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Abstract
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Let A to be an $ n\times n $ real matrix. The matrix. $ A^{D} $ is said to be the Drazin inverse of $ A $ if it satisfies the three conditons:
\[
A^{D}AA^{D}=A^{D},~~AA^{D}=A^{D}A,~~A^{k 1}A^{D}=A^{k},
\]
here $ k $, the index of $ A $, is the size of the largest Jordan block corresponding to the zero eigenvalue of $ A $, which is called $ ind(A) $.
In this dissertation, the krylove subspace methods were derived for the Drazin inverse solution of cosistent or inconsistent linear system of the form $ Ax=b $, where $ A\in\mathbb{R}^{n\times n} $ is a singular system and in general non symmetric matrix that has an arbitrary index.Also, We will show that, as is the case with nonsingular system, the krylov subspace methods developed here terminate in a finite number of setps that is at $ n-ind(A) $.
This method, we model after the generalized residual method $ (GCR) $ and denoted $ DGCR $. Then $ DGMRES $ method was implemented like $ DGMRES $ method. We demonstrate the use of this method with numerical examples.
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