SGGHRD 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 SGGHRD reduces the original


 problem to generalized Hessenberg form.

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 SGGBAL; 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 REAL 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 REAL 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 REAL 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 REAL 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.

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 2011

  This routine reduces A to Hessenberg and B to triangular form by


  an unblocked reduction, as described in <em>Matrix_Computations</em>,


  by Golub and Van Loan (Johns Hopkins Press.)