTINUE A( KP, KK ) = DCONJG( A( KP, KK ) ) R1 = DBLE( A( KK, KK ) ) A( KK, KK ) = DBLE( A( KP, KP ) ) A( KP, KP ) = R1 IF( KSTEP.EQ.2 ) THEN A( K, K ) = DBLE( A( K, K ) ) T = A( K+1, K ) A( K+1, K ) = A( KP, K ) A( KP, K ) = T END IF ELSE A( K, K ) = DBLE( A( K, K ) ) IF( KSTEP.EQ.2 ) $ A( K+1, K+1 ) = DBLE( A( K+1, K+1 ) ) 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)' = A - W(k)*(1/D(k))*W(k)' * R1 = ONE / DBLE( A( K, K ) ) CALL ZHER( UPLO, N-K, -R1, A( K+1, K ), 1, $ A( K+1, K+1 ), LDA ) * * Store L(k) in column K * CALL ZDSCAL( N-K, R1, 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 - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )' * * where L(k) and L(k+1) are the k-th and (k+1)-th * columns of L * D = DLAPY2( DBLE( A( K+1, K ) ), $ DIMAG( A( K+1, K ) ) ) D11 = DBLE( A( K+1, K+1 ) ) / D D22 = DBLE( A( K, K ) ) / D TT = ONE / ( D11*D22-ONE ) D21 = A( K+1, K ) / D D = TT / D * DO 80 J = K + 2, N WK = D*( D11*A( J, K )-D21*A( J, K+1 ) ) WKP1 = D*( D22*A( J, K+1 )-DCONJG( D21 )* $ A( J, K ) ) DO 70 I = J, N A( I, J ) = A( I, J ) - A( I, K )*DCONJG( WK ) - $ A( I, K+1 )*DCONJG( WKP1 ) 70 CONTINUE A( J, K ) = WK A( J, K+1 ) = WKP1 A( J, J ) = DCMPLX( DBLE( A( J, J ) ), 0.0D+0 ) 80 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 50 * END IF * 90 CONTINUE RETURN * * End of ZHETF2 * END