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The conjugate gradient method with a trivial modification is extendable to solving, given complex-valued matrix A and vector b, the system of linear equations = for the complex-valued vector x, where A is Hermitian (i.e., A' = A) and positive-definite matrix, and the symbol ' denotes the conjugate transpose.
A system of linear equations = consists of a known matrix and a known vector.To solve the system is to find the value of the unknown vector . [3] [5] A direct method for solving a system of linear equations is to take the inverse of the matrix , then calculate =.
The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector. The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986. [1] It is a generalization and improvement of the MINRES method due to Paige and Saunders in 1975.
Whereas linear conjugate gradient seeks a solution to the linear equation =, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at ...
In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point.
It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method. Unlike the original BiCG method, it doesn't require multiplication by the transpose of the system matrix.
Kantorovich in 1948 proposed calculating the smallest eigenvalue of a symmetric matrix by steepest descent using a direction = of a scaled gradient of a Rayleigh quotient = (,) / (,) in a scalar product (,) = ′, with the step size computed by minimizing the Rayleigh quotient in the linear span of the vectors and , i.e. in a locally optimal manner.
In particular, the gradient descent method would be slow. This can be seen in the diagram, where the green line is the result of always picking the local gradient direction. It zig-zags towards the minimum, but repeatedly overshoots.