*> \brief \b SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGEQRT2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGEQRT2( M, N, A, LDA, T, LDT, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDT, M, N * .. * .. Array Arguments .. * REAL A( LDA, * ), T( LDT, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SGEQRT2 computes a QR factorization of a real M-by-N matrix A, *> using the compact WY representation of Q. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= N. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL array, dimension (LDA,N) *> On entry, the real M-by-N matrix A. On exit, the elements on and *> above the diagonal contain the N-by-N upper triangular matrix R; the *> elements below the diagonal are the columns of V. See below for *> further details. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[out] T *> \verbatim *> T is REAL array, dimension (LDT,N) *> The N-by-N upper triangular factor of the block reflector. *> The elements on and above the diagonal contain the block *> reflector T; the elements below the diagonal are not used. *> See below for further details. *> \endverbatim *> *> \param[in] LDT *> \verbatim *> LDT is INTEGER *> The leading dimension of the array T. LDT >= max(1,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 V stores the elementary reflectors H(i) in the i-th column *> below the diagonal. For example, if M=5 and N=3, the matrix V is *> *> V = ( 1 ) *> ( v1 1 ) *> ( v1 v2 1 ) *> ( v1 v2 v3 ) *> ( v1 v2 v3 ) *> *> where the vi's represent the vectors which define H(i), which are returned *> in the matrix A. The 1's along the diagonal of V are not stored in A. The *> block reflector H is then given by *> *> H = I - V * T * V**T *> *> where V**T is the transpose of V. *> \endverbatim *> * ===================================================================== SUBROUTINE SGEQRT2( M, N, A, LDA, T, LDT, 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 INFO, LDA, LDT, M, N * .. * .. Array Arguments .. REAL A( LDA, * ), T( LDT, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER( ONE = 1.0, ZERO = 0.0 ) * .. * .. Local Scalars .. INTEGER I, K REAL AII, ALPHA * .. * .. External Subroutines .. EXTERNAL SLARFG, SGEMV, SGER, STRMV, XERBLA * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGEQRT2', -INFO ) RETURN END IF * K = MIN( M, N ) * DO I = 1, K * * Generate elem. refl. H(i) to annihilate A(i+1:m,i), tau(I) -> T(I,1) * CALL SLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, \$ T( I, 1 ) ) IF( I.LT.N ) THEN * * Apply H(i) to A(I:M,I+1:N) from the left * AII = A( I, I ) A( I, I ) = ONE * * W(1:N-I) := A(I:M,I+1:N)^H * A(I:M,I) [W = T(:,N)] * CALL SGEMV( 'T',M-I+1, N-I, ONE, A( I, I+1 ), LDA, \$ A( I, I ), 1, ZERO, T( 1, N ), 1 ) * * A(I:M,I+1:N) = A(I:m,I+1:N) + alpha*A(I:M,I)*W(1:N-1)^H * ALPHA = -(T( I, 1 )) CALL SGER( M-I+1, N-I, ALPHA, A( I, I ), 1, \$ T( 1, N ), 1, A( I, I+1 ), LDA ) A( I, I ) = AII END IF END DO * DO I = 2, N AII = A( I, I ) A( I, I ) = ONE * * T(1:I-1,I) := alpha * A(I:M,1:I-1)**T * A(I:M,I) * ALPHA = -T( I, 1 ) CALL SGEMV( 'T', M-I+1, I-1, ALPHA, A( I, 1 ), LDA, \$ A( I, I ), 1, ZERO, T( 1, I ), 1 ) A( I, I ) = AII * * T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I) * CALL STRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 ) * * T(I,I) = tau(I) * T( I, I ) = T( I, 1 ) T( I, 1) = ZERO END DO * * End of SGEQRT2 * END