*> \brief \b ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS). * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLAHEF + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, KB, LDA, LDW, N, NB * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX*16 A( LDA, * ), W( LDW, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLAHEF computes a partial factorization of a complex Hermitian *> 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**H U22**H ) *> *> A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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. *> Note that U**H denotes the conjugate transpose of U. *> *> ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code *> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or *> A22 (if UPLO = 'L'). *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> Specifies whether the upper or lower triangular part of the *> Hermitian 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] NB *> \verbatim *> NB is 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. *> \endverbatim *> *> \param[out] KB *> \verbatim *> KB is INTEGER *> The number of columns of A that were actually factored. *> KB is either NB-1 or NB, or N if N <= NB. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is 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, A contains details of the partial factorization. *> \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': *> Only the last KB elements of IPIV 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 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': *> Only the first KB elements of IPIV 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 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. *> \endverbatim *> *> \param[out] W *> \verbatim *> W is COMPLEX*16 array, dimension (LDW,NB) *> \endverbatim *> *> \param[in] LDW *> \verbatim *> LDW is INTEGER *> The leading dimension of the array W. LDW >= max(1,N). *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16HEcomputational * *> \par Contributors: * ================== *> *> \verbatim *> *> December 2016, Igor Kozachenko, *> Computer Science Division, *> University of California, Berkeley *> \endverbatim * * ===================================================================== SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) * * -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KB, LDA, LDW, N, NB * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX*16 A( LDA, * ), W( LDW, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D+0, 0.0D+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, R1, ROWMAX, T COMPLEX*16 D11, D21, D22, Z * .. * .. External Functions .. LOGICAL LSAME INTEGER IZAMAX EXTERNAL LSAME, IZAMAX * .. * .. External Subroutines .. EXTERNAL ZCOPY, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) ) * .. * .. 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 (note that conjg(W) is actually stored) * * 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 * KSTEP = 1 * * Copy column K of A to column KW of W and update it * CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 ) W( K, KW ) = DBLE( A( K, K ) ) IF( K.LT.N ) THEN CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA, \$ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 ) W( K, KW ) = DBLE( W( K, KW ) ) END IF * * 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( W( K, KW ) ) ) * * 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 = IZAMAX( K-1, W( 1, KW ), 1 ) COLMAX = CABS1( W( IMAX, KW ) ) 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 A( K, K ) = DBLE( A( K, K ) ) ELSE * * ============================================================ * * BEGIN pivot search * * Case(1) IF( ABSAKK.GE.ALPHA*COLMAX ) THEN * * no interchange, use 1-by-1 pivot block * KP = K ELSE * * BEGIN pivot search along IMAX row * * * Copy column IMAX to column KW-1 of W and update it * CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) ) CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, \$ W( IMAX+1, KW-1 ), 1 ) CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 ) IF( K.LT.N ) THEN CALL ZGEMV( 'No transpose', K, N-K, -CONE, \$ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW, \$ CONE, W( 1, KW-1 ), 1 ) W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) ) END IF * * JMAX is the column-index of the largest off-diagonal * element in row IMAX, and ROWMAX is its absolute value. * Determine only ROWMAX. * JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) ROWMAX = CABS1( W( JMAX, KW-1 ) ) IF( IMAX.GT.1 ) THEN JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 ) ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) ) END IF * * Case(2) IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN * * no interchange, use 1-by-1 pivot block * KP = K * * Case(3) ELSE IF( ABS( DBLE( 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 of W * CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) * * Case(4) ELSE * * interchange rows and columns K-1 and IMAX, use 2-by-2 * pivot block * KP = IMAX KSTEP = 2 END IF * * * END pivot search along IMAX row * END IF * * END pivot search * * ============================================================ * * KK is the column of A where pivoting step stopped * KK = K - KSTEP + 1 * * KKW is the column of W which corresponds to column KK of A * KKW = NB + KK - N * * Interchange rows and columns KP and KK. * Updated column KP is already stored in column KKW of W. * IF( KP.NE.KK ) THEN * * Copy non-updated column KK to column KP of submatrix A * at step K. No need to copy element into column K * (or K and K-1 for 2-by-2 pivot) of A, since these columns * will be later overwritten. * A( KP, KP ) = DBLE( A( KK, KK ) ) CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), \$ LDA ) CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA ) IF( KP.GT.1 ) \$ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) * * Interchange rows KK and KP in last K+1 to N columns of A * (columns K (or K and K-1 for 2-by-2 pivot) of A will be * later overwritten). Interchange rows KK and KP * in last KKW to NB columns of W. * IF( K.LT.N ) \$ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), \$ LDA ) CALL ZSWAP( 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(kw) = U(k)*D(k), * * where U(k) is the k-th column of U * * (1) Store subdiag. elements of column U(k) * and 1-by-1 block D(k) in column k of A. * (NOTE: Diagonal element U(k,k) is a UNIT element * and not stored) * A(k,k) := D(k,k) = W(k,kw) * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k) * * (NOTE: No need to use for Hermitian matrix * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal * element D(k,k) from W (potentially saves only one load)) CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) IF( K.GT.1 ) THEN * * (NOTE: No need to check if A(k,k) is NOT ZERO, * since that was ensured earlier in pivot search: * case A(k,k) = 0 falls into 2x2 pivot case(4)) * R1 = ONE / DBLE( A( K, K ) ) CALL ZDSCAL( K-1, R1, A( 1, K ), 1 ) * * (2) Conjugate column W(kw) * CALL ZLACGV( K-1, W( 1, KW ), 1 ) END IF * ELSE * * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold * * ( W(kw-1) W(kw) ) = ( 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 * * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2 * block D(k-1:k,k-1:k) in columns k-1 and k of A. * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT * block and not stored) * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw) * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) = * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) ) * IF( K.GT.2 ) THEN * * Factor out the columns of the inverse of 2-by-2 pivot * block D, so that each column contains 1, to reduce the * number of FLOPS when we multiply panel * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1). * * D**(-1) = ( d11 cj(d21) )**(-1) = * ( d21 d22 ) * * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) = * ( (-d21) ( d11 ) ) * * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) * * * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) = * ( ( -1 ) ( d11/conj(d21) ) ) * * = 1/(|d21|**2) * 1/(D22*D11-1) * * * * ( d21*( D11 ) conj(d21)*( -1 ) ) = * ( ( -1 ) ( D22 ) ) * * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) = * ( ( -1 ) ( D22 ) ) * * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) = * ( ( -1 ) ( D22 ) ) * * = ( conj(D21)*( D11 ) D21*( -1 ) ) * ( ( -1 ) ( D22 ) ), * * where D11 = d22/d21, * D22 = d11/conj(d21), * D21 = T/d21, * T = 1/(D22*D11-1). * * (NOTE: No need to check for division by ZERO, * since that was ensured earlier in pivot search: * (a) d21 != 0, since in 2x2 pivot case(4) * |d21| should be larger than |d11| and |d22|; * (b) (D22*D11 - 1) != 0, since from (a), * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.) * D21 = W( K-1, KW ) D11 = W( K, KW ) / DCONJG( D21 ) D22 = W( K-1, KW-1 ) / D21 T = ONE / ( DBLE( D11*D22 )-ONE ) D21 = T / D21 * * Update elements in columns A(k-1) and A(k) as * dot products of