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In this paper, we present homotopy analysis method (HAM) for solving system of linear equations and use of different H( x) in this method. The numerical results indicate that this method performs better than the homotopy perturbation method (HPM) for solving linear systems.

Approximating the solutions of the system of linear and nonlinear equations has widespread applications in applied mathematics [

Consider a linear system

where

First of all, the basic ideas of the homotopy analysis method are being discussed.

Let

and we define the operator

Let

from (2) and (3) we have

Obviously, at q = 0 and q = 1, one has

where

By using (5) we have

If the series (6) is convergent at q = 1, then using the relationship

Now we have the so-called mth-order deformation equation

where

and

By using (2) we obtain

Also by using (3) and (9) we have

and then

Finally by using (11) we obtain

Now with the initial guess

hence, by substituting (13) in (7) we obtain

and by factor of

Now we have to prove the convergence of (15).

Theorem 1. The sequence

Proof: Following ([

Now considering

then

let

so we have

since

which completes the proof.

In this section For solving the linear system (1) we apply different H(x) and the convergence of the method is checked. At first assume that A is a nonsingular diagonally dominate matrix and

where

and

Now we apply different H(x) and the convergence of the method is tested.

1) we propose

and show that

Theorem 2. If A is diagonally dominated and

Proof: By direct calculation we have

and first row is satisfied:

Since A is diagonally dominated, B is diagonally dominated and we have

Now by using (19) we obtain

This relation satisfis for other rows also and

2) We propose

and show that

Theorem 3. If A is diagonally dominated and

Proof: Following Theorem (2)

such that

and last row is satisfied:

This relation satisfis for other rows also

3) We propose

Theorem 4. If A is diagonally dominated and

Proof: Similar to proof of Theorems (2) and (3).

4) We propose

and show that

Theorem 5. If A is diagonally dominated and

Proof: Following Theorem (2) after expanding

This relation satisfis for other rows also

5) We propose

Theorem 6. If A is diagonally dominated and

Proof: Similar to proof of Theorems (3) and (5).

6) We propose

Theorem 7. If A is diagonally dominated and

Proof: Following Theorem (2) after expanding

This relation satisfis for other rows also

7) We propose

Theorem 8. If A is diagonally dominated and

Proof: Similar to proof of Theorems (3) and (7).

Now in the next section we apply

In this section, we present some numerical examples to apply HAM and HPM methods for solving linear system. We used of Matlab 2013 for numerical results.

Example 1. Consider the linear system

According to

In this example the matrices S and

Example 2. In this example we apply HAM method for solving the linear system

where A is a

Method | Iteration | Error | Spectral radius | Times (s) |
---|---|---|---|---|

HAM | 14 | 10^{−5} | 0.4004 | 0.013 |

HAM | 11 | 10^{−5} | 0.3057 | 0.010 |

HAM | 9 | 10^{−5} | 0.2168 | 0.010 |

HAM | 8 | 10^{−5} | 0.1806 | 0.014 |

HAM | 11 | 10^{−5} | 0.2918 | 0.010 |

HAM | 8 | 10^{−5} | 0.1806 | 0.014 |

HAM | 8 | 10^{−5} | 0.2918 | 0.011 |

HAM | 7 | 10^{−5} | 0.1667 | 0.011 |

Method | Iteration | Error | Spectral radius | Times (s) |
---|---|---|---|---|

HAM | 36 | 10^{−4} | 0.5258 | 16.378 |

HAM | 32 | 10^{−4} | 0.5167 | 19.141 |

HAM | 32 | 10^{−4} | 0.5157 | 17.176 |

HAM | 32 | 10^{−4} | 0.5169 | 18.010 |

HAM | 32 | 10^{−4} | 0.5162 | 20.112 |

HAM | 32 | 10^{−4} | 0.5261 | 19.200 |

HAM | 25 | 10^{−4} | 0.4143 | 11.731 |

HAM | 20 | 10^{−4} | 0.4092 | 9.175 |

From the numerical results, we have seen that the HAM method with different

We thank Islamic Azad University for support researcher plan entitled: “Combination of Iterative methods and semi analytic methods for solving linear systems” and the Editor and the referee for their comments.

Mohammad HasanKhani,JalilRashidinia,Sajjad ZiaBorujeni, (2015) Application of Different H(x) in Homotopy Analysis Methods for Solving Systems of Linear Equations. Advances in Linear Algebra & Matrix Theory,05,129-137. doi: 10.4236/alamt.2015.53012