*> \brief \b CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CGEBD2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. * REAL D( * ), E( * ) * COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CGEBD2 reduces a complex general m by n matrix A to upper or lower *> real bidiagonal form B by a unitary transformation: Q**H * A * P = B. *> *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows in the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns in the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX array, dimension (LDA,N) *> On entry, the m by n general matrix to be reduced. *> On exit, *> if m >= n, the diagonal and the first superdiagonal are *> overwritten with the upper bidiagonal matrix B; the *> elements below the diagonal, with the array TAUQ, represent *> the unitary matrix Q as a product of elementary *> reflectors, and the elements above the first superdiagonal, *> with the array TAUP, represent the unitary matrix P as *> a product of elementary reflectors; *> if m < n, the diagonal and the first subdiagonal are *> overwritten with the lower bidiagonal matrix B; the *> elements below the first subdiagonal, with the array TAUQ, *> represent the unitary matrix Q as a product of *> elementary reflectors, and the elements above the diagonal, *> with the array TAUP, represent the unitary matrix P 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,M). *> \endverbatim *> *> \param[out] D *> \verbatim *> D is REAL array, dimension (min(M,N)) *> The diagonal elements of the bidiagonal matrix B: *> D(i) = A(i,i). *> \endverbatim *> *> \param[out] E *> \verbatim *> E is REAL array, dimension (min(M,N)-1) *> The off-diagonal elements of the bidiagonal matrix B: *> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; *> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. *> \endverbatim *> *> \param[out] TAUQ *> \verbatim *> TAUQ is COMPLEX array, dimension (min(M,N)) *> The scalar factors of the elementary reflectors which *> represent the unitary matrix Q. See Further Details. *> \endverbatim *> *> \param[out] TAUP *> \verbatim *> TAUP is COMPLEX array, dimension (min(M,N)) *> The scalar factors of the elementary reflectors which *> represent the unitary matrix P. See Further Details. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX array, dimension (max(M,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 June 2017 * *> \ingroup complexGEcomputational * @precisions normal c -> s d z * *> \par Further Details: * ===================== *> *> \verbatim *> *> The matrices Q and P are represented as products of elementary *> reflectors: *> *> If m >= n, *> *> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) *> *> Each H(i) and G(i) has the form: *> *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H *> *> where tauq and taup are complex scalars, and v and u are complex *> vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in *> A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in *> A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). *> *> If m < n, *> *> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) *> *> Each H(i) and G(i) has the form: *> *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H *> *> where tauq and taup are complex scalars, v and u are complex vectors; *> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); *> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); *> tauq is stored in TAUQ(i) and taup in TAUP(i). *> *> The contents of A on exit are illustrated by the following examples: *> *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): *> *> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) *> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) *> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) *> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) *> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) *> ( v1 v2 v3 v4 v5 ) *> *> where d and e denote diagonal and off-diagonal elements of B, vi *> denotes an element of the vector defining H(i), and ui an element of *> the vector defining G(i). *> \endverbatim *> * ===================================================================== SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) * * -- LAPACK computational routine (version 3.7.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2017 * * .. Scalar Arguments .. INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. REAL D( * ), E( * ) COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX ZERO, ONE PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ), $ ONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. INTEGER I COMPLEX ALPHA * .. * .. External Subroutines .. EXTERNAL CLACGV, CLARF, CLARFG, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC CONJG, MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * 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 END IF IF( INFO.LT.0 ) THEN CALL XERBLA( 'CGEBD2', -INFO ) RETURN END IF * IF( M.GE.N ) THEN * * Reduce to upper bidiagonal form * DO 10 I = 1, N * * Generate elementary reflector H(i) to annihilate A(i+1:m,i) * ALPHA = A( I, I ) CALL CLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1, $ TAUQ( I ) ) D( I ) = ALPHA A( I, I ) = ONE * * Apply H(i)**H to A(i:m,i+1:n) from the left * IF( I.LT.N ) $ CALL CLARF( 'Left', M-I+1, N-I, A( I, I ), 1, $ CONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK ) A( I, I ) = D( I ) * IF( I.LT.N ) THEN * * Generate elementary reflector G(i) to annihilate * A(i,i+2:n) * CALL CLACGV( N-I, A( I, I+1 ), LDA ) ALPHA = A( I, I+1 ) CALL CLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), $ LDA, TAUP( I ) ) E( I ) = ALPHA A( I, I+1 ) = ONE * * Apply G(i) to A(i+1:m,i+1:n) from the right * CALL CLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) CALL CLACGV( N-I, A( I, I+1 ), LDA ) A( I, I+1 ) = E( I ) ELSE TAUP( I ) = ZERO END IF 10 CONTINUE ELSE * * Reduce to lower bidiagonal form * DO 20 I = 1, M * * Generate elementary reflector G(i) to annihilate A(i,i+1:n) * CALL CLACGV( N-I+1, A( I, I ), LDA ) ALPHA = A( I, I ) CALL CLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA, $ TAUP( I ) ) D( I ) = ALPHA A( I, I ) = ONE * * Apply G(i) to A(i+1:m,i:n) from the right * IF( I.LT.M ) $ CALL CLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, $ TAUP( I ), A( I+1, I ), LDA, WORK ) CALL CLACGV( N-I+1, A( I, I ), LDA ) A( I, I ) = D( I ) * IF( I.LT.M ) THEN * * Generate elementary reflector H(i) to annihilate * A(i+2:m,i) * ALPHA = A( I+1, I ) CALL CLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1, $ TAUQ( I ) ) E( I ) = ALPHA A( I+1, I ) = ONE * * Apply H(i)**H to A(i+1:m,i+1:n) from the left * CALL CLARF( 'Left', M-I, N-I, A( I+1, I ), 1, $ CONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA, $ WORK ) A( I+1, I ) = E( I ) ELSE TAUQ( I ) = ZERO END IF 20 CONTINUE END IF RETURN * * End of CGEBD2 * END