In mathematics, the generalized minimal residual method (GMRES) is an

iterative method In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an " ...

for the numerical solution of an indefinite nonsymmetric

system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of th ...

. The method approximates the solution by the vector in a

Krylov subspace In linear algebra, the order-''r'' Krylov subspace generated by an ''n''-by-''n'' matrix ''A'' and a vector ''b'' of dimension ''n'' is the linear subspace spanned by the images of ''b'' under the first ''r'' powers of ''A'' (starting from A^0=I) ...

with minimal residual. The

Arnoldi iteration In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non- Hermitian) matrices by c ...

is used to find this vector. The GMRES method was developed by

Yousef Saad Yousef Saad (born 1950) in Algiers, Algeria from Boghni, Tizi Ouzou, Kabylia is an I.T. Distinguished Professor of Computer Science in the Department of Computer Science and Engineering at the University of Minnesota.MINRES method due to Paige and Saunders in 1975. The MINRES method requires that the matrix is symmetric, but has the advantage that it only requires handling of three vectors. GMRES is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is applicable to non-linear systems.

The method

Denote the

Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...

of any vector v by

\, v\,

. Denote the (square) system of linear equations to be solved by

Ax = b.

The matrix ''A'' is assumed to be

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

of size ''m''-by-''m''. Furthermore, it is assumed that b is normalized, i.e., that

\, b\,  = 1

. The ''n''-th

for this problem is

K_n = K_n(A,r_0) = \operatorname \, \. \,

where

r_0 = b - A x_0

is the initial residual given an initial guess

x_0 \ne 0

. Clearly

r_0 = b

x_0 = 0

. GMRES approximates the exact solution of

Ax = b

by the vector

x_n \in x_0 + K_n

that minimizes the Euclidean norm of the residual

r_n= b - Ax_n

. The vectors

r_0,Ar_0,\ldots A^r_0

might be close to

linearly dependent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts ...

, so instead of this basis, the

is used to find orthonormal vectors

q_1, q_2, \ldots, q_n \,

which form a basis for

K_n

. In particular,

q_1=\, r_0\, _2^r_0

. Therefore, the vector

x_n \in x_0 + K_n

can be written as

x_n = x_0 + Q_n y_n

with

y_n \in \mathbb^n

, where

Q_n

is the ''m''-by-''n'' matrix formed by

q_1,\ldots,q_n

. In other words, finding the ''n''-th approximation of the solution (i.e.,

x_n

) is reduced to finding the vector

y_n

, which is determined via minimizing the residue as described

below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fred Belo ...

. The Arnoldi process also constructs

\tilde_n

, an (

n+1

)-by-

n

upper

Hessenberg matrix In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above ...

which satisfies

AQ_n = Q_ \tilde_n \,

an equality which is used to simplify the calculation of

y_n

(see ). Note that, for symmetric matrices, a symmetric tri-diagonal matrix is actually achieved, resulting in the MINRES method. Because columns of

Q_n

are orthonormal, we have

\begin
\left\,  r_n \right\, 
&= \left\,  b - A x_n \right\,  \\
&= \left\,  b - A(x_0 + Q_n y_n) \right\,  \\
&= \left\,  r_0 - A Q_n y_n \right\,  \\
&= \left\,  \beta q_1 - A Q_n y_n \right\,  \\
&= \left\,  \beta q_1 - Q_ \tilde_n y_n \right\,  \\
&= \left\,  Q_ (\beta e_1 - \tilde_n y_n) \right\,  \\
&= \left\,  \beta e_1 - \tilde_n y_n \right\, 
\end

where

e_1 = (1,0,0,\ldots,0)^T \,

is the first vector in the

standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For exampl ...

\mathbb^

, and

\beta = \, r_0\,  \, ,

r_0

being the first trial residual vector (usually

b

). Hence,

x_n

can be found by minimizing the Euclidean norm of the residual

r_n = \tilde_n y_n - \beta e_1.

This is a

linear least squares Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and ...

problem of size ''n''. This yields the GMRES method. On the

n

-th iteration: # calculate

q_n

with the Arnoldi method; # find the

y_n

which minimizes

\, r_n\,

; # compute

x_n = x_0 + Q_n y_n

; # repeat if the residual is not yet small enough. At every iteration, a matrix-vector product

A q_n

must be computed. This costs about

2m^2

floating-point operations for general dense matrices of size

m

, but the cost can decrease to

O(m)

for

sparse matrices In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix (mathematics), matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix ...

. In addition to the matrix-vector product,

O(nm)

floating-point operations must be computed at the ''n'' -th iteration.

