double precision
[SVD: auxiliary]

Functions

magma_int_t magma_dlabrd_gpu (magma_int_t m, magma_int_t n, magma_int_t nb, double *A, magma_int_t lda, double *dA, magma_int_t ldda, double *d, double *e, double *tauq, double *taup, double *X, magma_int_t ldx, double *dX, magma_int_t lddx, double *Y, magma_int_t ldy, double *dY, magma_int_t lddy)
 DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

Function Documentation

magma_int_t magma_dlabrd_gpu ( magma_int_t  m,
magma_int_t  n,
magma_int_t  nb,
double *  A,
magma_int_t  lda,
double *  dA,
magma_int_t  ldda,
double *  d,
double *  e,
double *  tauq,
double *  taup,
double *  X,
magma_int_t  ldx,
double *  dX,
magma_int_t  lddx,
double *  Y,
magma_int_t  ldy,
double *  dY,
magma_int_t  lddy 
)

DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by DGEBRD.

Parameters:
[in] m INTEGER The number of rows in the matrix A.
[in] n INTEGER The number of columns in the matrix A.
[in] nb INTEGER The number of leading rows and columns of A to be reduced.
[in,out] A DOUBLE_PRECISION array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.
If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details.
[in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M).
[in,out] dA DOUBLE_PRECISION array, dimension (LDDA,N) Copy of A on GPU.
[in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M).
[out] d DOUBLE_PRECISION array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).
[out] e DOUBLE_PRECISION array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.
[out] tauq DOUBLE_PRECISION array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out] taup DOUBLE_PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out] X DOUBLE_PRECISION array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.
[in] ldx INTEGER The leading dimension of the array X. LDX >= M.
[out] dX DOUBLE_PRECISION array, dimension (LDDX,NB) Copy of X on GPU.
[in] lddx INTEGER The leading dimension of the array dX. LDDX >= M.
[out] Y DOUBLE_PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.
[in] ldy INTEGER The leading dimension of the array Y. LDY >= N.
[out] dY DOUBLE_PRECISION array, dimension (LDDY,NB) Copy of Y on GPU.
[in] lddy INTEGER The leading dimension of the array dY. LDDY >= N.

Further Details --------------- The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'

where tauq and taup are real scalars, and v and u are real vectors.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 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).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y' - X*U'.

The contents of A on exit are illustrated by the following examples with nb = 2:

    m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

      (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
      (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
      (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
      (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
      (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
      (  v1  v2  a   a   a  )
    

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).


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