SUBROUTINE CLAVHE( UPLO, TRANS, DIAG, N, NRHS, A, LDA, IPIV, B,
$ LDB, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* CLAVHE performs one of the matrix-vector operations
* x := A*x or x := A^H*x,
* where x is an N element vector and A is one of the factors
* from the symmetric factorization computed by CHETRF.
* CHETRF produces a factorization of the form
* U * D * U^H or L * D * L^H,
* where U (or L) is a product of permutation and unit upper (lower)
* triangular matrices, U^H (or L^H) is the conjugate transpose of
* U (or L), and D is Hermitian and block diagonal with 1 x 1 and
* 2 x 2 diagonal blocks. The multipliers for the transformations
* and the upper or lower triangular parts of the diagonal blocks
* are stored in the leading upper or lower triangle of the 2-D
* array A.
*
* If TRANS = 'N' or 'n', CLAVHE multiplies either by U or U * D
* (or L or L * D).
* If TRANS = 'C' or 'c', CLAVHE multiplies either by U^H or D * U^H
* (or L^H or D * L^H ).
*
* Arguments
* ==========
*
* UPLO - CHARACTER*1
* On entry, UPLO specifies whether the triangular matrix
* stored in A is upper or lower triangular.
* UPLO = 'U' or 'u' The matrix is upper triangular.
* UPLO = 'L' or 'l' The matrix is lower triangular.
* Unchanged on exit.
*
* TRANS - CHARACTER*1
* On entry, TRANS specifies the operation to be performed as
* follows:
* TRANS = 'N' or 'n' x := A*x.
* TRANS = 'C' or 'c' x := A^H*x.
* Unchanged on exit.
*
* DIAG - CHARACTER*1
* On entry, DIAG specifies whether the diagonal blocks are
* assumed to be unit matrices:
* DIAG = 'U' or 'u' Diagonal blocks are unit matrices.
* DIAG = 'N' or 'n' Diagonal blocks are non-unit.
* Unchanged on exit.
*
* N - INTEGER
* On entry, N specifies the order of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* NRHS - INTEGER
* On entry, NRHS specifies the number of right hand sides,
* i.e., the number of vectors x to be multiplied by A.
* NRHS must be at least zero.
* Unchanged on exit.
*
* A - COMPLEX array, dimension( LDA, N )
* On entry, A contains a block diagonal matrix and the
* multipliers of the transformations used to obtain it,
* stored as a 2-D triangular matrix.
* Unchanged on exit.
*
* LDA - INTEGER
* On entry, LDA specifies the first dimension of A as declared
* in the calling ( sub ) program. LDA must be at least
* max( 1, N ).
* Unchanged on exit.
*
* IPIV - INTEGER array, dimension( N )
* On entry, IPIV contains the vector of pivot indices as
* determined by CSYTRF or CHETRF.
* If IPIV( K ) = K, no interchange was done.
* If IPIV( K ) <> K but IPIV( K ) > 0, then row K was inter-
* changed with row IPIV( K ) and a 1 x 1 pivot block was used.
* If IPIV( K ) < 0 and UPLO = 'U', then row K-1 was exchanged
* with row | IPIV( K ) | and a 2 x 2 pivot block was used.
* If IPIV( K ) < 0 and UPLO = 'L', then row K+1 was exchanged
* with row | IPIV( K ) | and a 2 x 2 pivot block was used.
*
* B - COMPLEX array, dimension( LDB, NRHS )
* On entry, B contains NRHS vectors of length N.
* On exit, B is overwritten with the product A * B.
*
* LDB - INTEGER
* On entry, LDB contains the leading dimension of B as
* declared in the calling program. LDB must be at least
* max( 1, N ).
* Unchanged on exit.
*
* INFO - INTEGER
* INFO is the error flag.
* On exit, a value of 0 indicates a successful exit.
* A negative value, say -K, indicates that the K-th argument
* has an illegal value.
*
* =====================================================================
*
* .. Parameters ..
COMPLEX ONE
PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL NOUNIT
INTEGER J, K, KP
COMPLEX D11, D12, D21, D22, T1, T2
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL CGEMV, CGERU, CLACGV, CSCAL, CSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.LSAME( TRANS, 'C' ) )
$ THEN
INFO = -2
ELSE IF( .NOT.LSAME( DIAG, 'U' ) .AND. .NOT.LSAME( DIAG, 'N' ) )
$ THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CLAVHE ', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
NOUNIT = LSAME( DIAG, 'N' )
*------------------------------------------
*
* Compute B := A * B (No transpose)
*
*------------------------------------------
IF( LSAME( TRANS, 'N' ) ) THEN
*
* Compute B := U*B
* where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1))
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Loop forward applying the transformations.
