DGEBRD reduces a general real M-by-N matrix A to upper or lower


 bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.


 If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

M

          M is INTEGER


          The number of rows in the matrix A.  M >= 0.

N

          N is INTEGER


          The number of columns in the matrix A.  N >= 0.

A

          A is DOUBLE PRECISION 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 orthogonal matrix Q as a product of elementary


            reflectors, and the elements above the first superdiagonal,


            with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of


            elementary reflectors, and the elements above the diagonal,


            with the array TAUP, represent the orthogonal matrix P as


            a product of elementary reflectors.


          See Further Details.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is DOUBLE PRECISION array, dimension (min(M,N))


          The diagonal elements of the bidiagonal matrix B:


          D(i) = A(i,i).

E

          E is DOUBLE PRECISION 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.

TAUQ

          TAUQ is DOUBLE PRECISION array, dimension (min(M,N))


          The scalar factors of the elementary reflectors which


          represent the orthogonal matrix Q. See Further Details.

TAUP

          TAUP is DOUBLE PRECISION array, dimension (min(M,N))


          The scalar factors of the elementary reflectors which


          represent the orthogonal matrix P. See Further Details.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of the array WORK.  LWORK >= max(1,M,N).


          For optimum performance LWORK >= (M+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

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2017

  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**T  and G(i) = I - taup * u * u**T


  where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T


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