version 1.13, 2012/12/14 14:22:50
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version 1.16, 2016/08/27 15:34:57
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*> \brief \b ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix, using the diagonal pivoting method. |
*> \brief \b ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS). |
* |
* |
* =========== DOCUMENTATION =========== |
* =========== DOCUMENTATION =========== |
* |
* |
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*> \verbatim |
*> \verbatim |
*> IPIV is INTEGER array, dimension (N) |
*> IPIV is INTEGER array, dimension (N) |
*> Details of the interchanges and the block structure of D. |
*> Details of the interchanges and the block structure of D. |
*> If UPLO = 'U', only the last KB elements of IPIV are set; |
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*> if UPLO = 'L', only the first KB elements are set. |
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*> |
*> |
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
*> If UPLO = 'U': |
*> interchanged and D(k,k) is a 1-by-1 diagonal block. |
*> Only the last KB elements of IPIV are set. |
*> 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) |
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = |
*> interchanged and D(k,k) is a 1-by-1 diagonal block. |
*> 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. |
*> If IPIV(k) = IPIV(k-1) < 0, then rows and columns |
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*> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) |
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*> is a 2-by-2 diagonal block. |
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*> |
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*> If UPLO = 'L': |
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*> Only the first KB elements of IPIV are set. |
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*> |
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*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
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*> interchanged and D(k,k) is a 1-by-1 diagonal block. |
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*> |
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*> If IPIV(k) = IPIV(k+1) < 0, then rows and columns |
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*> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) |
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*> is a 2-by-2 diagonal block. |
*> \endverbatim |
*> \endverbatim |
*> |
*> |
*> \param[out] W |
*> \param[out] W |
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*> \author Univ. of Colorado Denver |
*> \author Univ. of Colorado Denver |
*> \author NAG Ltd. |
*> \author NAG Ltd. |
* |
* |
*> \date September 2012 |
*> \date November 2013 |
* |
* |
*> \ingroup complex16HEcomputational |
*> \ingroup complex16HEcomputational |
* |
* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> \verbatim |
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*> |
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*> November 2013, Igor Kozachenko, |
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*> Computer Science Division, |
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*> University of California, Berkeley |
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*> \endverbatim |
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* |
* ===================================================================== |
* ===================================================================== |
SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) |
SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) |
* |
* |
* -- LAPACK computational routine (version 3.4.2) -- |
* -- LAPACK computational routine (version 3.5.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* September 2012 |
* November 2013 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
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IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 ) |
IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 ) |
$ GO TO 30 |
$ GO TO 30 |
* |
* |
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KSTEP = 1 |
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* |
* Copy column K of A to column KW of W and update it |
* Copy column K of A to column KW of W and update it |
* |
* |
CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 ) |
CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 ) |
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W( K, KW ) = DBLE( W( K, KW ) ) |
W( K, KW ) = DBLE( W( K, KW ) ) |
END IF |
END IF |
* |
* |
KSTEP = 1 |
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* |
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* Determine rows and columns to be interchanged and whether |
* Determine rows and columns to be interchanged and whether |
* a 1-by-1 or 2-by-2 pivot block will be used |
* a 1-by-1 or 2-by-2 pivot block will be used |
* |
* |
ABSAKK = ABS( DBLE( W( K, KW ) ) ) |
ABSAKK = ABS( DBLE( W( K, KW ) ) ) |
* |
* |
* IMAX is the row-index of the largest off-diagonal element in |
* IMAX is the row-index of the largest off-diagonal element in |
* column K, and COLMAX is its absolute value |
* column K, and COLMAX is its absolute value. |