Convergence

The ''n''th iterate minimizes the residual in the Krylov subspace

K_n

. Since every subspace is contained in the next subspace, the residual does not increase. After ''m'' iterations, where ''m'' is the size of the matrix ''A'', the Krylov space ''K''_''m'' is the whole of R^''m'' and hence the GMRES method arrives at the exact solution. However, the idea is that after a small number of iterations (relative to ''m''), the vector ''x''_''n'' is already a good approximation to the exact solution. This does not happen in general. Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every nonincreasing sequence ''a''₁, ..., ''a''_''m''−1, ''a''_''m'' = 0, one can find a matrix ''A'' such that the = ''a''_''n'' for all ''n'', where ''r''_''n'' is the residual defined above. In particular, it is possible to find a matrix for which the residual stays constant for ''m'' − 1 iterations, and only drops to zero at the last iteration. In practice, though, GMRES often performs well. This can be proven in specific situations. If the symmetric part of ''A'', that is

(A^T + A)/2

, is positive definite, then

\, r_n\,  \leq \left( 1-\frac \right)^ \, r_0\, ,

where

\lambda_(M)

and

\lambda_(M)

denote the smallest and largest

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

of the matrix

M

, respectively. If ''A'' is

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

and positive definite, then we even have

\, r_n\,  \leq \left( \frac \right)^ \, r_0\, .

where

\kappa_2(A)

denotes the condition number of ''A'' in the Euclidean norm. In the general case, where ''A'' is not positive definite, we have

\frac \le \inf_ \, p(A)\,  \le \kappa_2(V) \inf_ \max_ , p(\lambda), , \,

where ''P''_''n'' denotes the set of polynomials of degree at most ''n'' with ''p''(0) = 1, ''V'' is the matrix appearing in the spectral decomposition of ''A'', and ''σ''(''A'') is the

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

of ''A''. Roughly speaking, this says that fast convergence occurs when the eigenvalues of ''A'' are clustered away from the origin and ''A'' is not too far from normality. All these inequalities bound only the residuals instead of the actual error, that is, the distance between the current iterate ''x''_''n'' and the exact solution.

Extensions of the method

Like other iterative methods, GMRES is usually combined with a

preconditioning In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing ...

method in order to speed up convergence. The cost of the iterations grow as O(''n''²), where ''n'' is the iteration number. Therefore, the method is sometimes restarted after a number, say ''k'', of iterations, with ''x''_''k'' as initial guess. The resulting method is called GMRES(''k'') or Restarted GMRES. For non-positive definite matrices, this method may suffer from stagnation in convergence as the restarted subspace is often close to the earlier subspace. The shortcomings of GMRES and restarted GMRES are addressed by the recycling of Krylov subspace in the GCRO type methods such as GCROT and GCRODR. Recycling of Krylov subspaces in GMRES can also speed up convergence when sequences of linear systems need to be solved.

Comparison with other solvers

The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding

method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does. Another class of methods builds on the unsymmetric Lanczos iteration, in particular the BiCG method. These use a three-term recurrence relation, but they do not attain the minimum residual, and hence the residual does not decrease monotonically for these methods. Convergence is not even guaranteed. The third class is formed by methods like CGS and BiCGSTAB. These also work with a three-term recurrence relation (hence, without optimality) and they can even terminate prematurely without achieving convergence. The idea behind these methods is to choose the generating polynomials of the iteration sequence suitably. None of these three classes is the best for all matrices; there are always examples in which one class outperforms the other. Therefore, multiple solvers are tried in practice to see which one is the best for a given problem.

Solving the least squares problem

One part of the GMRES method is to find the vector

y_n

which minimizes

\left\,  \tilde_n y_n - \beta e_1 \right\, .

Note that

\tilde_n

is an (''n'' + 1)-by-''n'' matrix, hence it gives an over-constrained linear system of ''n''+1 equations for ''n'' unknowns. The minimum can be computed using a

QR decomposition In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...

: find an (''n'' + 1)-by-(''n'' + 1)

orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identi ...

Ω_''n'' and an (''n'' + 1)-by-''n'' upper

triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are z ...

\tilde_n

such that

\Omega_n \tilde_n = \tilde_n.

The triangular matrix has one more row than it has columns, so its bottom row consists of zero. Hence, it can be decomposed as

\tilde_n = \begin R_n \\ 0 \end,

where

R_n

is an ''n''-by-''n'' (thus square) triangular matrix. The QR decomposition can be updated cheaply from one iteration to the next, because the Hessenberg matrices differ only by a row of zeros and a column:

\tilde_ = \begin \tilde_n & h_ \\ 0 & h_ \end,

where ''h''_''n+1'' = (''h''_1,''n''+1, ..., ''h''_{''n''+1,''n''+1})^T. This implies that premultiplying the Hessenberg matrix with Ω_''n'', augmented with zeroes and a row with multiplicative identity, yields almost a triangular matrix:

\begin \Omega_n & 0 \\ 0 & 1 \end \tilde_ = \begin R_n & r_ \\ 0 & \rho \\ 0 & \sigma \end

This would be triangular if σ is zero. To remedy this, one needs the Givens rotation

G_n = \begin I_ & 0 & 0 \\ 0 & c_n & s_n \\ 0 & -s_n & c_n \end

where

c_n = \frac \quad\text\quad s_n = \frac.

