Questions tagged [conjugate-gradient]

For questions related to conjugate gradient (method). It is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite.

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.

The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization.

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Solving simple quadratic function using conjugate gradient method - why are my directions linearly dependent?

I have a simple function: $$ f(x)=x_1^2 + 2 x_2^2 -2 x_1 x_2 - 2 x_2 + 2 x_1$$ And I'd like to solve it using conjugate gradient method at initial point $x^0=(2,2)$. I found the hessian $$ Q= \begin{bmatrix} 1 & -1 \\ -1 & 2 \\…
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Deriving the basis vectors of the Krylov subspace

I am struggling to follow the derivation of the basis vectors of the Krylov subspace starting from 22:27 of this video. If the next step of a search is given as $$ x^k = x^{k - 1} + \alpha_{k - 1} r^{k - 1} $$ where $x^{k- 1}$ is the previous step,…
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