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* Determine both COLMAX and IMAX. |
* |
* |
IF( K.GT.1 ) THEN |
IF( K.GT.1 ) THEN |
IMAX = IZAMAX( K-1, W( 1, KW ), 1 ) |
IMAX = IZAMAX( K-1, W( 1, KW ), 1 ) |
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* |
* |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
* |
* |
* Column K is zero: set INFO and continue |
* Column K is zero or underflow: set INFO and continue |
* |
* |
IF( INFO.EQ.0 ) |
IF( INFO.EQ.0 ) |
$ INFO = K |
$ INFO = K |
KP = K |
KP = K |
A( K, K ) = DBLE( A( K, K ) ) |
A( K, K ) = DBLE( A( K, K ) ) |
ELSE |
ELSE |
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* |
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* ============================================================ |
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* |
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* BEGIN pivot search |
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* |
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* Case(1) |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
* |
* |
* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
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KP = K |
KP = K |
ELSE |
ELSE |
* |
* |
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* BEGIN pivot search along IMAX row |
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* |
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* |
* Copy column IMAX to column KW-1 of W and update it |
* Copy column IMAX to column KW-1 of W and update it |
* |
* |
CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) |
CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) |
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END IF |
END IF |
* |
* |
* JMAX is the column-index of the largest off-diagonal |
* JMAX is the column-index of the largest off-diagonal |
* element in row IMAX, and ROWMAX is its absolute value |
* element in row IMAX, and ROWMAX is its absolute value. |
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* Determine only ROWMAX. |
* |
* |
JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) |
JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) |
ROWMAX = CABS1( W( JMAX, KW-1 ) ) |
ROWMAX = CABS1( W( JMAX, KW-1 ) ) |
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ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) ) |
ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) ) |
END IF |
END IF |
* |
* |
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* Case(2) |
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN |
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN |
* |
* |
* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
* |
* |
KP = K |
KP = K |
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* |
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* Case(3) |
ELSE IF( ABS( DBLE( W( IMAX, KW-1 ) ) ).GE.ALPHA*ROWMAX ) |
ELSE IF( ABS( DBLE( W( IMAX, KW-1 ) ) ).GE.ALPHA*ROWMAX ) |
$ THEN |
$ THEN |
* |
* |
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* |
* |
KP = IMAX |
KP = IMAX |
* |
* |
* copy column KW-1 of W to column KW |
* copy column KW-1 of W to column KW of W |
* |
* |
CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) |
CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) |
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* |
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* Case(4) |
ELSE |
ELSE |
* |
* |
* interchange rows and columns K-1 and IMAX, use 2-by-2 |
* interchange rows and columns K-1 and IMAX, use 2-by-2 |
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KP = IMAX |
KP = IMAX |
KSTEP = 2 |
KSTEP = 2 |
END IF |
END IF |
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* |
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* |
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* END pivot search along IMAX row |
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* |
END IF |
END IF |
* |
* |
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* END pivot search |
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* |
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* ============================================================ |
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* |
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* KK is the column of A where pivoting step stopped |
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* |
KK = K - KSTEP + 1 |
KK = K - KSTEP + 1 |
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* |
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* KKW is the column of W which corresponds to column KK of A |
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* |
KKW = NB + KK - N |
KKW = NB + KK - N |
* |
* |
* Updated column KP is already stored in column KKW of W |
* Interchange rows and columns KP and KK. |
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* Updated column KP is already stored in column KKW of W. |
* |
* |
IF( KP.NE.KK ) THEN |
IF( KP.NE.KK ) THEN |
* |
* |
* Copy non-updated column KK to column KP |
* Copy non-updated column KK to column KP of submatrix A |
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* at step K. No need to copy element into column K |
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* (or K and K-1 for 2-by-2 pivot) of A, since these columns |
