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*> \brief \b ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS).
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*
* =========== DOCUMENTATION ===========
*
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* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
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*
*> \htmlonly
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*> Download ZHETF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetf2.f">
*> [ZIP]</a>
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*> [TXT]</a>
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*> \endhtmlonly
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*
* Definition:
* ===========
*
* SUBROUTINE ZHETF2( UPLO, N, A, LDA, IPIV, INFO )
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*
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* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX*16 A( LDA, * )
* ..
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*
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*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZHETF2 computes the factorization of a complex Hermitian matrix A
*> using the Bunch-Kaufman diagonal pivoting method:
*>
*> A = U*D*U**H or A = L*D*L**H
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, U**H is the conjugate transpose of U, and D is
*> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> \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, the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L (see below for further details).
*> \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.
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*>
*> If UPLO = 'U':
*> 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':
*> 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.
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*> \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, D(k,k) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, and division by zero will occur if it
*> is used to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
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*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
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*
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*> \date November 2013
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*
*> \ingroup complex16HEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', then A = U*D*U**H, where
*> U = P(n)*U(n)* ... *P(k)U(k)* ...,
*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I v 0 ) k-s
*> U(k) = ( 0 I 0 ) s
*> ( 0 0 I ) n-k
*> k-s s n-k
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*> and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*> If UPLO = 'L', then A = L*D*L**H, where
*> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*> ( I 0 0 ) k-1
*> L(k) = ( 0 I 0 ) s
*> ( 0 v I ) n-k-s+1
*> k-1 s n-k-s+1
*>
*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*> 09-29-06 - patch from
*> Bobby Cheng, MathWorks
*>
*> Replace l.210 and l.393
*> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*> by
*> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
*>
*> 01-01-96 - Based on modifications by
*> J. Lewis, Boeing Computer Services Company
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*> \endverbatim
*
* =====================================================================
SUBROUTINE ZHETF2 ( UPLO , N , A , LDA , IPIV , INFO )
*
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* -- LAPACK computational routine (version 3.5.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* November 2013
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*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO , LDA , N
* ..
* .. Array Arguments ..
INTEGER IPIV ( * )
COMPLEX * 16 A ( LDA , * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO , ONE
PARAMETER ( ZERO = 0.0D+0 , ONE = 1.0D+0 )
DOUBLE PRECISION EIGHT , SEVTEN
PARAMETER ( EIGHT = 8.0D+0 , SEVTEN = 1 7.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I , IMAX , J , JMAX , K , KK , KP , KSTEP
DOUBLE PRECISION ABSAKK , ALPHA , COLMAX , D , D11 , D22 , R1 , ROWMAX ,
$ TT
COMPLEX * 16 D12 , D21 , T , WK , WKM1 , WKP1 , ZDUM
* ..
* .. External Functions ..
LOGICAL LSAME , DISNAN
INTEGER IZAMAX
DOUBLE PRECISION DLAPY2
EXTERNAL LSAME , IZAMAX , DLAPY2 , DISNAN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA , ZDSCAL , ZHER , ZSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS , DBLE , DCMPLX , DCONJG , DIMAG , MAX , SQRT
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1 ( ZDUM ) = ABS ( DBLE ( ZDUM ) ) + ABS ( DIMAG ( ZDUM ) )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME ( UPLO , 'U' )
IF ( . NOT . UPPER . AND . . NOT . LSAME ( UPLO , 'L' ) ) THEN
INFO = - 1
ELSE IF ( N . LT . 0 ) THEN
INFO = - 2
ELSE IF ( LDA . LT . MAX ( 1 , N ) ) THEN
INFO = - 4
END IF
IF ( INFO . NE . 0 ) THEN
CALL XERBLA ( 'ZHETF2' , - INFO )
RETURN
END IF
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE + SQRT ( SEVTEN ) ) / EIGHT
*
IF ( UPPER ) THEN
*
* Factorize A as U*D*U**H using the upper triangle of A
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2
*
K = N
10 CONTINUE
*
* If K < 1, exit from loop
*
IF ( K . LT . 1 )
$ GO TO 90
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS ( DBLE ( A ( K , K ) ) )
*
* IMAX is the row-index of the largest off-diagonal element in
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* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
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*
IF ( K . GT . 1 ) THEN
IMAX = IZAMAX ( K - 1 , A ( 1 , K ) , 1 )
COLMAX = CABS1 ( A ( IMAX , K ) )
ELSE
COLMAX = ZERO
END IF
*
IF ( ( MAX ( ABSAKK , COLMAX ) . EQ . ZERO ) . OR . DISNAN ( ABSAKK ) ) THEN
*
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* Column K is zero or underflow, or contains a NaN:
* set INFO and continue
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*
IF ( INFO . EQ . 0 )
$ INFO = K
KP = K
A ( K , K ) = DBLE ( A ( K , K ) )
ELSE
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*
* ============================================================
*
* Test for interchange
*
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IF ( ABSAKK . GE . ALPHA * COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
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* element in row IMAX, and ROWMAX is its absolute value.
* Determine only ROWMAX.
