DGGHD3 reduces a pair of real matrices (A,B) to generalized upper


 Hessenberg form using orthogonal transformations, where A is a


 general matrix and B is upper triangular.  The form of the


 generalized eigenvalue problem is


    A*x = lambda*B*x,


 and B is typically made upper triangular by computing its QR


 factorization and moving the orthogonal matrix Q to the left side


 of the equation.


 This subroutine simultaneously reduces A to a Hessenberg matrix H:


    Q**T*A*Z = H


 and transforms B to another upper triangular matrix T:


    Q**T*B*Z = T


 in order to reduce the problem to its standard form


    H*y = lambda*T*y


 where y = Z**T*x.


 The orthogonal matrices Q and Z are determined as products of Givens


 rotations.  They may either be formed explicitly, or they may be


 postmultiplied into input matrices Q1 and Z1, so that


      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T


      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T


 If Q1 is the orthogonal matrix from the QR factorization of B in the


 original equation A*x = lambda*B*x, then DGGHD3 reduces the original


 problem to generalized Hessenberg form.


 This is a blocked variant of DGGHRD, using matrix-matrix


 multiplications for parts of the computation to enhance performance.

COMPQ

          COMPQ is CHARACTER*1


          = 'N': do not compute Q;


          = 'I': Q is initialized to the unit matrix, and the


                 orthogonal matrix Q is returned;


          = 'V': Q must contain an orthogonal matrix Q1 on entry,


                 and the product Q1*Q is returned.

COMPZ

          COMPZ is CHARACTER*1


          = 'N': do not compute Z;


          = 'I': Z is initialized to the unit matrix, and the


                 orthogonal matrix Z is returned;


          = 'V': Z must contain an orthogonal matrix Z1 on entry,


                 and the product Z1*Z is returned.

N

          N is INTEGER


          The order of the matrices A and B.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          ILO and IHI mark the rows and columns of A which are to be


          reduced.  It is assumed that A is already upper triangular


          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are


          normally set by a previous call to DGGBAL; otherwise they


          should be set to 1 and N respectively.


          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

          A is DOUBLE PRECISION array, dimension (LDA, N)


          On entry, the N-by-N general matrix to be reduced.


          On exit, the upper triangle and the first subdiagonal of A


          are overwritten with the upper Hessenberg matrix H, and the


          rest is set to zero.

LDA

          LDA is INTEGER


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

B

          B is DOUBLE PRECISION array, dimension (LDB, N)


          On entry, the N-by-N upper triangular matrix B.


          On exit, the upper triangular matrix T = Q**T B Z.  The


          elements below the diagonal are set to zero.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)


          On entry, if COMPQ = 'V', the orthogonal matrix Q1,


          typically from the QR factorization of B.


          On exit, if COMPQ='I', the orthogonal matrix Q, and if


          COMPQ = 'V', the product Q1*Q.


          Not referenced if COMPQ='N'.

LDQ

          LDQ is INTEGER


          The leading dimension of the array Q.


          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)


          On entry, if COMPZ = 'V', the orthogonal matrix Z1.


          On exit, if COMPZ='I', the orthogonal matrix Z, and if


          COMPZ = 'V', the product Z1*Z.


          Not referenced if COMPZ='N'.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.


          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)


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

LWORK

          LWORK is INTEGER


          The length of the array WORK.  LWORK >= 1.


          For optimum performance LWORK >= 6*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.

January 2015

  This routine reduces A to Hessenberg form and maintains B in


  using a blocked variant of Moler and Stewart's original algorithm,


  as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti


  (BIT 2008).