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* will be later overwritten. |
* |
* |
A( KP, KP ) = DBLE( A( KK, KK ) ) |
A( KP, KP ) = DBLE( A( KK, KK ) ) |
CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), |
CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), |
$ LDA ) |
$ LDA ) |
CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA ) |
CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA ) |
CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) |
IF( KP.GT.1 ) |
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$ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) |
* |
* |
* Interchange rows KK and KP in last KK columns of A and W |
* Interchange rows KK and KP in last K+1 to N columns of A |
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* (columns K (or K and K-1 for 2-by-2 pivot) of A will be |
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* later overwritten). Interchange rows KK and KP |
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* in last KKW to NB columns of W. |
* |
* |
IF( KK.LT.N ) |
IF( K.LT.N ) |
$ CALL ZSWAP( N-KK, A( KK, KK+1 ), LDA, A( KP, KK+1 ), |
$ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), |
$ LDA ) |
$ LDA ) |
CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), |
CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), |
$ LDW ) |
$ LDW ) |
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* |
* |
IF( KSTEP.EQ.1 ) THEN |
IF( KSTEP.EQ.1 ) THEN |
* |
* |
* 1-by-1 pivot block D(k): column KW of W now holds |
* 1-by-1 pivot block D(k): column kw of W now holds |
* |
* |
* W(k) = U(k)*D(k) |
* W(kw) = U(k)*D(k), |
* |
* |
* where U(k) is the k-th column of U |
* where U(k) is the k-th column of U |
* |
* |
* Store U(k) in column k of A |
* (1) Store subdiag. elements of column U(k) |
* |
* and 1-by-1 block D(k) in column k of A. |
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* (NOTE: Diagonal element U(k,k) is a UNIT element |
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* and not stored) |
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* A(k,k) := D(k,k) = W(k,kw) |
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* A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k) |
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* |
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* (NOTE: No need to use for Hermitian matrix |
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* A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal |
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* element D(k,k) from W (potentially saves only one load)) |
CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) |
CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) |
R1 = ONE / DBLE( A( K, K ) ) |
IF( K.GT.1 ) THEN |
CALL ZDSCAL( K-1, R1, A( 1, K ), 1 ) |
|
* |
* |
* Conjugate W(k) |
* (NOTE: No need to check if A(k,k) is NOT ZERO, |
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* since that was ensured earlier in pivot search: |
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* case A(k,k) = 0 falls into 2x2 pivot case(4)) |
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* |
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R1 = ONE / DBLE( A( K, K ) ) |
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CALL ZDSCAL( K-1, R1, A( 1, K ), 1 ) |
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* |
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* (2) Conjugate column W(kw) |
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* |
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CALL ZLACGV( K-1, W( 1, KW ), 1 ) |
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END IF |
* |
* |
CALL ZLACGV( K-1, W( 1, KW ), 1 ) |
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ELSE |
ELSE |
* |
* |
* 2-by-2 pivot block D(k): columns KW and KW-1 of W now |
* 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold |
* hold |
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* |
* |
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) |
* ( 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 |
* where U(k) and U(k-1) are the k-th and (k-1)-th columns |
* of U |
* of U |
* |
* |
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* (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2 |
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* block D(k-1:k,k-1:k) in columns k-1 and k of A. |
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* (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT |
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* block and not stored) |
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* A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw) |
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* A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) = |
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* = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) ) |
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* |
IF( K.GT.2 ) THEN |
IF( K.GT.2 ) THEN |
* |
* |
* Store U(k) and U(k-1) in columns k and k-1 of A |
* Factor out the columns of the inverse of 2-by-2 pivot |
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* block D, so that each column contains 1, to reduce the |
