SUBROUTINE ZHETF2( UPLO, N, A, LDA, IPIV, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * )
* ..
*
* Purpose
* =======
*
* ZHETF2 computes the factorization of a complex Hermitian matrix A
* using the Bunch-Kaufman diagonal pivoting method:
*
* A = U*D*U' or A = L*D*L'
*
* where U (or L) is a product of permutation and unit upper (lower)
* triangular matrices, U' 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.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* Hermitian matrix A is stored:
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) 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).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (output) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D.
* 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 UPLO = 'U' and 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' and 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.
*
* INFO (output) 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.
*
* Further Details
* ===============
*
* 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
*
* If UPLO = 'U', then A = U*D*U', 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', 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).
*
* =====================================================================
*
* .. 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 )
* ..
* .. 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' 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
* column K, and COLMAX is its absolute value
*
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
*
* Column K is zero or contains a NaN: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
A( K, K ) = DBLE( A( K, K ) )
ELSE
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
* element in row IMAX, and ROWMAX is its absolute value
*
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
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
END IF
*
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)' = A - W(k)*1/D(k)*W(k)'
*
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) )'
* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
*
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' 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
* column K, and COLMAX is its absolute value
*
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
*
* Column K is zero or contains a NaN: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
A( K, K ) = DBLE( A( K, K ) )
ELSE
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
* element in row IMAX, and ROWMAX is its absolute value
*
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
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
END IF
*
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)' = A - W(k)*(1/D(k))*W(k)'
*
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) )'
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'
*
* 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