The computation of the roots of positive definite matrices arises in nuclear magnetic resonance, control theory, lattice quantum chromo-dynamics (QCD), and several other areas of applications. The Cauchy integral theorem which arises in complex analysis can be used for computing

It is well known that contour integrals which form a component of the Cauchy integral theorem have an important role in complex analysis. The trapezoid rule is popular for the approximation of integrals due to its exponential accuracy if particular conditions are satisfied. It has been established in [

Kellems [

In this paper, we investigate computation of the

The Cauchy integral theorem states that the value of

In order to calculate the integral (

The function

Now we suppose the Random matrix and use trapezoid rule with

Error in trapezoid rule for computing

Matrix factorizations are utilized to compute the

One of the most applicable factorization of matrices is Schur decomposition which is presented in the following theorem [

Let

One of the most famous algorithms to compute matrix roots is Smith's algorithm proposed in [

Compute the Schur factorization

Matrix

Else operate column-by-column on

This algorithm uses

This factorization is also called spectral decomposition and is presented as follows [

Let

Utilizing the property of

This factorization is also called spectral decomposition and is presented as follows [

Let

Applying the Hessenberg factorization and the orthogonality of

In this section we present some numerical experiment to illustrate the theory which is developed. All the computations have been carried out using MATLAB 7.10(Ra). We assume positive definite matrices with positive nonzero eigenvalues. These matrices, which are given in MATLAB gallery, are used to compute roots of matrices. Recall that if

For the first experiment, consider

Comparison residual error for computing the

Schur decomposition | Eigenvalue decomposition | Hessenberg decomposition | |
---|---|---|---|

2 | |||

16 | |||

52 | |||

128 | |||

2012 |

Comparison time in seconds for computing the

Schur decomposition | Eigenvalue decomposition | Hessenberg decomposition | |
---|---|---|---|

2 | 0.054855 | 0.028401 | 0.017409 |

16 | 0.161744 | 0.226524 | 0.141455 |

52 | 0.404749 | 0.761385 | 0.663878 |

128 | 0.968970 | 1.914147 | 1.803490 |

2012 | 0.013331 | 0.025015 | 0.018379 |

In this example consider

Comparison absolute error for different numbers of points for different matrices.

No. of points | Schur decomposition | Eigenvalue decomposition | Hessenberg decomposition |
---|---|---|---|

2 | 4.235476891477593 | 3.617696711330309 | 9.018271218363070 |

4 | 7.642822555078070 | 2.268548808462573 | 85.569768110351674 |

8 | 6.509731621793472 | 16.059530670734521 | 0.951463218971532 |

16 | 0.035910610643974 | 0.342929548483509 | 0.001435059307873 |

32 | 0.000002770887073 | 0.342929548483509 | 0.000000210760699 |

64 | 0.000000000000375 | 0.003429295484835 | 0.000000000000349 |

128 | 0.000000000000002 | 0.003429295484835 | 0.000000000000002 |

256 | 0.000000000000002 | 0.003429295484835 | 0.000000000000002 |

512 | 0.000000000000004 | 0.003429295484835 | 0.000000000000003 |

1024 | 0.000000000000001 | 0.000342929548484 | 0.000000000000000 |

In the last test Random matrix, Lehmer matrix, and Hilbert matrix are supposed. We have computed the 5th, 17th, 64th, and 128th root of these matrices using trapezoidal rule and Smith's algorithm. Furthermore, in this example the absolute errors are estimated. As shown in the figures, the accuracy is measured in either the Frobenius norm or the 2-norm for different matrices. The relations between dimension of

According to Figure ^{20} points in trapezoid rule can achieve error of

Residual error for computing

Time in seconds for computing

Residual error for computing

Time in seconds for computing

Residual error for computing

Time in seconds for computing

In this paper, we have studied the use of trapezoidal rule in conjunction with the Cauchy integral theorem to compute the

The authors would like to thank Dr. Anthony Kellems and the anonymous referees for their helpful comments which improved the presentation.

^{α}, log(A)