The Minimal Residual Method or MINRES is a Krylov subspace method for the iterative solution of symmetric linear equation systems. It was proposed by mathematicians Christopher Conway Paige and Michael Alan Saunders in 1975. In contrast to the popular CG method, the MINRES method does not assume that the matrix is positive semidefinite, only the symmetry of the matrix is mandatory.
GMRES vs. MINRES The GMRES method is essentially a generalization of MINRES for arbitrary matrices. Both minimize the 2-norm of the residual and do the same calculations in exact arithmetic when the matrix is symmetric. MINRES is a short-recurrence method with a constant memory requirement, whereas GMRES requires storing the whole Krylov space, so its memory requirement is roughly proportional to the number of iterations. On the other hand, GMRES tends to suffer less from loss of orthogonality.
Properties of the MINRES method The MINRES method iteratively calculates an approximate solution of a linear system of equations of the form
A x = b , {\displaystyle Ax=b,}
where A ∈ R n × n {\displaystyle A\in \mathbb {R} ^{n\times n}} is a symmetric matrix and b ∈ R n {\displaystyle b\in \mathbb {R} ^{n}} a vector. For this, the norm of the residual r ( x ) := b − A x {\displaystyle r(x):=b-Ax} in a k {\displaystyle k} -dimensional Krylov subspace
V k = x 0 + span { r 0 , A r 0 … , A k − 1 r 0 } {\displaystyle V_{k}=x_{0}+\operatorname {span} \{r_{0},Ar_{0}\ldots ,A^{k-1}r_{0}\}}
is minimized. Here x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} is an initial value (often zero) and r 0 := r ( x 0 ) {\displaystyle r_{0}:=r(x_{0})} . More precisely, we define the approximate solutions x k {\displaystyle x_{k}} through
x k := a r g m i n x ∈ V k ‖ r ( x ) ‖ , {\displaystyle x_{k}:=\mathrm {argmin} _{x\in V_{k}}\|r(x)\|,}
where ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the standard Euclidean norm on R n {\displaystyle \mathbb {R} ^{n}} . Because of the symmetry of A {\displaystyle A} , unlike in the GMRES method, it is possible to carry out this minimization process recursively, storing only two previous steps (short recurrence). This saves memory.
MINRES algorithm Note: The MINRES method is more complicated than the algebraically equivalent Conjugate Residual method. The Conjugate Residual (CR) method was therefore produced below as a substitute. It differs from MINRES is that in MINRES, the columns of a basis of the Krylov space (denoted below by p k {\displaystyle p_{k}} ) can be orthogonalized, whereas in CR their images (below labeled with s k {\displaystyle s_{k}} ) can be orthogonalized via the Lanczos recursion. There are more efficient and preconditioned variants with fewer AXPYs. Compare with the article. First you choose x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} arbitrary and compute
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