numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
| .. | |||
| lapack/SRC/clahef_rk.f | 41414B | -rw-r--r-- |
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*> \brief \b CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CLAHEF_RK + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clahef_rk.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clahef_rk.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clahef_rk.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, * INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, KB, LDA, LDW, N, NB * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX A( LDA, * ), E( * ), W( LDW, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> CLAHEF_RK computes a partial factorization of a complex Hermitian *> matrix A using the bounded Bunch-Kaufman (rook) 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. *> *> CLAHEF_RK is an auxiliary routine called by CHETRF_RK. 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 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 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. *> *> 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 submatrix A(1:N,N-KB+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,N-KB+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 submatrix A(1:N,N-KB+1:N). *> If -IPIV(k-1) = k-1, no interchange occurred. *> *> c) In both cases a) and b) is 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 submatrix A(1:N,1:KB). *> 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 submatrix A(1:N,1:KB). *> 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 submatrix A(1:N,1:KB). *> If -IPIV(k+1) = k+1, no interchange occurred. *> *> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k. *> *> d) NOTE: Any entry IPIV(k) is always NONZERO on output. *> \endverbatim *> *> \param[out] W *> \verbatim *> W is COMPLEX 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, 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. * *> \ingroup lahef_rk * *> \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 *> *> \endverbatim * * ===================================================================== SUBROUTINE CLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, $ INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KB, LDA, LDW, N, NB * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX A( LDA, * ), W( LDW, * ), E( * ) * .. * * ===================================================================== * * .. 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 ) COMPLEX CONE, CZERO PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ), $ CZERO = ( 0.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL DONE INTEGER IMAX, ITEMP, II, J, JMAX, K, KK, KKW, $ KP, KSTEP, KW, P REAL ABSAKK, ALPHA, COLMAX, STEMP, R1, ROWMAX, T, $ SFMIN COMPLEX D11, D21, D22, Z * .. * .. External Functions .. LOGICAL LSAME INTEGER ICAMAX REAL SLAMCH EXTERNAL LSAME, ICAMAX, SLAMCH * .. * .. External Subroutines .. EXTERNAL CCOPY, CSSCAL, CGEMM, CGEMV, CLACGV, $ CSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, CONJG, AIMAG, MAX, MIN, REAL, SQRT * .. * .. Statement Functions .. REAL CABS1 * .. * .. Statement Function definitions .. CABS1( Z ) = ABS( REAL( Z ) ) + ABS( AIMAG( Z ) ) * .. * .. Executable Statements .. * INFO = 0 * * Initialize ALPHA for use in choosing pivot block size. * ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT * * Compute machine safe minimum * SFMIN = SLAMCH( 'S' ) * 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) * * Initialize 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 in steps of 1 or 2 * K = N 10 CONTINUE * * KW is the column of W which corresponds to column K of A * 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 P = K * * Copy column K of A to column KW of W and update it * IF( K.GT.1 ) $ CALL CCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 ) W( K, KW ) = REAL( A( K, K ) ) IF( K.LT.N ) THEN CALL CGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), $ LDA, $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 ) W( K, KW ) = REAL( 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( REAL( 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 = ICAMAX( 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 ) = REAL( W( K, KW ) ) IF( K.GT.1 ) $ CALL CCOPY( K-1, W( 1, KW ), 1, A( 1, K ), 1 ) * * 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 * * Lop until pivot found * DONE = .FALSE. * 12 CONTINUE * * BEGIN pivot search loop body * * * Copy column IMAX to column KW-1 of W and update it * IF( IMAX.GT.1 ) $ CALL CCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, $ KW-1 ), $ 1 ) W( IMAX, KW-1 ) = REAL( A( IMAX, IMAX ) ) * CALL CCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, $ W( IMAX+1, KW-1 ), 1 ) CALL CLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 ) * IF( K.LT.N ) THEN CALL CGEMV( '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 ) = REAL( 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 both ROWMAX and JMAX. * IF( IMAX.NE.K ) THEN JMAX = IMAX + ICAMAX( K-IMAX, W( IMAX+1, KW-1 ), $ 1 ) ROWMAX = CABS1( W( JMAX, KW-1 ) ) ELSE ROWMAX = ZERO END IF * IF( IMAX.GT.1 ) THEN ITEMP = ICAMAX( IMAX-1, W( 1, KW-1 ), 1 ) STEMP = CABS1( W( ITEMP, KW-1 ) ) IF( STEMP.GT.ROWMAX ) THEN ROWMAX = STEMP 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( REAL( W( IMAX,KW-1 ) ) ) $ .LT.