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* number of FLOPS when we multiply panel |
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* ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1). |
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* |
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* D**(-1) = ( d11 cj(d21) )**(-1) = |
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* ( d21 d22 ) |
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* |
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* = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) = |
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* ( (-d21) ( d11 ) ) |
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* |
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* = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) * |
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* |
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* * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) = |
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* ( ( -1 ) ( d11/conj(d21) ) ) |
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* |
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* = 1/(|d21|**2) * 1/(D22*D11-1) * |
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* |
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* * ( d21*( D11 ) conj(d21)*( -1 ) ) = |
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* ( ( -1 ) ( D22 ) ) |
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* |
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* = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) = |
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* ( ( -1 ) ( D22 ) ) |
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* |
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* = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) = |
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* ( ( -1 ) ( D22 ) ) |
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* |
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* = ( conj(D21)*( D11 ) D21*( -1 ) ) |
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* ( ( -1 ) ( D22 ) ), |
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* |
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* where D11 = d22/d21, |
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* D22 = d11/conj(d21), |
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* D21 = T/d21, |
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* T = 1/(D22*D11-1). |
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* |
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* (NOTE: No need to check for division by ZERO, |
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* since that was ensured earlier in pivot search: |
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* (a) d21 != 0, since in 2x2 pivot case(4) |
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* |d21| should be larger than |d11| and |d22|; |
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* (b) (D22*D11 - 1) != 0, since from (a), |
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* both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.) |
* |
* |
D21 = W( K-1, KW ) |
D21 = W( K-1, KW ) |
D11 = W( K, KW ) / DCONJG( D21 ) |
D11 = W( K, KW ) / DCONJG( D21 ) |
D22 = W( K-1, KW-1 ) / D21 |
D22 = W( K-1, KW-1 ) / D21 |
T = ONE / ( DBLE( D11*D22 )-ONE ) |
T = ONE / ( DBLE( D11*D22 )-ONE ) |
D21 = T / D21 |
D21 = T / D21 |
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* |
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* Update elements in columns A(k-1) and A(k) as |
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* dot products of rows of ( W(kw-1) W(kw) ) and columns |
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* of D**(-1) |
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* |
DO 20 J = 1, K - 2 |
DO 20 J = 1, K - 2 |
A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) |
A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) |
A( J, K ) = DCONJG( D21 )* |
A( J, K ) = DCONJG( D21 )* |
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A( K-1, K ) = W( K-1, KW ) |
A( K-1, K ) = W( K-1, KW ) |
A( K, K ) = W( K, KW ) |
A( K, K ) = W( K, KW ) |
* |
* |
* Conjugate W(k) and W(k-1) |
* (2) Conjugate columns W(kw) and W(kw-1) |
* |
* |
CALL ZLACGV( K-1, W( 1, KW ), 1 ) |
CALL ZLACGV( K-1, W( 1, KW ), 1 ) |
CALL ZLACGV( K-2, W( 1, KW-1 ), 1 ) |
CALL ZLACGV( K-2, W( 1, KW-1 ), 1 ) |
|
* |
END IF |
END IF |
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* |
END IF |
END IF |
* |
* |
* Store details of the interchanges in IPIV |
* Store details of the interchanges in IPIV |
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50 CONTINUE |
50 CONTINUE |
* |
* |
* Put U12 in standard form by partially undoing the interchanges |
* Put U12 in standard form by partially undoing the interchanges |
* in columns k+1:n |
* in columns k+1:n looping backwards from k+1 to n |
* |
* |
J = K + 1 |
J = K + 1 |
60 CONTINUE |
60 CONTINUE |
JJ = J |
* |
JP = IPIV( J ) |
* Undo the interchanges (if any) of rows JJ and JP at each |
IF( JP.LT.0 ) THEN |
* step J |
JP = -JP |
* |
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* (Here, J is a diagonal index) |
