*> \brief \b ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZHETF2_RK + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, LDA, N * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX*16 A( LDA, * ), E ( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> ZHETF2_RK computes the factorization of a complex Hermitian matrix A *> using the bounded Bunch-Kaufman (rook) diagonal pivoting method: *> *> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), *> *> where U (or L) is unit upper (or lower) triangular matrix, *> U**H (or L**H) is the conjugate of U (or L), P is a permutation *> matrix, P**T is the transpose of P, 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. *> For more information see Further Details section. *> \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,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, contains: *> a) ONLY diagonal elements of the Hermitian block diagonal *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); *> (superdiagonal (or subdiagonal) elements of D *> are stored on exit in array E), and *> b) If UPLO = 'U': factor U in the superdiagonal part of A. *> If UPLO = 'L': factor L in the subdiagonal part of A. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] E *> \verbatim *> E is COMPLEX*16 array, dimension (N) *> On exit, contains the superdiagonal (or subdiagonal) *> elements of the Hermitian block diagonal matrix D *> with 1-by-1 or 2-by-2 diagonal blocks, where *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. *> *> NOTE: For 1-by-1 diagonal block D(k), where *> 1 <= k <= N, the element E(k) is set to 0 in both *> UPLO = 'U' or UPLO = 'L' cases. *> \endverbatim *> *> \param[out] IPIV *> \verbatim *> IPIV is INTEGER array, dimension (N) *> IPIV describes the permutation matrix P in the factorization *> of matrix A as follows. The absolute value of IPIV(k) *> represents the index of row and column that were *> interchanged with the k-th row and column. The value of UPLO *> describes the order in which the interchanges were applied. *> Also, the sign of IPIV represents the block structure of *> the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 *> diagonal blocks which correspond to 1 or 2 interchanges *> at each factorization step. For more info see Further *> Details section. *> *> If UPLO = 'U', *> ( in factorization order, k decreases from N to 1 ): *> a) A single positive entry IPIV(k) > 0 means: *> D(k,k) is a 1-by-1 diagonal block. *> If IPIV(k) != k, rows and columns k and IPIV(k) were *> interchanged in the matrix A(1:N,1:N); *> If IPIV(k) = k, no interchange occurred. *> *> b) A pair of consecutive negative entries *> IPIV(k) < 0 and IPIV(k-1) < 0 means: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block. *> (NOTE: negative entries in IPIV appear ONLY in pairs). *> 1) If -IPIV(k) != k, rows and columns *> k and -IPIV(k) were interchanged *> in the matrix A(1:N,1:N). *> If -IPIV(k) = k, no interchange occurred. *> 2) If -IPIV(k-1) != k-1, rows and columns *> k-1 and -IPIV(k-1) were interchanged *> in the matrix A(1:N,1:N). *> If -IPIV(k-1) = k-1, no interchange occurred. *> *> c) In both cases a) and b), always ABS( IPIV(k) ) <= k. *> *> d) NOTE: Any entry IPIV(k) is always NONZERO on output. *> *> If UPLO = 'L', *> ( in factorization order, k increases from 1 to N ): *> a) A single positive entry IPIV(k) > 0 means: *> D(k,k) is a 1-by-1 diagonal block. *> If IPIV(k) != k, rows and columns k and IPIV(k) were *> interchanged in the matrix A(1:N,1:N). *> If IPIV(k) = k, no interchange occurred. *> *> b) A pair of consecutive negative entries *> IPIV(k) < 0 and IPIV(k+1) < 0 means: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block. *> (NOTE: negative entries in IPIV appear ONLY in pairs). *> 1) If -IPIV(k) != k, rows and columns *> k and -IPIV(k) were interchanged *> in the matrix A(1:N,1:N). *> If -IPIV(k) = k, no interchange occurred. *> 2) If -IPIV(k+1) != k+1, rows and columns *> k-1 and -IPIV(k-1) were interchanged *> in the matrix A(1:N,1:N). *> If -IPIV(k+1) = k+1, no interchange occurred. *> *> c) In both cases a) and b), always ABS( IPIV(k) ) >= k. *> *> d) NOTE: Any entry IPIV(k) is always NONZERO on output. *> \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, the matrix A is singular, because: *> If UPLO = 'U': column k in the upper *> triangular part of A contains all zeros. *> If UPLO = 'L': column k in the lower *> triangular part of A contains all zeros. *> *> Therefore D(k,k) is exactly zero, and superdiagonal *> elements of column k of U (or subdiagonal elements of *> column k of L ) are all zeros. 