ZGEBD2 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.

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 COMPLEX*16 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.

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 COMPLEX*16 array, dimension (min(M,N))


          The scalar factors of the elementary reflectors which


          represent the unitary matrix Q. See Further Details.

TAUP

          TAUP is COMPLEX*16 array, dimension (min(M,N))


          The scalar factors of the elementary reflectors which


          represent the unitary matrix P. See Further Details.

WORK

          WORK is COMPLEX*16 array, dimension (max(M,N))

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.

June 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**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).