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JJ = J |
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JP = IPIV( J ) |
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IF( JP.LT.0 ) THEN |
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JP = -JP |
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* (Here, J is a diagonal index) |
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J = J + 1 |
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END IF |
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* (NOTE: Here, J is used to determine row length. Length N-J+1 |
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* of the rows to swap back doesn't include diagonal element) |
J = J + 1 |
J = J + 1 |
END IF |
IF( JP.NE.JJ .AND. J.LE.N ) |
J = J + 1 |
$ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) |
IF( JP.NE.JJ .AND. J.LE.N ) |
IF( J.LT.N ) |
$ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) |
|
IF( J.LE.N ) |
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$ GO TO 60 |
$ GO TO 60 |
* |
* |
* Set KB to the number of columns factorized |
* Set KB to the number of columns factorized |
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IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N ) |
IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N ) |
$ GO TO 90 |
$ GO TO 90 |
* |
* |
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KSTEP = 1 |
|
* |
* Copy column K of A to column K of W and update it |
* Copy column K of A to column K of W and update it |
* |
* |
W( K, K ) = DBLE( A( K, K ) ) |
W( K, K ) = DBLE( A( K, K ) ) |
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$ W( K, 1 ), LDW, CONE, W( K, K ), 1 ) |
$ W( K, 1 ), LDW, CONE, W( K, K ), 1 ) |
W( K, K ) = DBLE( W( K, K ) ) |
W( K, K ) = DBLE( W( K, K ) ) |
* |
* |
KSTEP = 1 |
|
* |
|
* Determine rows and columns to be interchanged and whether |
* Determine rows and columns to be interchanged and whether |
* a 1-by-1 or 2-by-2 pivot block will be used |
* a 1-by-1 or 2-by-2 pivot block will be used |
* |
* |
ABSAKK = ABS( DBLE( W( K, K ) ) ) |
ABSAKK = ABS( DBLE( W( K, K ) ) ) |
* |
* |
* IMAX is the row-index of the largest off-diagonal element in |
* IMAX is the row-index of the largest off-diagonal element in |
* column K, and COLMAX is its absolute value |
* column K, and COLMAX is its absolute value. |
|
* Determine both COLMAX and IMAX. |
* |
* |
IF( K.LT.N ) THEN |
IF( K.LT.N ) THEN |
IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 ) |
IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 ) |
Line 512
|
Line 648
|
* |
* |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
* |
* |
* Column K is zero: set INFO and continue |
* Column K is zero or underflow: set INFO and continue |
* |
* |
IF( INFO.EQ.0 ) |
IF( INFO.EQ.0 ) |
$ INFO = K |
$ INFO = K |
KP = K |
KP = K |
A( K, K ) = DBLE( A( K, K ) ) |
A( K, K ) = DBLE( A( K, K ) ) |
ELSE |
ELSE |
|
* |
|
* ============================================================ |
|
* |
|
* BEGIN pivot search |
|
* |
|
* Case(1) |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
* |
* |
* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
Line 526
|
Line 668
|
KP = K |
KP = K |
ELSE |
ELSE |
* |
* |
|
* BEGIN pivot search along IMAX row |
|
* |
|
* |
* Copy column IMAX to column K+1 of W and update it |
* Copy column IMAX to column K+1 of W and update it |
* |
* |
CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) |
CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) |
Line 540
|
Line 685
|
W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) ) |
W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) ) |
* |
* |
* JMAX is the column-index of the largest off-diagonal |
* JMAX is the column-index of the largest off-diagonal |
* element in row IMAX, and ROWMAX is its absolute value |
* element in row IMAX, and ROWMAX is its absolute value. |
|
* Determine only ROWMAX. |
* |
* |
JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 ) |
JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 ) |
ROWMAX = CABS1( W( JMAX, K+1 ) ) |
ROWMAX = CABS1( W( JMAX, K+1 ) ) |
Line 549
|
Line 695
|
ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) ) |
ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) ) |
END IF |
END IF |
* |
* |
|
* Case(2) |
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN |
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN |
* |
* |
* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
* |
* |
KP = K |
KP = K |
|
* |
|
* Case(3) |
ELSE IF( ABS( DBLE( W( IMAX, K+1 ) ) ).GE.ALPHA*ROWMAX ) |
ELSE IF( ABS( DBLE( W( IMAX, K+1 ) ) ).GE.ALPHA*ROWMAX ) |
$ THEN |
$ THEN |
* |
* |
Line 562
|
Line 711
|
* |
* |
KP = IMAX |
KP = IMAX |
* |
* |
* copy column K+1 of W to column K |
* copy column K+1 of W to column K of W |
* |
* |
CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) |
CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) |
|
* |
|
* Case(4) |
ELSE |
ELSE |
* |
* |
* interchange rows and columns K+1 and IMAX, use 2-by-2 |
* interchange rows and columns K+1 and IMAX, use 2-by-2 |
Line 573
|
Line 724
|
KP = IMAX |
KP = IMAX |
KSTEP = 2 |
KSTEP = 2 |
END IF |
END IF |
|
* |
|
* |
|
* END pivot search along IMAX row |
|
* |
END IF |
END IF |
* |
* |
|
* END pivot search |
|
* |
|