*
K = 1
10 CONTINUE
IF( K.GT.N )
$ GO TO 30
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 pivot block
*
* Multiply by the diagonal element if forming U * D.
*
IF( NOUNIT )
$ CALL CSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
*
* Multiply by P(K) * inv(U(K)) if K > 1.
*
IF( K.GT.1 ) THEN
*
* Apply the transformation.
*
CALL CGERU( K-1, NRHS, ONE, A( 1, K ), 1, B( K, 1 ),
$ LDB, B( 1, 1 ), LDB )
*
* Interchange if P(K) != I.
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
K = K + 1
ELSE
*
* 2 x 2 pivot block
*
* Multiply by the diagonal block if forming U * D.
*
IF( NOUNIT ) THEN
D11 = A( K, K )
D22 = A( K+1, K+1 )
D12 = A( K, K+1 )
D21 = CONJG( D12 )
DO 20 J = 1, NRHS
T1 = B( K, J )
T2 = B( K+1, J )
B( K, J ) = D11*T1 + D12*T2
B( K+1, J ) = D21*T1 + D22*T2
20 CONTINUE
END IF
*
* Multiply by P(K) * inv(U(K)) if K > 1.
*
IF( K.GT.1 ) THEN
*
* Apply the transformations.
*
CALL CGERU( K-1, NRHS, ONE, A( 1, K ), 1, B( K, 1 ),
$ LDB, B( 1, 1 ), LDB )
CALL CGERU( K-1, NRHS, ONE, A( 1, K+1 ), 1,
$ B( K+1, 1 ), LDB, B( 1, 1 ), LDB )
*
* Interchange if P(K) != I.
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K )
$ CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
K = K + 2
END IF
GO TO 10
30 CONTINUE
*
* Compute B := L*B
* where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m)) .
*
ELSE
*
* Loop backward applying the transformations to B.
*
K = N
40 CONTINUE
IF( K.LT.1 )
$ GO TO 60
*
* Test the pivot index. If greater than zero, a 1 x 1
* pivot was used, otherwise a 2 x 2 pivot was used.
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 pivot block:
*
* Multiply by the diagonal element if forming L * D.
*
IF( NOUNIT )
$ CALL CSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
*
* Multiply by P(K) * inv(L(K)) if K < N.
*
IF( K.NE.N ) THEN
KP = IPIV( K )
*
* Apply the transformation.
*
CALL CGERU( N-K, NRHS, ONE, A( K+1, K ), 1,
$ B( K, 1 ), LDB, B( K+1, 1 ), LDB )
*
* Interchange if a permutation was applied at the
* K-th step of the factorization.
*
IF( KP.NE.K )
$ CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
K = K - 1
*
ELSE
*
* 2 x 2 pivot block:
*
* Multiply by the diagonal block if forming L * D.
*
IF( NOUNIT ) THEN
D11 = A( K-1, K-1 )
D22 = A( K, K )
D21 = A( K, K-1 )
D12 = CONJG( D21 )
DO 50 J = 1, NRHS
T1 = B( K-1, J )
T2 = B( K, J )
B( K-1, J ) = D11*T1 + D12*T2
B( K, J ) = D21*T1 + D22*T2
50 CONTINUE
END IF
*
* Multiply by P(K) * inv(L(K)) if K < N.
*
IF( K.NE.N ) THEN
*
* Apply the transformation.
*
CALL CGERU( N-K, NRHS, ONE, A( K+1, K ), 1,
$ B( K, 1 ), LDB, B( K+1, 1 ), LDB )
CALL CGERU( N-K, NRHS, ONE, A( K+1, K-1 ), 1,
$ B( K-1, 1 ), LDB, B( K+1, 1 ), LDB )
*
* Interchange if a permutation was applied at the
* K-th step of the factorization.
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K )
$ CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
K = K - 2
END IF
GO TO 40
60 CONTINUE
END IF
*--------------------------------------------------
*
* Compute B := A^H * B (conjugate transpose)
*
*--------------------------------------------------
ELSE
*
* Form B := U^H*B
* where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1))
* and U^H = inv(U^H(1))*P(1)* ... *inv(U^H(m))*P(m)
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Loop backward applying the transformations.
*
K = N
70 IF( K.LT.1 )
$ GO TO 90
*
* 1 x 1 pivot block.
*
IF( IPIV( K ).GT.0 ) THEN
IF( K.GT.1 ) THEN
*
* Interchange if P(K) != I.
