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Numerical Linear Algebra with Applications by William Ford

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21.7 GMRES

Assume A is a real n × n matrix, b is an n × 1 vector, and we want to solve the system Ax = b. Assume that x0 is an initial guess for the solution, and r0 = bAx0 is the corresponding residual. The GMRES method looks for a solution of the form xm= x0 + Qmym, ymentitymwhere the columns of Qmare an n-dimensional orthogonal basis for the Krylov subspace Km(A, r0) = {r0, Ar0,…, Am−1r0}. The vector ymis chosen so the residual

rm2=bA(x0+Qmym)2=r0AQmym2

si175_e

has minimal norm over Km(A, r0). This is a least-squares problem. We must find ...

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