rows of ( W(kw-1) W(kw) ) and columns * of D**(-1) * DO 20 J = 1, K - 2 A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) A( J, K ) = DCONJG( 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 ) * * (2) Conjugate columns W(kw) and W(kw-1) * CALL ZLACGV( K-1, W( 1, KW ), 1 ) CALL ZLACGV( K-2, W( 1, KW-1 ), 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 * * 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**H = A11 - U12*W**H * * computing blocks of NB columns at a time (note that conjg(W) is * actually stored) * 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 A( JJ, JJ ) = DBLE( A( JJ, JJ ) ) CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE, \$ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE, \$ A( J, JJ ), 1 ) A( JJ, JJ ) = DBLE( A( JJ, JJ ) ) 40 CONTINUE * * Update the rectangular superdiagonal block * CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K, \$ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW, \$ CONE, A( 1, J ), LDA ) 50 CONTINUE * * Put U12 in standard form by partially undoing the interchanges * in columns k+1:n looping backwards from k+1 to n * J = K + 1 60 CONTINUE * * Undo the interchanges (if any) of rows JJ and JP at each * step J * * (Here, J is a diagonal index) JJ = J JP = IPIV( J ) IF( JP.LT.0 ) THEN JP = -JP * (Here, J is a diagonal index) J = J + 1 END IF * (NOTE: Here, J is used to determine row length. Length N-J+1 * of the rows to swap back doesn't include diagonal element) J = J + 1 IF( JP.NE.JJ .AND. J.LE.N ) \$ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) IF( J.LT.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 (note that conjg(W) is actually stored) * * 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 * KSTEP = 1 * * Copy column K of A to column K of W and update it * W( K, K ) = DBLE( A( K, K ) ) IF( K.LT.N ) \$ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 ) CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA, \$ W( K, 1 ), LDW, CONE, W( K, K ), 1 ) W( K, K ) = DBLE( W( K, 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( DBLE( W( 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 + IZAMAX( N-K, W( K+1, K ), 1 ) COLMAX = CABS1( W( 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 A( K, K ) = DBLE( A( K, K ) ) ELSE * * ============================================================ * * BEGIN pivot search * * Case(1) IF( ABSAKK.GE.ALPHA*COLMAX ) THEN * * no interchange, use 1-by-1 pivot block * KP = K ELSE * * BEGIN pivot search along IMAX row * * * Copy column IMAX to column K+1 of W and update it * CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 ) W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) ) IF( IMAX.LT.N ) \$ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1, \$ W( IMAX+1, K+1 ), 1 ) CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), \$ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ), \$ 1 ) W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) ) * * JMAX is the column-index of the largest off-diagonal * element in row IMAX, and ROWMAX is its absolute value. * Determine only ROWMAX. * JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 ) ROWMAX = CABS1( W( JMAX, K+1 ) ) IF( IMAX.LT.N ) THEN JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 ) ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) ) END IF * * Case(2) IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN * * no interchange, use 1-by-1 pivot block * KP = K * * Case(3) ELSE IF( ABS( DBLE( 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 of W * CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) * * Case(4) ELSE * * interchange rows and columns K+1 and IMAX, use 2-by-2 * pivot block * KP = IMAX KSTEP = 2 END IF * * * END pivot search along IMAX row * END IF * * END pivot search * * ============================================================ * * KK is the column of A where pivoting step stopped * KK = K + KSTEP - 1 * * Interchange rows and columns KP and KK. * Updated column KP is already stored in column KK of W. * IF( KP.NE.KK ) THEN * * Copy non-updated column KK to column KP of submatrix A * at step K. No need to copy element into column K * (or K and K+1 for 2-by-2 pivot) of A, since these columns * will be later overwritten. * A( KP, KP ) = DBLE( A( KK, KK ) ) CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), \$ LDA ) CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA ) IF( KP.LT.N ) \$ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) * * Interchange rows KK and