*> \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
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \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.
*
*> \date December 2016
*
*> \ingroup complexHEcomputational
*
*> \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 (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 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, JB, JJ, 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)
*
* 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 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
*
* 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 ) = REAL( A( JJ, JJ ) )
CALL CGEMV( '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 ) = REAL( A( JJ, JJ ) )
40 CONTINUE
*
* Update the rectangular superdiagonal block
*
IF( J.GE.2 )
$ CALL CGEMM( '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
*
* 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)
*
* Initilize 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
*
* 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 ) = REAL( A( JJ, JJ ) )
CALL CGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
$ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
$ A( JJ, JJ ), 1 )
A( JJ, JJ ) = REAL( A( JJ, JJ ) )
100 CONTINUE
*
* Update the rectangular subdiagonal block
*
IF( J+JB.LE.N )
$ CALL CGEMM( '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
*
* Set KB to the number of columns factorized
*
KB = K - 1
*
END IF
RETURN
*
* End of CLAHEF_RK
*
END