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Apply the transformation
* y = y - B' conjg(x),
* where x is a column of A and y is a row of B.
*
CALL CLACGV( NRHS, B( K, 1 ), LDB )
CALL CGEMV( 'Conjugate', K-1, NRHS, ONE, B, LDB,
$ A( 1, K ), 1, ONE, B( K, 1 ), LDB )
CALL CLACGV( NRHS, B( K, 1 ), LDB )
END IF
IF( NOUNIT )
$ CALL CSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
K = K - 1
*
* 2 x 2 pivot block.
*
ELSE
IF( K.GT.2 ) THEN
*
* Interchange if P(K) != I.
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K-1 )
$ CALL CSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ),
$ LDB )
*
* Apply the transformations
* y = y - B' conjg(x),
* where x is a block column of A and y is a block
* row of B.
*
CALL CLACGV( NRHS, B( K, 1 ), LDB )
CALL CGEMV( 'Conjugate', K-2, NRHS, ONE, B, LDB,
$ A( 1, K ), 1, ONE, B( K, 1 ), LDB )
CALL CLACGV( NRHS, B( K, 1 ), LDB )
*
CALL CLACGV( NRHS, B( K-1, 1 ), LDB )
CALL CGEMV( 'Conjugate', K-2, NRHS, ONE, B, LDB,
$ A( 1, K-1 ), 1, ONE, B( K-1, 1 ), LDB )
CALL CLACGV( NRHS, B( K-1, 1 ), LDB )
END IF
*
* Multiply by the diagonal block if non-unit.
*
IF( NOUNIT ) THEN
D11 = A( K-1, K-1 )
D22 = A( K, K )
D12 = A( K-1, K )
D21 = CONJG( D12 )
DO 80 J = 1, NRHS
T1 = B( K-1, J )
T2 = B( K, J )
B( K-1, J ) = D11*T1 + D12*T2
B( K, J ) = D21*T1 + D22*T2
80 CONTINUE
END IF
K = K - 2
END IF
GO TO 70
90 CONTINUE
*
* Form B := L^H*B
* where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m))
* and L^H = inv(L^H(m))*P(m)* ... *inv(L^H(1))*P(1)
*
ELSE
*
* Loop forward applying the L-transformations.
*
K = 1
100 CONTINUE
IF( K.GT.N )
$ GO TO 120
*
* 1 x 1 pivot block
*
IF( IPIV( K ).GT.0 ) THEN
IF( K.LT.N ) THEN
*
* Interchange if P(K) != I.
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Apply the transformation
*
CALL CLACGV( NRHS, B( K, 1 ), LDB )
CALL CGEMV( 'Conjugate', N-K, NRHS, ONE, B( K+1, 1 ),
$ LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
CALL CLACGV( NRHS, B( K, 1 ), LDB )
END IF
IF( NOUNIT )
$ CALL CSCAL( NRHS, A( K, K ), B( K, 1 ), LDB )
K = K + 1
*
* 2 x 2 pivot block.
*
ELSE
IF( K.LT.N-1 ) THEN
*
* Interchange if P(K) != I.
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K+1 )
$ CALL CSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ),
$ LDB )
*
* Apply the transformation
*
CALL CLACGV( NRHS, B( K+1, 1 ), LDB )
CALL CGEMV( 'Conjugate', N-K-1, NRHS, ONE,
$ B( K+2, 1 ), LDB, A( K+2, K+1 ), 1, ONE,
$ B( K+1, 1 ), LDB )
CALL CLACGV( NRHS, B( K+1, 1 ), LDB )
*
CALL CLACGV( NRHS, B( K, 1 ), LDB )
CALL CGEMV( 'Conjugate', N-K-1, NRHS, ONE,
$ B( K+2, 1 ), LDB, A( K+2, K ), 1, ONE,
$ B( K, 1 ), LDB )
CALL CLACGV( NRHS, B( K, 1 ), LDB )
END IF
*
* Multiply by the diagonal block if non-unit.
*
IF( NOUNIT ) THEN
D11 = A( K, K )
D22 = A( K+1, K+1 )
D21 = A( K+1, K )
D12 = CONJG( D21 )
DO 110 J = 1, NRHS
T1 = B( K, J )
T2 = B( K+1, J )
B( K, J ) = D11*T1 + D12*T2
B( K+1, J ) = D21*T1 + D22*T2
110 CONTINUE
END IF
K = K + 2
END IF
GO TO 100
120 CONTINUE
END IF
*
END IF
RETURN
*
* End of CLAVHE
*
END