Conjugate residual method
Encyclopedia
The conjugate residual method is an iterative numeric method used for solving systems of linear equations. It's a Krylov subspace method very similar to the much more popular conjugate gradient method
, with similar construction and convergence properties.
This method is used to solve linear equations of the form
where A is an invertible and Hermitian matrix, and b is nonzero.
The conjugate residual method differs from the closely related conjugate gradient method
primarily in that it involves somewhat more computation but is applicable to problems that aren't positive definite; in fact the only requirement (besides the obvious invertible A and nonzero b) is that A be Hermitian (or, with real numbers, symmetric). This makes the conjugate residual method applicable to problems which intuitively require finding saddle points instead of minima, such as numeric optimization with Lagrange multiplier constraints.
Given an (arbitrary) initial estimate of the solution
, the method is outlined below:
the iteration may be stopped once
has been deemed converged. Note that the only difference between this and the conjugate gradient method is the calculation of 
and 
(plus the optional recursive calculation of 
at the end).
The preconditioner
must be symmetric. Note that the residual vector here is different from the residual vector without preconditioning.
Conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too...
, with similar construction and convergence properties.
This method is used to solve linear equations of the form
where A is an invertible and Hermitian matrix, and b is nonzero.
The conjugate residual method differs from the closely related conjugate gradient method
Conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too...
primarily in that it involves somewhat more computation but is applicable to problems that aren't positive definite; in fact the only requirement (besides the obvious invertible A and nonzero b) is that A be Hermitian (or, with real numbers, symmetric). This makes the conjugate residual method applicable to problems which intuitively require finding saddle points instead of minima, such as numeric optimization with Lagrange multiplier constraints.
Given an (arbitrary) initial estimate of the solution
, the method is outlined below:
the iteration may be stopped once
has been deemed converged. Note that the only difference between this and the conjugate gradient method is the calculation of and (plus the optional recursive calculation of at the end).Preconditioning
By making a few substitutions and variable changes, a preconditioned conjugate residual method may be derived in the same way as done for the conjugate gradient method:
The preconditioner
Preconditioner
In mathematics, preconditioning is a procedure of an application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solution. Preconditioning is typically related to reducing a condition number of the problem...
must be symmetric. Note that the residual vector here is different from the residual vector without preconditioning.