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. *> *> NOTE: INFO only stores the first occurrence of *> a singularity, any subsequent occurrence of singularity *> is not stored in INFO even though the factorization *> always completes. *> \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 Further Details: * ===================== *> *> \verbatim *> TODO: put further details *> \endverbatim * *> \par Contributors: * ================== *> *> \verbatim *> *> December 2016, 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 ZHETF2_RK( UPLO, N, A, LDA, E, IPIV, 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, LDA, N * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX*16 A( LDA, * ), E( * ) * .. * * ====================================================================== * * .. 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 ) COMPLEX*16 CZERO PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL DONE, UPPER INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP, \$ P DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, DTEMP, \$ ROWMAX, TT, SFMIN COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, Z * .. * .. External Functions .. * LOGICAL LSAME INTEGER IZAMAX DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL LSAME, IZAMAX, DLAMCH, DLAPY2 * .. * .. 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( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) ) * .. * .. 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_RK', -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**H using the upper triangle of A * * Initilize the first entry of array E, where superdiagonal * elements of D are stored * E( 1 ) = CZERO * * 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 34 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( DBLE( 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 = IZAMAX( K-1, A( 1, K ), 1 ) COLMAX = CABS1( 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 A( K, K ) = DBLE( A( K, K ) ) * * Set E( K ) to zero * IF( K.GT.1 ) \$ E( K ) = CZERO * ELSE * * ============================================================ * * BEGIN pivot search * * Case(1) * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX * (used to handle NaN and Inf) * 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 + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), \$ LDA ) ROWMAX = CABS1( A( IMAX, JMAX ) ) ELSE ROWMAX = ZERO END IF * IF( IMAX.GT.1 ) THEN ITEMP = IZAMAX( IMAX-1, A( 1, IMAX ), 1 ) DTEMP = CABS1( A( ITEMP, IMAX ) ) IF( DTEMP.GT.ROWMAX ) THEN ROWMAX = DTEMP JMAX = ITEMP END IF END IF * * Case(2) * Equivalent to testing for * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX * (used to handle NaN and Inf) * IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) ) \$ .LT.ALPHA*ROWMAX ) ) THEN * * interchange rows and columns K and IMAX, * use 1-by-1 pivot block * KP = IMAX DONE = .TRUE. * * Case(3) * 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. * * Case(4) ELSE * * Pivot not found: set params and repeat * P = IMAX COLMAX = ROWMAX IMAX = JMAX END IF * * END pivot search loop body * IF( .NOT.DONE ) GOTO 12 * END IF * * END pivot search * * ============================================================ * * KK is the column of A where pivoting step stopped * KK = K - KSTEP + 1 * * For only a 2x2 pivot, interchange rows and columns K and P * in the leading submatrix A(1:k,1:k) * IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN * (1) Swap columnar parts IF( P.GT.1 ) \$ CALL ZSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 ) * (2) Swap and conjugate middle parts DO 14 J = P + 1, K - 1 T = DCONJG( A( J, K ) ) A( J, K ) = DCONJG( A( P, J ) ) A( P, J ) = T 14 