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 CCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) * 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 * * Copy updated JMAXth (next IMAXth) column to Kth of W * CALL CCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) * 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 * * KKW is the column of W which corresponds to column KK of A * KKW = NB + KK - N * * Interchange rows and columns P and K. * Updated column P is already stored in column KW of W. * IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN * * Copy non-updated column K to column P of submatrix A * at step K. No need to copy element into columns * K and K-1 of A for 2-by-2 pivot, since these columns * will be later overwritten. * A( P, P ) = REAL( A( K, K ) ) CALL CCOPY( K-1-P, A( P+1, K ), 1, A( P, P+1 ), $ LDA ) CALL CLACGV( K-1-P, A( P, P+1 ), LDA ) IF( P.GT.1 ) $ CALL CCOPY( P-1, A( 1, K ), 1, A( 1, P ), 1 ) * * Interchange rows K and P in the last K+1 to N columns of A * (columns K and K-1 of A for 2-by-2 pivot will be * later overwritten). Interchange rows K and P * in last KKW to NB columns of W. * IF( K.LT.N ) $ CALL CSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), $ LDA ) CALL CSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), $ LDW ) END IF * * 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 ) = REAL( A( KK, KK ) ) CALL CCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), $ LDA ) CALL CLACGV( KK-1-KP, A( KP, KP+1 ), LDA ) IF( KP.GT.1 ) $ CALL CCOPY( 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 CSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), $ LDA ) CALL CSWAP( 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 ) = REAL( W( K, K) ) to separately copy diagonal * element D(k,k) from W (potentially saves only one load)) CALL CCOPY( 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(3)) * * Handle division by a small number * T = REAL( A( K, K ) ) IF( ABS( T ).GE.SFMIN ) THEN R1 = ONE / T CALL CSSCAL( K-1, R1, A( 1, K ), 1 ) ELSE DO 14 II = 1, K-1 A( II, K ) = A( II, K ) / T 14 CONTINUE END IF * * (2) Conjugate column W(kw) * CALL CLACGV( K-1, W( 1, KW ), 1 ) * * Store the superdiagonal element of D in array E * E( K ) = CZERO * 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 ) ) * * Handle division by a small number. (NOTE: order of * operations is important) * * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) ) * ( (( -1 ) ) (( D22 ) ) ), * * where D11 = d22/d21, * D22 = d11/conj(d21), * D21 = 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 in 2x2 pivot case(4), * since |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 ) / CONJG( D21 ) D22 = W( K-1, KW-1 ) / D21 T = ONE / ( REAL( D11*D22 )-ONE ) * * 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 ) = T*( ( D11*W( J, KW-1 )-W( J, KW ) ) / $ D21 ) A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) / $ CONJG( D21 ) ) 20 CONTINUE END IF * * Copy diagonal elements of D(K) to A, * copy superdiagonal element of D(K) to E(K) and * ZERO out superdiagonal entry of A * A( K-1, K-1 ) = W( K-1, KW-1 ) A( K-1, K ) = CZERO A( K, K ) = W( K, KW ) E( K ) = W( K-1, KW ) E( K-1 ) = CZERO * * (2) Conjugate columns W(kw) and W(kw-1) * CALL CLACGV( K-1, W( 1, KW ), 1 ) CALL CLACGV( K-2, W( 1, KW-1 ), 1 ) * 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 * 30 CONTINUE * * Update the upper triangle of A11 (= A(1:k,1:k)) as * * A11 := A11 - U12*D*U12**H = A11 - U12*W**H * * (note that conjg(W) is actually stored) * CALL CGEMMTR( 'Upper', 'No transpose', 'Transpose', K, N-K, $ -CONE, A( 1, K+1 ), LDA, W( 1, KW+1 ), LDW, $ CONE, A( 1, 1 ), LDA ) * * 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) * * Initialize the unused last entry of the subdiagonal array E. * E( N ) = CZERO * * 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 P = K * * Copy column K of A to column K of W and update column K of W * W( K, K ) = REAL( A( K, K ) ) IF( K.LT.N ) $ CALL CCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 ) IF( K.GT.1 ) THEN CALL CGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 ) W( K, K ) = REAL( W( K, K ) ) 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( REAL( 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 + ICAMAX( 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 ) = REAL( W( K, K ) ) IF( K.LT.N ) $ CALL CCOPY( N-K, W( K+1, K ), 1, A( K+1, K ), 1 ) * * 