*> \brief \b SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGEHD2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) * * .. Scalar Arguments .. * INTEGER IHI, ILO, INFO, LDA, N * .. * .. Array Arguments .. * REAL A( LDA, * ), TAU( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SGEHD2 reduces a real general matrix A to upper Hessenberg form H by *> an orthogonal similarity transformation: Q**T * A * Q = H . *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] ILO *> \verbatim *> ILO is INTEGER *> \endverbatim *> *> \param[in] IHI *> \verbatim *> IHI is 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 SGEBAL; otherwise they should be *> set to 1 and N respectively. See Further Details. *> 1 <= ILO <= IHI <= max(1,N). *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is REAL array, dimension (N-1) *> The scalar factors of the elementary reflectors (see Further *> Details). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, the i-th argument had an illegal value. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup realGEcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> 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**T *> *> 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). *> \endverbatim *> * ===================================================================== SUBROUTINE SGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, 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 .. INTEGER IHI, ILO, INFO, LDA, N * .. * .. Array Arguments .. REAL A( LDA, * ), TAU( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE PARAMETER ( ONE = 1.0E+0 ) * .. * .. Local Scalars .. INTEGER I REAL AII * .. * .. External Subroutines .. EXTERNAL SLARF, SLARFG, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 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 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEHD2', -INFO ) RETURN END IF * DO 10 I = ILO, IHI - 1 * * Compute elementary reflector H(i) to annihilate A(i+2:ihi,i) * CALL SLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, \$ TAU( I ) ) AII = A( I+1, I ) A( I+1, I ) = ONE * * Apply H(i) to A(1:ihi,i+1:ihi) from the right * CALL SLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ), \$ A( 1, I+1 ), LDA, WORK ) * * Apply H(i) to A(i+1:ihi,i+1:n) from the left * CALL SLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ), \$ A( I+1, I+1 ), LDA, WORK ) * A( I+1, I ) = AII 10 CONTINUE * RETURN * * End of SGEHD2 * END