LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

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Functions/Subroutines  
subroutine  sgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO) 
SGGHRD 
subroutine sgghrd  (  character  COMPQ, 
character  COMPZ,  
integer  N,  
integer  ILO,  
integer  IHI,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( ldq, * )  Q,  
integer  LDQ,  
real, dimension( ldz, * )  Z,  
integer  LDZ,  
integer  INFO  
) 
SGGHRD
Download SGGHRD + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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. 
[in]  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. 
[in]  N  N is INTEGER The order of the matrices A and B. N >= 0. 
[in]  ILO  ILO is INTEGER 
[in]  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:ILO1 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. 
[in,out]  A  A is REAL array, dimension (LDA, N) On entry, the NbyN 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in,out]  B  B is REAL array, dimension (LDB, N) On entry, the NbyN upper triangular matrix B. On exit, the upper triangular matrix T = Q**T B Z. The elements below the diagonal are set to zero. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  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'. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. 
[in,out]  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'. 
[in]  LDZ  LDZ is INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
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.)
Definition at line 207 of file sgghrd.f.