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 * 72 CONTINUE * * BEGIN pivot search loop body * * * Copy column IMAX to column k+1 of W and update it * CALL CCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), $ 1) CALL CLACGV( IMAX-K, W( K, K+1 ), 1 ) W( IMAX, K+1 ) = REAL( A( IMAX, IMAX ) ) * IF( IMAX.LT.N ) $ CALL CCOPY( N-IMAX, A( IMAX+1, IMAX ), 1, $ W( IMAX+1, K+1 ), 1 ) * IF( K.GT.1 ) THEN CALL CGEMV( '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 ) = REAL( W( IMAX, K+1 ) ) END IF * * 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 + ICAMAX( IMAX-K, W( K, K+1 ), 1 ) ROWMAX = CABS1( W( JMAX, K+1 ) ) ELSE ROWMAX = ZERO END IF * IF( IMAX.LT.N ) THEN ITEMP = IMAX + ICAMAX( N-IMAX, W( IMAX+1, K+1 ), $ 1) STEMP = CABS1( W( ITEMP, K+1 ) ) IF( STEMP.GT.ROWMAX ) THEN ROWMAX = STEMP JMAX = ITEMP END IF END IF * * Case(2) * Equivalent to testing for * ABS( REAL( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX * (used to handle NaN and Inf) * IF( .NOT.( ABS( REAL( W( IMAX,K+1 ) ) ) $ .LT.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 CCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), $ 1 ) * 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 * * Copy updated JMAXth (next IMAXth) column to Kth of W * CALL CCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), $ 1 ) * END IF * * * End pivot search loop body * IF( .NOT.DONE ) GOTO 72 * END IF * * END pivot search * * ============================================================ * * KK is the column of A where pivoting step stopped * KK = K + KSTEP - 1 * * Interchange rows and columns P and K (only for 2-by-2 pivot). * Updated column P is already stored in column K of W. * IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN * * Copy non-updated column KK-1 to column P of submatrix A * at step K. No need to copy element into columns * K and K+1 of A for 2-by-2 pivot, since these columns * will be later overwritten. * A( P, P ) = REAL( A( K, K ) ) CALL CCOPY( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA ) CALL CLACGV( P-K-1, A( P, K+1 ), LDA ) IF( P.LT.N ) $ CALL CCOPY( N-P, A( P+1, K ), 1, A( P+1, P ), 1 ) * * Interchange rows K and P in first K-1 columns of A * (columns K and K+1 of A for 2-by-2 pivot will be * later overwritten). Interchange rows K and P * in first KK columns of W. * IF( K.GT.1 ) $ CALL CSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA ) CALL CSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW ) END IF * * 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 ) = REAL( A( KK, KK ) ) CALL CCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), $ LDA ) CALL CLACGV( KP-KK-1, A( KP, KK+1 ), LDA ) IF( KP.LT.N ) $ CALL CCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), $ 1 ) * * Interchange rows KK and KP in first K-1 columns of A * (column 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 CSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) CALL CSWAP( 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 ) = REAL( W( K, K) ) to separately copy diagonal * element D(k,k) from W (potentially saves only one load)) CALL CCOPY( 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(3)) * * Handle division by a small number * T = REAL( A( K, K ) ) IF( ABS( T ).GE.SFMIN ) THEN R1 = ONE / T CALL CSSCAL( N-K, R1, A( K+1, K ), 1 ) ELSE DO 74 II = K + 1, N A( II, K ) = A( II, K ) / T 74 CONTINUE END IF * * (2) Conjugate column W(k) * CALL CLACGV( N-K, W( K+1, K ), 1 ) * * 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 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 ) ) * * Handle division by a small number. (NOTE: order of * operations is important) * * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) ) * ( (( -1 ) ) (( D22 ) ) ), * * where D11 = d22/d21, * D22 = d11/conj(d21), * D21 = 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 in 2x2 pivot case(4), * since |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 ) / CONJG( D21 ) T = ONE / ( REAL( D11*D22 )-ONE ) * * 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 ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) / $ CONJG( D21 ) ) A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) / $ D21 ) 80 CONTINUE END IF * * Copy diagonal elements of D(K) to A, * copy subdiagonal element of D(K) to E(K) and * ZERO out subdiagonal entry of A * A( K, K ) = W( K, K ) A( K+1, K ) = CZERO A( K+1, K+1 ) = W( K+1, K+1 ) E( K ) = W( K+1, K ) E( K+1 ) = CZERO * * (2) Conjugate columns W(k) and W(k+1) * CALL CLACGV( N-K, W( K+1, K ), 1 ) CALL CLACGV( N-K-1, W( K+2, K+1 ), 1 ) * 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 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 * * (note that conjg(W) is actually stored) * CALL CGEMMTR( 'Lower', 'No transpose', 'Transpose', N-K+1, $ K-1, -CONE, A( K, 1 ), LDA, W( K, 1 ), LDW, $ CONE, A( K, K ), LDA ) * * Set KB to the number of columns factorized * KB = K - 1 * END IF RETURN * * End of CLAHEF_RK * END