Description Usage Arguments Details Value Preconditioners Warning References See Also Examples

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

1 | ```
pcgsolve(A, b, preconditioner = "Jacobi", tol = 1e-6, maxIter = 1000)
``` |

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

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

`preconditioner` |
string, method for preconditioning: |

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

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

When the condition number for *A* 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} A x = {C}^{-1} b*

where the symmetric and positive-definite matrix *C* approximates *A* and *{C}^{-1} A * improves the condition number of *A*.

Common choices for the preconditioner include: Jacobi preconditioning, symmetric successive over-relaxation (SSOR), and incomplete Cholesky factorization [2].

Returns a vector representing solution x.

`Jacobi`

: The Jacobi preconditioner is the diagonal of the matrix A, with an assumption that all diagonal elements are non-zero.

`SSOR`

: The symmetric successive over-relaxation preconditioner, implemented as *M = (D+L) D^{-1} (D+L)^T*. [1]

`ICC`

: The incomplete Cholesky factorization preconditioner. [2]

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

[1] David Young. <e2><80><9c>Iterative methods for solving partial difference equations of elliptic type<e2><80><9d>. In: Transactions of the American Mathematical Society 76.1 (1954), pp. 92<e2><80><93>111.

[2] David S Kershaw. <e2><80><9c>The incomplete Cholesky<e2><80><94>conjugate gradient method for the iter- ative solution of systems of linear equations<e2><80><9d>. In: Journal of computational physics 26.1 (1978), pp. 43<e2><80><93>65.

1 2 3 4 5 6 |

```
[,1]
[1,] 0.09074421
[2,] 0.63682348
```

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