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<H1>DSGHIEP</H1>
Dense Generalized Hermitian Indefinite Eigenvalue Problem.
<P>
<H3><FONT COLOR="#883300">Notes</FONT></H3>
The problem is expressed as A*X = B*X*Lambda, where both A and B are
real symmetric (or complex Hermitian) and possibly indefinite. Lambda
is a diagonal matrix whose diagonal elements are the arguments of <A HREF="../DS/DSSolve.html#DSSolve">DSSolve</A>().
After solve, A is overwritten with Lambda. Note that in the case of real
scalars, A is overwritten with a real representation of Lambda, i.e.,
complex conjugate eigenvalue pairs are stored as a 2x2 block in the
quasi-diagonal matrix.
<P>
In the intermediate state A is reduced to tridiagonal form and B is
transformed into a signature matrix. In compact storage format, these
matrices are stored in T and D, respectively.
<P>
<H3><FONT COLOR="#883300">Used DS matrices</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B><A HREF="../DS/DSMatType.html#DSMatType">DS_MAT_A</A> </B></TD><TD> - first problem matrix
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B><A HREF="../DS/DSMatType.html#DSMatType">DS_MAT_B</A> </B></TD><TD> - second problem matrix
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B><A HREF="../DS/DSMatType.html#DSMatType">DS_MAT_T</A> </B></TD><TD> - symmetric tridiagonal matrix of the reduced pencil
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B><A HREF="../DS/DSMatType.html#DSMatType">DS_MAT_D</A> </B></TD><TD> - diagonal matrix (signature) of the reduced pencil
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B><A HREF="../DS/DSMatType.html#DSMatType">DS_MAT_Q</A> </B></TD><TD> - pseudo-orthogonal transformation that reduces (A,B) to
tridiagonal-diagonal form (intermediate step) or a real basis of eigenvectors
</TD></TR></TABLE>
<P>
<H3><FONT COLOR="#883300">Implemented methods</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>0 </B></TD><TD> - QR iteration plus inverse iteration for the eigenvectors
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>1 </B></TD><TD> - HZ iteration
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>2 </B></TD><TD> - QR iteration plus pseudo-orthogonalization for the eigenvectors
</TD></TR></TABLE>
<P>
<H3><FONT COLOR="#883300">References</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>1. </B></TD><TD> - C. Campos and J. E. Roman, "Restarted Q-Arnoldi-type methods exploiting
symmetry in quadratic eigenvalue problems", BIT Numer. Math. 56(4):1213-1236, 2016.
</TD></TR></TABLE>
<P>
<H3><FONT COLOR="#883300">See Also</FONT></H3>
<A HREF="../DS/DSCreate.html#DSCreate">DSCreate</A>(), <A HREF="../DS/DSSetType.html#DSSetType">DSSetType</A>(), <A HREF="../DS/DSType.html#DSType">DSType</A>
<BR><P><B></B><H3><FONT COLOR="#883300">Level</FONT></H3>beginner<BR>
<H3><FONT COLOR="#883300">Location</FONT></H3>
</B><A HREF="../../../src/sys/classes/ds/impls/ghiep/dsghiep.c.html#DSGHIEP">src/sys/classes/ds/impls/ghiep/dsghiep.c</A>
<BR><BR><A HREF="./index.html">Index of all DS routines</A>
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