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*
JMAX = IMAX + IZAMAX ( K - IMAX , A ( IMAX , IMAX + 1 ) , LDA )
ROWMAX = CABS1 ( A ( IMAX , JMAX ) )
IF ( IMAX . GT . 1 ) THEN
JMAX = IZAMAX ( IMAX - 1 , A ( 1 , IMAX ) , 1 )
ROWMAX = MAX ( ROWMAX , CABS1 ( A ( JMAX , IMAX ) ) )
END IF
*
IF ( ABSAKK . GE . ALPHA * COLMAX * ( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
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*
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ELSE IF ( ABS ( DBLE ( A ( IMAX , IMAX ) ) ) . GE . ALPHA * ROWMAX )
$ THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K-1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
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*
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END IF
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*
* ============================================================
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*
KK = K - KSTEP + 1
IF ( KP . NE . KK ) THEN
*
* Interchange rows and columns KK and KP in the leading
* submatrix A(1:k,1:k)
*
CALL ZSWAP ( KP - 1 , A ( 1 , KK ) , 1 , A ( 1 , KP ) , 1 )
DO 20 J = KP + 1 , KK - 1
T = DCONJG ( A ( J , KK ) )
A ( J , KK ) = DCONJG ( A ( KP , J ) )
A ( KP , J ) = T
20 CONTINUE
A ( KP , KK ) = DCONJG ( A ( KP , KK ) )
R1 = DBLE ( A ( KK , KK ) )
A ( KK , KK ) = DBLE ( A ( KP , KP ) )
A ( KP , KP ) = R1
IF ( KSTEP . EQ . 2 ) THEN
A ( K , K ) = DBLE ( A ( K , K ) )
T = A ( K - 1 , K )
A ( K - 1 , K ) = A ( KP , K )
A ( KP , K ) = T
END IF
ELSE
A ( K , K ) = DBLE ( A ( K , K ) )
IF ( KSTEP . EQ . 2 )
$ A ( K - 1 , K - 1 ) = DBLE ( A ( K - 1 , K - 1 ) )
END IF
*
* Update the leading submatrix
*
IF ( KSTEP . EQ . 1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = U(k)*D(k)
*
* where U(k) is the k-th column of U
*
* Perform a rank-1 update of A(1:k-1,1:k-1) as
*
* A := A - U(k)*D(k)*U(k)**H = A - W(k)*1/D(k)*W(k)**H
*
R1 = ONE / DBLE ( A ( K , K ) )
CALL ZHER ( UPLO , K - 1 , - R1 , A ( 1 , K ) , 1 , A , LDA )
*
* Store U(k) in column k
*
CALL ZDSCAL ( K - 1 , R1 , A ( 1 , K ) , 1 )
ELSE
*
* 2-by-2 pivot block D(k): columns k and k-1 now hold
*
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
* where U(k) and U(k-1) are the k-th and (k-1)-th columns
* of U
*
* Perform a rank-2 update of A(1:k-2,1:k-2) as
*
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**H
* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**H
*
IF ( K . GT . 2 ) THEN
*
D = DLAPY2 ( DBLE ( A ( K - 1 , K ) ) ,
$ DIMAG ( A ( K - 1 , K ) ) )
D22 = DBLE ( A ( K - 1 , K - 1 ) ) / D
D11 = DBLE ( A ( K , K ) ) / D
TT = ONE / ( D11 * D22 - ONE )
D12 = A ( K - 1 , K ) / D
D = TT / D
*
DO 40 J = K - 2 , 1 , - 1
WKM1 = D * ( D11 * A ( J , K - 1 ) - DCONJG ( D12 ) *
$ A ( J , K ) )
WK = D * ( D22 * A ( J , K ) - D12 * A ( J , K - 1 ) )
DO 30 I = J , 1 , - 1
A ( I , J ) = A ( I , J ) - A ( I , K ) * DCONJG ( WK ) -
$ A ( I , K - 1 ) * DCONJG ( WKM1 )
30 CONTINUE
A ( J , K ) = WK
A ( J , K - 1 ) = WKM1
A ( J , J ) = DCMPLX ( DBLE ( A ( J , J ) ) , 0.0D+0 )
40 CONTINUE
*
END IF
*
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF ( KSTEP . EQ . 1 ) THEN
IPIV ( K ) = KP
ELSE
IPIV ( K ) = - KP
IPIV ( K - 1 ) = - KP
END IF
*
* Decrease K and return to the start of the main loop
*
K = K - KSTEP
GO TO 10
*
ELSE
*
* Factorize A as L*D*L**H using the lower triangle of A
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2
*
K = 1
50 CONTINUE
*
* If K > N, exit from loop
*
IF ( K . GT . N )
$ GO TO 90
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS ( DBLE ( A ( K , K ) ) )
*
* IMAX is the row-index of the largest off-diagonal element in
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* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
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*
IF ( K . LT . N ) THEN
IMAX = K + IZAMAX ( N - K , A ( K + 1 , K ) , 1 )
COLMAX = CABS1 ( A ( IMAX , K ) )
ELSE
COLMAX = ZERO
END IF
*
IF ( ( MAX ( ABSAKK , COLMAX ) . EQ . ZERO ) . OR . DISNAN ( ABSAKK ) ) THEN
*
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* Column K is zero or underflow, or contains a NaN:
* set INFO and continue
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*
IF ( INFO . EQ . 0 )
$ INFO = K
KP = K
A ( K , K ) = DBLE ( A ( K , K ) )
ELSE
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*
* ============================================================
*
* Test for interchange
*
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IF ( ABSAKK . GE . ALPHA * COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
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* element in row IMAX, and ROWMAX is its absolute value.
