knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

Functions in this package serve the purpose of solving for $\boldsymbol{x}$ in $\boldsymbol{Ax=b}$, where $\boldsymbol{A}$ is a $n \times n$ symmetric and positive definite matrix, $\boldsymbol{b}$ is a $n \times 1$ column vector.

To improve scalability of conjugate gradient methods for larger matrices, the C++ Armadillo templated linear algebra library is used for the implementation. The package also provides flexibility to have user-specified preconditioner options to cater for different optimization needs.

This vignette will walk through some simple examples for using main functions in the package.

`cgsolve()`

: Conjugate gradient methodThe idea of conjugate gradient method is to find a set of mutually conjugate directions for the unconstrained problem $$\arg \min_x f(x)$$ where $f(x) = 0.5 y^T \Sigma y - yx + z$ and $z$ is a constant. The problem is equivalent to solving $\Sigma x = y$.

This function implements an iterative procedure to reduce the number of matrix-vector multiplications. The conjugate gradient method improves memory efficiency and computational complexity, especially when $\Sigma$ is relatively sparse.

**Example**: Solve $Ax = b$ where $A = \begin{bmatrix}
4 & 1 \
1 & 3
\end{bmatrix}$, $b = \begin{bmatrix}
1 \
2
\end{bmatrix}$.

test_A <- matrix(c(4,1,1,3), ncol = 2) test_b <- matrix(1:2, ncol = 1) cgsolve(test_A, test_b, 1e-6, 1000)

`pcgsolve()`

: Preconditioned conjugate gradient methodWhen the condition number for $\Sigma$ is large, the conjugate gradient (CG) method may fail to converge in a reasonable number of iterations. The Preconditioned Conjugate Gradient (PCG) Method applies a precondition matrix $C$ and approaches the problem by solving: $$C^{-1} \Sigma x = C^{-1} y$$ where the symmetric and positive-definite matrix $C$ approximates $\Sigma$ and $C^{-1} \Sigma$ improves the condition number of $\Sigma$.

Choices for the preconditioner include: Jacobi preconditioning (`Jacobi`

), symmetric successive over-relaxation (`SSOR`

), and incomplete Cholesky factorization (`ICC`

).

**Example revisited**: Now we solve the same problem using incomplete Cholesky factorization of $A$ as preconditioner.

test_A <- matrix(c(4,1,1,3), ncol = 2) test_b <- matrix(1:2, ncol = 1) pcgsolve(test_A, test_b, "ICC")

*Check Github repo and cPCG: Efficient and Customized Preconditioned Conjugate Gradient Method for more information.*

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