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.

## 1. cgsolve(): Conjugate gradient method

The 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)


## 2. pcgsolve(): Preconditioned conjugate gradient method

When 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.