SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGEHRD reduces a real general matrix A to upper Hessenberg form H by
* an orthogonal similarity transformation: Q' * A * Q = H .
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that A is already upper triangular in rows
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
* set by a previous call to DGEBAL; otherwise they should be
* set to 1 and N respectively. See Further Details.
* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the N-by-N general matrix to be reduced.
* On exit, the upper triangle and the first subdiagonal of A
* are overwritten with the upper Hessenberg matrix H, and the
* elements below the first subdiagonal, with the array TAU,
* represent the orthogonal matrix Q as a product of elementary
* reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* TAU (output) DOUBLE PRECISION array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
* zero.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The length of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is the
* optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of (ihi-ilo) elementary
* reflectors
*
* Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
* exit in A(i+2:ihi,i), and tau in TAU(i).
*
* The contents of A are illustrated by the following example, with
* n = 7, ilo = 2 and ihi = 6:
*
* on entry, on exit,
*
* ( a a a a a a a ) ( a a h h h h a )
* ( a a a a a a ) ( a h h h h a )
* ( a a a a a a ) ( h h h h h h )
* ( a a a a a a ) ( v2 h h h h h )
* ( a a a a a a ) ( v2 v3 h h h h )
* ( a a a a a a ) ( v2 v3 v4 h h h )
* ( a ) ( a )
*
* where a denotes an element of the original matrix A, h denotes a
* modified element of the upper Hessenberg matrix H, and vi denotes an
* element of the vector defining H(i).
*
* This file is a slight modification of LAPACK-3.0's DGEHRD
* subroutine incorporating improvements proposed by Quintana-Orti and
* Van de Geijn (2005).
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0,
$ ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
$ NBMIN, NH, NX
DOUBLE PRECISION EI
* ..
* .. Local Arrays ..
DOUBLE PRECISION T( LDT, NBMAX )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB, DTRMM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
LWKOPT = N*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEHRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
*
DO 10 I = 1, ILO - 1
TAU( I ) = ZERO
10 CONTINUE
DO 20 I = MAX( 1, IHI ), N - 1
TAU( I ) = ZERO
20 CONTINUE
*
* Quick return if possible
*
NH = IHI - ILO + 1
IF( NH.LE.1 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Determine the block size
*
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
NBMIN = 2
IWS = 1
IF( NB.GT.1 .AND. NB.LT.NH ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code)
*
NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
IF( NX.LT.NH ) THEN
*
* Determine if workspace is large enough for blocked code
*
IWS = N*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code
*
NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI,
$ -1 ) )
IF( LWORK.GE.N*NBMIN ) THEN
NB = LWORK / N
ELSE
NB = 1
END IF
END IF
END IF
END IF
LDWORK = N
*
IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
*
* Use unblocked code below
*
I = ILO
*
ELSE
*
* Use blocked code
*
DO 40 I = ILO, IHI - 1 - NX, NB
IB = MIN( NB, IHI-I )
*
* Reduce columns i:i+ib-1 to Hessenberg form, returning the
* matrices V and T of the block reflector H = I - V*T*V'
* which performs the reduction, and also the matrix Y = A*V*T
*
CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
$ WORK, LDWORK )
*
* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
* right, computing A := A - Y * V'. V(i+ib,ib-1) must be set
* to 1
*
EI = A( I+IB, I+IB-1 )
A( I+IB, I+IB-1 ) = ONE
CALL DGEMM( 'No transpose', 'Transpose',
$ IHI, IHI-I-IB+1,
$ IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
$ A( 1, I+IB ), LDA )
A( I+IB, I+IB-1 ) = EI
*
* Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
* right
*
CALL DTRMM( 'Right', 'Lower', 'Transpose',
$ 'Unit', I, IB-1,
$ ONE, A( I+1, I ), LDA, WORK, LDWORK )
DO 30 J = 0, IB-2
CALL DAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
$ A( 1, I+J+1 ), 1 )
30 CONTINUE
*
* Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
* left
*
CALL DLARFB( 'Left', 'Transpose', 'Forward',
$ 'Columnwise',
$ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,
$ A( I+1, I+IB ), LDA, WORK, LDWORK )
40 CONTINUE
END IF
*
* Use unblocked code to reduce the rest of the matrix
*
CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
WORK( 1 ) = IWS
*
RETURN
*
* End of DGEHRD
*
END