CONTINUE * (3) Swap and conjugate corner elements at row-col interserction A( P, K ) = DCONJG( A( P, K ) ) * (4) Swap diagonal elements at row-col intersection R1 = DBLE( A( K, K ) ) A( K, K ) = DBLE( A( P, P ) ) A( P, P ) = R1 * * Convert upper triangle of A into U form by applying * the interchanges in columns k+1:N. * IF( K.LT.N ) \$ CALL ZSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA ) * END IF * * For both 1x1 and 2x2 pivots, interchange rows and * columns KK and KP in the leading submatrix A(1:k,1:k) * IF( KP.NE.KK ) THEN * (1) Swap columnar parts IF( KP.GT.1 ) \$ CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) * (2) Swap and conjugate middle parts DO 15 J = KP + 1, KK - 1 T = DCONJG( A( J, KK ) ) A( J, KK ) = DCONJG( A( KP, J ) ) A( KP, J ) = T 15 CONTINUE * (3) Swap and conjugate corner elements at row-col interserction A( KP, KK ) = DCONJG( A( KP, KK ) ) * (4) Swap diagonal elements at row-col intersection R1 = DBLE( A( KK, KK ) ) A( KK, KK ) = DBLE( A( KP, KP ) ) A( KP, KP ) = R1 * IF( KSTEP.EQ.2 ) THEN * (*) Make sure that diagonal element of pivot is real A( K, K ) = DBLE( A( K, K ) ) * (5) Swap row elements T = A( K-1, K ) A( K-1, K ) = A( KP, K ) A( KP, K ) = T END IF * * Convert upper triangle of A into U form by applying * the interchanges in columns k+1:N. * IF( K.LT.N ) \$ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), \$ LDA ) * ELSE * (*) Make sure that diagonal element of pivot is real 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 * 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( DBLE( 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 / DBLE( A( K, K ) ) CALL ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) * * Store U(k) in column k * CALL ZDSCAL( K-1, D11, A( 1, K ), 1 ) ELSE * * Store L(k) in column K * D11 = DBLE( 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 ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) END IF * * Store the superdiagonal element of D in array E * E( K ) = CZERO * 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 * D = |A12| D = DLAPY2( DBLE( A( K-1, K ) ), \$ DIMAG( A( K-1, K ) ) ) D11 = A( K, K ) / D D22 = A( K-1, K-1 ) / D D12 = A( K-1, K ) / D TT = ONE / ( D11*D22-ONE ) * DO 30 J = K - 2, 1, -1 * * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J * WKM1 = TT*( D11*A( J, K-1 )-DCONJG( D12 )* \$ A( J, K ) ) WK = TT*( D22*A( J, K )-D12*A( J, K-1 ) ) * * Perform a rank-2 update of A(1:k-2,1:k-2) * DO 20 I = J, 1, -1 A( I, J ) = A( I, J ) - \$ ( A( I, K ) / D )*DCONJG( WK ) - \$ ( A( I, K-1 ) / D )*DCONJG( WKM1 ) 20 CONTINUE * * Store U(k) and U(k-1) in cols k and k-1 for row J * A( J, K ) = WK / D A( J, K-1 ) = WKM1 / D * (*) Make sure that diagonal element of pivot is real A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO ) * 30 CONTINUE * END IF * * Copy superdiagonal elements of D(K) to E(K) and * ZERO out superdiagonal entry of A * E( K ) = A( K-1, K ) E( K-1 ) = CZERO A( K-1, K ) = CZERO * END IF * * End column K is nonsingular * 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 * 34 CONTINUE * ELSE * * Factorize A as L*D*L**H using the lower triangle of A * * Initilize the unused last entry of the subdiagonal array E. * E( N ) = CZERO * * 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 64 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( DBLE( 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 + IZAMAX( N-K, A( K+1, K ), 1 ) COLMAX = CABS1( 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 A( K, K ) = DBLE( A( K, K ) ) * * Set E( K ) to zero * IF( K.LT.N ) \$ E( K ) = CZERO * ELSE * * ============================================================ * * BEGIN pivot search * * Case(1) * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX * (used to handle NaN and Inf) * 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 + IZAMAX( IMAX-K, A( IMAX, K ), LDA ) ROWMAX = CABS1( A( IMAX, JMAX ) ) ELSE ROWMAX = ZERO END IF * IF( IMAX.LT.N ) THEN ITEMP = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), \$ 1 ) DTEMP = CABS1( A( ITEMP, IMAX ) ) IF( DTEMP.GT.ROWMAX ) THEN ROWMAX = DTEMP JMAX = ITEMP END IF END IF * * Case(2) * Equivalent