Description Usage Arguments Details Value Warning References See Also Examples

Conjugate gradient method for solving system of linear equations Ax = b, where A is symmetric and positive definite, b is a column vector.

1 | ```
cgsolve(A, b, tol = 1e-6, maxIter = 1000)
``` |

`A` |
matrix, symmetric and positive definite. |

`b` |
vector, with same dimension as number of rows of A. |

`tol` |
numeric, threshold for convergence, default is |

`maxIter` |
numeric, maximum iteration, default is |

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 b^T A b - bx + z* and *z* is a constant. The problem is equivalent to solving *Ax = b*.

This function implements an iterative procedure to reduce the number of matrix-vector multiplications [1]. The conjugate gradient method improves memory efficiency and computational complexity, especially when *A* is relatively sparse.

Returns a vector representing solution x.

Users need to check that input matrix A is symmetric and positive definite before applying the function.

[1] Yousef Saad. Iterative methods for sparse linear systems. Vol. 82. siam, 2003.

1 2 3 4 5 6 |

```
[,1]
[1,] 0.09090909
[2,] 0.63636364
```

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