KP in first K-1 columns of A * (columns K (or K and K+1 for 2-by-2 pivot) of A will be * later overwritten). Interchange rows KK and KP * in first KK columns of W. * IF( K.GT.1 ) \$ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) CALL ZSWAP( 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 * * (1) Store subdiag. elements of column L(k) * and 1-by-1 block D(k) in column k of A. * (NOTE: Diagonal element L(k,k) is a UNIT element * and not stored) * A(k,k) := D(k,k) = W(k,k) * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k) * * (NOTE: No need to use for Hermitian matrix * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal * element D(k,k) from W (potentially saves only one load)) CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) IF( K.LT.N ) THEN * * (NOTE: No need to check if A(k,k) is NOT ZERO, * since that was ensured earlier in pivot search: * case A(k,k) = 0 falls into 2x2 pivot case(4)) * R1 = ONE / DBLE( A( K, K ) ) CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 ) * * (2) Conjugate column W(k) * CALL ZLACGV( N-K, W( 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 * * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2 * block D(k:k+1,k:k+1) in columns k and k+1 of A. * (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT * block and not stored) * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1) * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) = * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) ) * IF( K.LT.N-1 ) THEN * * Factor out the columns of the inverse of 2-by-2 pivot * block D, so that each column contains 1, to reduce the * number of FLOPS when we multiply panel * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1). * * D**(-1) = ( d11 cj(d21) )**(-1) = * ( d21 d22 ) * * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) = * ( (-d21) ( d11 ) ) * * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) * * * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) = * ( ( -1 ) ( d11/conj(d21) ) ) * * = 1/(|d21|**2) * 1/(D22*D11-1) * * * * ( d21*( D11 ) conj(d21)*( -1 ) ) = * ( ( -1 ) ( D22 ) ) * * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) = * ( ( -1 ) ( D22 ) ) * * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) = * ( ( -1 ) ( D22 ) ) * * = ( conj(D21)*( D11 ) D21*( -1 ) ) * ( ( -1 ) ( D22 ) ), * * where D11 = d22/d21, * D22 = d11/conj(d21), * D21 = T/d21, * T = 1/(D22*D11-1). * * (NOTE: No need to check for division by ZERO, * since that was ensured earlier in pivot search: * (a) d21 != 0, since in 2x2 pivot case(4) * |d21| should be larger than |d11| and |d22|; * (b) (D22*D11 - 1) != 0, since from (a), * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.) * D21 = W( K+1, K ) D11 = W( K+1, K+1 ) / D21 D22 = W( K, K ) / DCONJG( D21 ) T = ONE / ( DBLE( D11*D22 )-ONE ) D21 = T / D21 * * Update elements in columns A(k) and A(k+1) as * dot products of rows of ( W(k) W(k+1) ) and columns * of D**(-1) * DO 80 J = K + 2, N A( J, K ) = DCONJG( 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 ) * * (2) Conjugate columns W(k) and W(k+1) * CALL ZLACGV( N-K, W( K+1, K ), 1 ) CALL ZLACGV( N-K-1, W( K+2, K+1 ), 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**H = A22 - L21*W**H * * computing blocks of NB columns at a time (note that conjg(W) is * actually stored) * 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 A( JJ, JJ ) = DBLE( A( JJ, JJ ) ) CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE, \$ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE, \$ A( JJ, JJ ), 1 ) A( JJ, JJ ) = DBLE( A( JJ, JJ ) ) 100 CONTINUE * * Update the rectangular subdiagonal block * IF( J+JB.LE.N ) \$ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, \$ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ), \$ LDW, CONE, A( J+JB, J ), LDA ) 110 CONTINUE * * Put L21 in standard form by partially undoing the interchanges * of rows in columns 1:k-1 looping backwards from k-1 to 1 * J = K - 1 120 CONTINUE * * Undo the interchanges (if any) of rows JJ and JP at each * step J * * (Here, J is a diagonal index) JJ = J JP = IPIV( J ) IF( JP.LT.0 ) THEN JP = -JP * (Here, J is a diagonal index) J = J - 1 END IF * (NOTE: Here, J is used to determine row length. Length J * of the rows to swap back doesn't include diagonal element) J = J - 1 IF( JP.NE.JJ .AND. J.GE.1 ) \$ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) IF( J.GT.1 ) \$ GO TO 120 * * Set KB to the number of columns factorized * KB = K - 1 * END IF RETURN * * End of ZLAHEF * END