* ============================================================ |
|
* |
|
* KK is the column of A where pivoting step stopped |
|
* |
KK = K + KSTEP - 1 |
KK = K + KSTEP - 1 |
* |
* |
* Updated column KP is already stored in column KK of W |
* Interchange rows and columns KP and KK. |
|
* Updated column KP is already stored in column KK of W. |
* |
* |
IF( KP.NE.KK ) THEN |
IF( KP.NE.KK ) THEN |
* |
* |
* Copy non-updated column KK to column KP |
* 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 ) = DBLE( A( KK, KK ) ) |
A( KP, KP ) = DBLE( A( KK, KK ) ) |
CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), |
CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), |
Line 590
|
Line 755
|
IF( KP.LT.N ) |
IF( KP.LT.N ) |
$ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) |
$ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) |
* |
* |
* Interchange rows KK and KP in first KK columns of A and W |
* Interchange rows KK and KP in first K-1 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 first KK columns of W. |
* |
* |
CALL ZSWAP( KK-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) |
IF( K.GT.1 ) |
|
$ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) |
CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) |
CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) |
END IF |
END IF |
* |
* |
Line 600
|
Line 769
|
* |
* |
* 1-by-1 pivot block D(k): column k of W now holds |
* 1-by-1 pivot block D(k): column k of W now holds |
* |
* |
* W(k) = L(k)*D(k) |
* W(k) = L(k)*D(k), |
* |
* |
* where L(k) is the k-th column of L |
* where L(k) is the k-th column of L |
* |
* |
* Store L(k) in column k of A |
* (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 ) = DBLE( W( K, K) ) to separately copy diagonal |
|
* element D(k,k) from W (potentially saves only one load)) |
CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) |
CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) |
IF( K.LT.N ) THEN |
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(4)) |
|
* |
R1 = ONE / DBLE( A( K, K ) ) |
R1 = ONE / DBLE( A( K, K ) ) |
CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 ) |
CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 ) |
* |
* |
* Conjugate W(k) |
* (2) Conjugate column W(k) |
* |
* |
CALL ZLACGV( N-K, W( K+1, K ), 1 ) |
CALL ZLACGV( N-K, W( K+1, K ), 1 ) |
END IF |
END IF |
|
* |
ELSE |
ELSE |
* |
* |
* 2-by-2 pivot block D(k): columns k and k+1 of W now hold |
* 2-by-2 pivot block D(k): columns k and k+1 of W now hold |
Line 624
|
Line 807
|
* where L(k) and L(k+1) are the k-th and (k+1)-th columns |
* where L(k) and L(k+1) are the k-th and (k+1)-th columns |
* of L |
* 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 |
IF( K.LT.N-1 ) THEN |
* |
* |
* Store L(k) and L(k+1) in columns k and k+1 of A |
* 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 ) ) |
|
* |
|
* = ( conj(D21)*( D11 ) D21*( -1 ) ) |
|
* ( ( -1 ) ( D22 ) ), |
|
* |
|
* where D11 = d22/d21, |
|
* D22 = d11/conj(d21), |
|
* D21 = T/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, since in 2x2 pivot case(4) |
|
* |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 ) |
D21 = W( K+1, K ) |
D11 = W( K+1, K+1 ) / D21 |
D11 = W( K+1, K+1 ) / D21 |
D22 = W( K, K ) / DCONJG( D21 ) |
D22 = W( K, K ) / DCONJG( D21 ) |
T = ONE / ( DBLE( D11*D22 )-ONE ) |
T = ONE / ( DBLE( D11*D22 )-ONE ) |
D21 = T / D21 |
D21 = T / D21 |
|
* |
|
* 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 |
DO 80 J = K + 2, N |
A( J, K ) = DCONJG( D21 )* |
A( J, K ) = DCONJG( D21 )* |
$ ( D11*W( J, K )-W( J, K+1 ) ) |
$ ( D11*W( J, K )-W( J, K+1 ) ) |
Line 646
|
Line 882
|
A( K+1, K ) = W( K+1, K ) |
A( K+1, K ) = W( K+1, K ) |
A( K+1, K+1 ) = W( K+1, K+1 ) |
A( K+1, K+1 ) = W( K+1, K+1 ) |
* |
* |
* Conjugate W(k) and W(k+1) |
* (2) Conjugate columns W(k) and W(k+1) |
* |
* |
CALL ZLACGV( N-K, W( K+1, K ), 1 ) |
CALL ZLACGV( N-K, W( K+1, K ), 1 ) |
CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 ) |
CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 ) |
|
* |
END IF |
END IF |
|
* |
END IF |
END IF |
* |
* |
* Store details of the interchanges in IPIV |
* Store details of the interchanges in IPIV |
Line 698
|
Line 936
|
110 CONTINUE |
110 CONTINUE |
* |
* |
* Put L21 in standard form by partially undoing the interchanges |
* Put L21 in standard form by partially undoing the interchanges |
* in columns 1:k-1 |
* of rows in columns 1:k-1 looping backwards from k-1 to 1 |
* |
* |
J = K - 1 |
J = K - 1 |
120 CONTINUE |
120 CONTINUE |
JJ = J |
* |
JP = IPIV( J ) |
* Undo the interchanges (if any) of rows JJ and JP at each |
IF( JP.LT.0 ) THEN |
* step J |
JP = -JP |
* |
|
* (Here, J is a diagonal index) |
|
JJ = J |
|
JP = IPIV( J ) |
|
IF( JP.LT.0 ) THEN |
|
JP = -JP |
|
* (Here, J is a diagonal index) |
|
J = J - 1 |
|
END IF |
|
* (NOTE: Here, J is used to determine row length. Length J |
|
* of the rows to swap back doesn't include diagonal element) |
J = J - 1 |
J = J - 1 |
END IF |
IF( JP.NE.JJ .AND. J.GE.1 ) |
J = J - 1 |
$ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) |
IF( JP.NE.JJ .AND. J.GE.1 ) |
IF( J.GT.1 ) |
$ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) |
|
IF( J.GE.1 ) |
|
$ GO TO 120 |
$ GO TO 120 |
* |
* |
* Set KB to the number of columns factorized |
* Set KB to the number of columns factorized |