* Determine only ROWMAX.
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*
JMAX = K - 1 + IZAMAX ( IMAX - K , A ( IMAX , K ) , LDA )
ROWMAX = CABS1 ( A ( IMAX , JMAX ) )
IF ( IMAX . LT . N ) THEN
JMAX = IMAX + IZAMAX ( N - IMAX , A ( IMAX + 1 , IMAX ) , 1 )
ROWMAX = MAX ( ROWMAX , CABS1 ( A ( JMAX , IMAX ) ) )
END IF
*
IF ( ABSAKK . GE . ALPHA * COLMAX * ( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
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*
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ELSE IF ( ABS ( DBLE ( A ( IMAX , IMAX ) ) ) . GE . ALPHA * ROWMAX )
$ THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K+1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
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*
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END IF
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*
* ============================================================
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*
KK = K + KSTEP - 1
IF ( KP . NE . KK ) THEN
*
* Interchange rows and columns KK and KP in the trailing
* submatrix A(k:n,k:n)
*
IF ( KP . LT . N )
$ CALL ZSWAP ( N - KP , A ( KP + 1 , KK ) , 1 , A ( KP + 1 , KP ) , 1 )
DO 60 J = KK + 1 , KP - 1
T = DCONJG ( A ( J , KK ) )
A ( J , KK ) = DCONJG ( A ( KP , J ) )
A ( KP , J ) = T
60 CONTINUE
A ( KP , KK ) = DCONJG ( A ( KP , KK ) )
R1 = DBLE ( A ( KK , KK ) )
A ( KK , KK ) = DBLE ( A ( KP , KP ) )
A ( KP , KP ) = R1
IF ( KSTEP . EQ . 2 ) THEN
A ( K , K ) = DBLE ( A ( K , K ) )
T = A ( K + 1 , K )
A ( K + 1 , K ) = A ( KP , K )
A ( KP , K ) = T
END IF
ELSE
A ( K , K ) = DBLE ( A ( K , K ) )
IF ( KSTEP . EQ . 2 )
$ A ( K + 1 , K + 1 ) = DBLE ( A ( K + 1 , K + 1 ) )
END IF
*
* Update the trailing submatrix
*
IF ( KSTEP . EQ . 1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = L(k)*D(k)
*
* where L(k) is the k-th column of L
*
IF ( K . LT . N ) THEN
*
* Perform a rank-1 update of A(k+1:n,k+1:n) as
*
* A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H
*
R1 = ONE / DBLE ( A ( K , K ) )
CALL ZHER ( UPLO , N - K , - R1 , A ( K + 1 , K ) , 1 ,
$ A ( K + 1 , K + 1 ) , LDA )
*
* Store L(k) in column K
*
CALL ZDSCAL ( N - K , R1 , A ( K + 1 , K ) , 1 )
END IF
ELSE
*
* 2-by-2 pivot block D(k)
*
IF ( K . LT . N - 1 ) THEN
*
* Perform a rank-2 update of A(k+2:n,k+2:n) as
*
* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H
*
* where L(k) and L(k+1) are the k-th and (k+1)-th
* columns of L
*
D = DLAPY2 ( DBLE ( A ( K + 1 , K ) ) ,
$ DIMAG ( A ( K + 1 , K ) ) )
D11 = DBLE ( A ( K + 1 , K + 1 ) ) / D
D22 = DBLE ( A ( K , K ) ) / D
TT = ONE / ( D11 * D22 - ONE )
D21 = A ( K + 1 , K ) / D
D = TT / D
*
DO 80 J = K + 2 , N
WK = D * ( D11 * A ( J , K ) - D21 * A ( J , K + 1 ) )
WKP1 = D * ( D22 * A ( J , K + 1 ) - DCONJG ( D21 ) *
$ A ( J , K ) )
DO 70 I = J , N
A ( I , J ) = A ( I , J ) - A ( I , K ) * DCONJG ( WK ) -
$ A ( I , K + 1 ) * DCONJG ( WKP1 )
70 CONTINUE
A ( J , K ) = WK
A ( J , K + 1 ) = WKP1
A ( J , J ) = DCMPLX ( DBLE ( A ( J , J ) ) , 0.0D+0 )
80 CONTINUE
END IF
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF ( KSTEP . EQ . 1 ) THEN
IPIV ( K ) = KP
ELSE
IPIV ( K ) = - KP
IPIV ( K + 1 ) = - KP
END IF
*
* Increase K and return to the start of the main loop
*
K = K + KSTEP
GO TO 50
*
END IF
*
90 CONTINUE
RETURN
*
* End of ZHETF2
*
END