to testing for * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX * (used to handle NaN and Inf) * IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) ) \$ .LT.ALPHA*ROWMAX ) ) THEN * * interchange rows and columns K and IMAX, * use 1-by-1 pivot block * KP = IMAX DONE = .TRUE. * * Case(3) * 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. * * Case(4) ELSE * * Pivot not found: set params and repeat * P = IMAX COLMAX = ROWMAX IMAX = JMAX END IF * * * END pivot search loop body * IF( .NOT.DONE ) GOTO 42 * END IF * * END pivot search * * ============================================================ * * KK is the column of A where pivoting step stopped * KK = K + KSTEP - 1 * * For only a 2x2 pivot, interchange rows and columns K and P * in the trailing submatrix A(k:n,k:n) * IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN * (1) Swap columnar parts IF( P.LT.N ) \$ CALL ZSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 ) * (2) Swap and conjugate middle parts DO 44 J = K + 1, P - 1 T = DCONJG( A( J, K ) ) A( J, K ) = DCONJG( A( P, J ) ) A( P, J ) = T 44 CONTINUE * (3) Swap and conjugate corner elements at row-col interserction A( P, K ) = DCONJG( A( P, K ) ) * (4) Swap diagonal elements at row-col intersection R1 = DBLE( A( K, K ) ) A( K, K ) = DBLE( A( P, P ) ) A( P, P ) = R1 * * Convert lower triangle of A into L form by applying * the interchanges in columns 1:k-1. * IF ( K.GT.1 ) \$ CALL ZSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA ) * END IF * * For both 1x1 and 2x2 pivots, interchange rows and * columns KK and KP in the trailing submatrix A(k:n,k:n) * IF( KP.NE.KK ) THEN * (1) Swap columnar parts IF( KP.LT.N ) \$ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) * (2) Swap and conjugate middle parts DO 45 J = KK + 1, KP - 1 T = DCONJG( A( J, KK ) ) A( J, KK ) = DCONJG( A( KP, J ) ) A( KP, J ) = T 45 CONTINUE * (3) Swap and conjugate corner elements at row-col interserction A( KP, KK ) = DCONJG( A( KP, KK ) ) * (4) Swap diagonal elements at row-col intersection R1 = DBLE( A( KK, KK ) ) A( KK, KK ) = DBLE( A( KP, KP ) ) A( KP, KP ) = R1 * IF( KSTEP.EQ.2 ) THEN * (*) Make sure that diagonal element of pivot is real A( K, K ) = DBLE( A( K, K ) ) * (5) Swap row elements T = A( K+1, K ) A( K+1, K ) = A( KP, K ) A( KP, K ) = T END IF * * Convert lower triangle of A into L form by applying * the interchanges in columns 1:k-1. * IF ( K.GT.1 ) \$ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) * ELSE * (*) Make sure that diagonal element of pivot is real 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 of A 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 * * Handle division by a small number * IF( ABS( DBLE( 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 / DBLE( A( K, K ) ) CALL ZHER( UPLO, N-K, -D11, A( K+1, K ), 1, \$ A( K+1, K+1 ), LDA ) * * Store L(k) in column k * CALL ZDSCAL( N-K, D11, A( K+1, K ), 1 ) ELSE * * Store L(k) in column k * D11 = DBLE( 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 ZHER( UPLO, N-K, -D11, A( K+1, K ), 1, \$ A( K+1, K+1 ), LDA ) END IF * * Store the subdiagonal element of D in array E * E( K ) = CZERO * 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 * D = |A21| 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 D21 = A( K+1, K ) / D TT = ONE / ( D11*D22-ONE ) * DO 60 J = K + 2, N * * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J * WK = TT*( D11*A( J, K )-D21*A( J, K+1 ) ) WKP1 = TT*( D22*A( J, K+1 )-DCONJG( D21 )* \$ 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 ) / D )*DCONJG( WK ) - \$ ( A( I, K+1 ) / D )*DCONJG( WKP1 ) 50 CONTINUE * * Store L(k) and L(k+1) in cols k and k+1 for row J * A( J, K ) = WK / D A( J, K+1 ) = WKP1 / D * (*) Make sure that diagonal element of pivot is real A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO ) * 60 CONTINUE * END IF * * Copy subdiagonal elements of D(K) to E(K) and * ZERO out subdiagonal entry of A * E( K ) = A( K+1, K ) E( K+1 ) = CZERO A( K+1, K ) = CZERO * END IF * * End column K is nonsingular * 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 * 64 CONTINUE * END IF * RETURN * * End of ZHETF2_RK * END