*> \brief \b DLASYF_RK computes a partial factorization of a real symmetric 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 DLASYF_RK + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLASYF_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( * ) * DOUBLE PRECISION A( LDA, * ), E( * ), W( LDW, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> DLASYF_RK computes a partial factorization of a real symmetric *> 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**T U22**T ) *> *> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) 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. *> *> DLASYF_RK is an auxiliary routine called by DSYTRF_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 *> symmetric 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 DOUBLE PRECISION array, dimension (LDA,N) *> On entry, the symmetric 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 symmetric 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 DOUBLE PRECISION array, dimension (N) *> On exit, contains the superdiagonal (or subdiagonal) *> elements of the symmetric 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 symmetric 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 DOUBLE PRECISION 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 doubleSYcomputational * *> \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 DLASYF_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( * ) DOUBLE PRECISION A( LDA, * ), E( * ), W( LDW, * ) * .. * * ===================================================================== * * .. 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 DONE INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, K, KK, KW, KKW, $ KP, KSTEP, P, II DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, $ DTEMP, R1, ROWMAX, T, SFMIN * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH * .. * .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT * .. * .. Executable Statements .. * INFO = 0 * * Initialize ALPHA for use in choosing pivot block size. * ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT * * Compute machine safe minimum * SFMIN = DLAMCH( '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 * * Initilize the first entry of array E, where superdiagonal * elements of D are stored * E( 1 ) = ZERO * * 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 * CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 ) IF( K.LT.N ) $ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), $ LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 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( 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 = IDAMAX( K-1, W( 1, KW ), 1 ) COLMAX = ABS( 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 CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) * * Set E( K ) to zero * IF( K.GT.1 ) $ E( K ) = ZERO * ELSE * * ============================================================ * * Test for interchange * * 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 * 12 CONTINUE * * Begin pivot search loop body * * * Copy column IMAX to column KW-1 of W and update it * CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, $ W( IMAX+1, KW-1 ), 1 ) * IF( K.LT.N ) $ CALL DGEMV( 'No transpose', K, N-K, -ONE, $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW, $ ONE, W( 1, KW-1 ), 1 ) * * 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 + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ), $ 1 ) ROWMAX = ABS( W( JMAX, KW-1 ) ) ELSE ROWMAX = ZERO END IF * IF( IMAX.GT.1 ) THEN ITEMP = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 ) DTEMP = ABS( W( ITEMP, KW-1 ) ) IF( DTEMP.GT.ROWMAX ) THEN ROWMAX = DTEMP JMAX = ITEMP END IF END IF * * Equivalent to testing for * ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX * (used to handle NaN and Inf) * IF( .NOT.(ABS( 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 DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) * DONE = .TRUE. * * 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. 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 DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) * END IF * * End pivot search loop body * IF( .NOT. DONE ) GOTO 12 * END IF * * ============================================================ * KK = K - KSTEP + 1 * * KKW is the column of W which corresponds to column KK of A * KKW = NB + KK - N * IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN * * Copy non-updated column K to column P * CALL DCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA ) CALL DCOPY( P, A( 1, K ), 1, A( 1, P ), 1 ) * * Interchange rows K and P in last N-K+1 columns of A * and last N-K+2 columns of W * CALL DSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA ) CALL DSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW ) END IF * * Updated column KP is already stored in column KKW of W * IF( KP.NE.KK ) THEN * * Copy non-updated column KK to column KP * A( KP, K ) = A( KK, K ) CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), $ LDA ) CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 ) * * Interchange rows KK and KP in last N-KK+1 columns * of A and W * CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA ) CALL DSWAP( 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(k) = U(k)*D(k) * * where U(k) is the k-th column of U * * Store U(k) in column k of A * CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) IF( K.GT.1 ) THEN IF( ABS( A( K, K ) ).GE.SFMIN ) THEN R1 = ONE / A( K, K ) CALL DSCAL( K-1, R1, A( 1, K ), 1 ) ELSE IF( A( K, K ).NE.ZERO ) THEN DO 14 II = 1, K - 1 A( II, K ) = A( II, K ) / A( K, K ) 14 CONTINUE END IF * * Store the superdiagonal element of D in array E * E( K ) = ZERO * END IF * ELSE * * 2-by-2 pivot block D(k): columns KW and KW-1 of W 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 * IF( K.GT.2 ) THEN * * Store U(k) and U(k-1) in columns k and k-1 of A * D12 = W( K-1, KW ) D11 = W( K, KW ) / D12 D22 = W( K-1, KW-1 ) / D12 T = ONE / ( D11*D22-ONE ) DO 20 J = 1, K - 2 A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) / $ D12 ) A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) / $ D12 ) 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 ) = ZERO A( K, K ) = W( K, KW ) E( K ) = W( K-1, KW ) E( K-1 ) = ZERO * 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**T = A11 - U12*W**T * * computing blocks of NB columns at a time * 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 CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE, $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE, $ A( J, JJ ), 1 ) 40 CONTINUE * * Update the rectangular superdiagonal block * IF( J.GE.2 ) $ CALL DGEMM( 'No transpose', 'Transpose', J-1, JB, $ N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ), $ LDW, ONE, 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 * * Initilize the unused last entry of the subdiagonal array E. * E( N ) = ZERO * * 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 it * CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 ) IF( K.GT.1 ) $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), $ LDA, W( K, 1 ), LDW, ONE, W( K, K ), 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( 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 + IDAMAX( N-K, W( K+1, K ), 1 ) COLMAX = ABS( 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 CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) * * Set E( K ) to zero * IF( K.LT.N ) $ E( K ) = ZERO * ELSE * * ============================================================ * * Test for interchange * * 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 DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1) CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, $ W( IMAX, K+1 ), 1 ) IF( K.GT.1 ) $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW, $ ONE, W( K, K+1 ), 1 ) * * 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 + IDAMAX( IMAX-K, W( K, K+1 ), 1 ) ROWMAX = ABS( W( JMAX, K+1 ) ) ELSE ROWMAX = ZERO END IF * IF( IMAX.LT.N ) THEN ITEMP = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1) DTEMP = ABS( W( ITEMP, K+1 ) ) IF( DTEMP.GT.ROWMAX ) THEN ROWMAX = DTEMP JMAX = ITEMP END IF END IF * * Equivalent to testing for * ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX * (used to handle NaN and Inf) * IF( .NOT.( ABS( 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 DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) * DONE = .TRUE. * * 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. 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 DCOPY( 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 * * ============================================================ * KK = K + KSTEP - 1 * IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN * * Copy non-updated column K to column P * CALL DCOPY( P-K, A( K, K ), 1, A( P, K ), LDA ) CALL DCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 ) * * Interchange rows K and P in first K columns of A * and first K+1 columns of W * CALL DSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA ) CALL DSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW ) END IF * * Updated column KP is already stored in column KK of W * IF( KP.NE.KK ) THEN * * Copy non-updated column KK to column KP * A( KP, K ) = A( KK, K ) CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA ) CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 ) * * Interchange rows KK and KP in first KK columns of A and W * CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) CALL DSWAP( 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 * * Store L(k) in column k of A * CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) IF( K.LT.N ) THEN IF( ABS( A( K, K ) ).GE.SFMIN ) THEN R1 = ONE / A( K, K ) CALL DSCAL( N-K, R1, A( K+1, K ), 1 ) ELSE IF( A( K, K ).NE.ZERO ) THEN DO 74 II = K + 1, N A( II, K ) = A( II, K ) / A( K, K ) 74 CONTINUE END IF * * Store the subdiagonal element of D in array E * E( K ) = ZERO * 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 * IF( K.LT.N-1 ) THEN * * Store L(k) and L(k+1) in columns k and k+1 of A * D21 = W( K+1, K ) D11 = W( K+1, K+1 ) / D21 D22 = W( K, K ) / D21 T = ONE / ( D11*D22-ONE ) DO 80 J = K + 2, N A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) / $ 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 ) = ZERO A( K+1, K+1 ) = W( K+1, K+1 ) E( K ) = W( K+1, K ) E( K+1 ) = ZERO * 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**T = A22 - L21*W**T * * computing blocks of NB columns at a time * 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 CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE, $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE, $ A( JJ, JJ ), 1 ) 100 CONTINUE * * Update the rectangular subdiagonal block * IF( J+JB.LE.N ) $ CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), $ LDW, ONE, A( J+JB, J ), LDA ) 110 CONTINUE * * Set KB to the number of columns factorized * KB = K - 1 * END IF * RETURN * * End of DLASYF_RK * END