With this Givens rotation, we form

\Omega_ = G_n \begin \Omega_n & 0 \\ 0 & 1 \end.

Indeed,

\Omega_ \tilde_ = \begin R_n & r_ \\ 0 & r_ \\ 0 & 0 \end

is a triangular matrix with

r_ = \sqrt

. Given the QR decomposition, the minimization problem is easily solved by noting that

\begin
\left\,  \tilde_n y_n - \beta e_1 \right\,  
&= \left\,  \Omega_n (\tilde_n y_n - \beta e_1) \right\,  \\
&= \left\,  \tilde_n y_n - \beta \Omega_n e_1 \right\, . \end

Denoting the vector

\beta\Omega_ne_1

\tilde_n = \begin g_n \\ \gamma_n \end

with ''g''_''n'' ∈ R^''n'' and γ_''n'' ∈ R, this is

\begin
\left\,  \tilde_n y_n - \beta e_1 \right\,  
&= \left\,  \tilde_n y_n - \beta \Omega_n e_1 \right\,  \\
&= \left\,  \begin R_n \\ 0 \end y_n - \begin g_n \\ \gamma_n \end \right\, . 
\end

The vector ''y'' that minimizes this expression is given by

y_n = R_n^ g_n.

Again, the vectors

g_n

are easy to update.

Example code

Regular GMRES (MATLAB / GNU Octave)

function

, e The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

= gmres(A, b, x, max_iterations, threshold) n = length(A); m = max_iterations; % use x as the initial vector r = b - A * x; b_norm = norm(b); error = norm(r) / b_norm; % initialize the 1D vectors sn = zeros(m, 1); cs = zeros(m, 1); %e1 = zeros(n, 1); e1 = zeros(m+1, 1); e1(1) = 1; e = rror r_norm = norm(r); Q(:,1) = r / r_norm; % Note: this is not the beta scalar in section "The method" above but % the beta scalar multiplied by e1 beta = r_norm * e1; for k = 1:m % run arnoldi (1:k+1, k), Q(:, k+1)= arnoldi(A, Q, k); % eliminate the last element in H ith row and update the rotation matrix (1:k+1, k), cs(k), sn(k)= apply_givens_rotation(H(1:k+1,k), cs, sn, k); % update the residual vector beta(k + 1) = -sn(k) * beta(k); beta(k) = cs(k) * beta(k); error = abs(beta(k + 1)) / b_norm; % save the error e = ; error if (error <= threshold) break; end end % if threshold is not reached, k = m at this point (and not m+1) % calculate the result y = H(1:k, 1:k) \ beta(1:k); x = x + Q(:, 1:k) * y; end %----------------------------------------------------% % Arnoldi Function % %----------------------------------------------------% function

, q The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

= arnoldi(A, Q, k) q = A*Q(:,k); % Krylov Vector for i = 1:k % Modified Gram-Schmidt, keeping the Hessenberg matrix h(i) = q' * Q(:, i); q = q - h(i) * Q(:, i); end h(k + 1) = norm(q); q = q / h(k + 1); end %---------------------------------------------------------------------% % Applying Givens Rotation to H col % %---------------------------------------------------------------------% function , cs_k, sn_k= apply_givens_rotation(h, cs, sn, k) % apply for ith column for i = 1:k-1 temp = cs(i) * h(i) + sn(i) * h(i + 1); h(i+1) = -sn(i) * h(i) + cs(i) * h(i + 1); h(i) = temp; end % update the next sin cos values for rotation s_k, sn_k= givens_rotation(h(k), h(k + 1)); % eliminate H(i + 1, i) h(k) = cs_k * h(k) + sn_k * h(k + 1); h(k + 1) = 0.0; end %%----Calculate the Givens rotation matrix----%% function s, sn= givens_rotation(v1, v2) % if (v1

0) % cs = 0; % sn = 1; % else t = sqrt(v1^2 + v2^2); % cs = abs(v1) / t; % sn = cs * v2 / v1; cs = v1 / t; % see http://www.netlib.org/eispack/comqr.f sn = v2 / t; % end end

References

* * *
Dongarra et al., ''Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods''
2nd Edition, SIAM, Philadelphia, 1994 * {{Numerical linear algebra Numerical linear algebra Articles with example pseudocode

The method

Convergence

Extensions of the method

Comparison with other solvers

Solving the least squares problem

Example code

Regular GMRES (MATLAB / GNU Octave)

See also

References