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The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the ...
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method [1] for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. The intent of this article is to document the important steps in these derivations.
In optimization, a gradient method is an algorithm to solve problems of the form with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradient.
For high-dimensional problems, the exact computation of the Hessian is usually prohibitively expensive, and even its storage can be problematic, requiring () memory (but see the limited-memory L-BFGS quasi-Newton method). The conjugate gradient method can also be derived using optimal control theory. [6]
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem
An incomplete Cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method. The Cholesky factorization of a positive definite matrix A is A = LL* where L is a lower triangular matrix. An incomplete Cholesky factorization is given by a sparse lower triangular matrix K that is in some sense close to L.
Typical examples involve using non-linear iterative methods, e.g., the conjugate gradient method, as a part of the preconditioner construction. Such preconditioners may be practically very efficient, however, their behavior is hard to predict theoretically.
Here is an example gradient method that uses a line search in step 5: Set iteration counter = ... One example of the former is conjugate gradient method.