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A Novel Algorithm for determinant calculation of N×N matrix
Citation for published version:
Taheri, SM, Boostanpour, J & Mohammadi, B 2013, 'A Novel Algorithm for determinant calculation of N×N
matrix'. in Proceedings of the 2013 International Conference on Advances in Computing, Communications
and Informatics, ICACCI 2013., 6637270, pp. 764-769, 2013 2nd International Conference on Advances in
Computing, Communications and Informatics, ICACCI 2013, Mysore, United Kingdom, 22-25 August.,
10.1109/ICACCI.2013.6637270
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10.1109/ICACCI.2013.6637270
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Download date: 14. Oct. 2016
A Novel Algorithm for Determinant Calculation of
N×N Matrix
Sayed Mostafa Taheri (Author) M’IEEE
Jafar Boostanpour (Co-Author)
Department of Electronics and Electrical Engineering
Ferdowsi University of Mashhad (FUM)
Mashhad, I.R.IRAN
[email protected]
Electrical Engineering Department
Ferdowsi University of Mashhad (FUM)
Mashhad, I.R.IRAN
[email protected]
Bahareh Mohammadi (Co-Author)
Electrical Engineering Department
Barthawa Institute of Technology
Mashhad, I.R.IRAN
Abstract— Evaluation of the determinant of a square
matrix is an important issue in science and engineering for
different types of applications. Several methods have been
already proposed, but there are some limitation and also
drawbacks using each method in particular applications.
In this paper we present a novel method for determining
the determinant of a square matrix which can be especially
faster, and more useful when the order of the matrix
increases. The paper is presented in five sections. I.
Introduction. II. Review III. The Novel Algorithm (TaBe)
where we introduce our new method. IV. Mathematical
Proof, here we illustrate the mathematical proof. V.
Advantages VI. Conclusion, and finally References.
Keywords— Matrix Determinant, Algebra,
Computation, Mathematical Algorithms.
I.
Mathematical
INTRODUCTION
Working with matrix is one of the most applicable and
common issues in science and engineering. Almost every
computational method in science and engineering has to either
use matrix theory or at least to be expressed and viewed by
matrices. Matrices are the well-known main tools in many
methods and applications in order to lead the algorithm to
achieve the desired objectives. Many famous softwares in
mathematics and engineering are based on matrix
manipulation; for instance the giant MATLAB (which its
name is derived from Matrix Laboratory).
One of the most important problems in matrix operations, is
to find the determinant of an n order matrix; either containing
numbers (real, complex, etc.) or characters, symbols and
parameters. This step is a critical step in matrix operations,
and is usually cumbersome especially when n increases.
Several methods have been already presented which can be
used due to the particular application. Nearly all of the
presented methods show that the mass and volume of the
calculations increase substantially with n enhancement. For
instance, in the common method of determinant calculation,
we need to find the determinant of n * [n-1] × [n-1]
matrices for an n×n matrix. In Laplace expansion (or cofactor
expansion) you can evaluate the determinant of an n×n matrix
by knowing the 2×2 determinant. Det (A) is a weighted sum
of the determinants of sub-matrices of A, each of order
(n-1) × (n-1) etc. What is obvious is that the probability of
making mistakes for high order matrices increases since the
algorithms is somewhat complicated. In this paper, we
actually introduce a systematic method which can be useful
for both humans and machines (having absolutely simple and
fast code).
II.
REVIEW
While there are many way of calculating determinant of an
N×N square matrix, few methods are presented over the time
to develop or improve the efficiency of the traditional
techniques.
The most important ways of calculating determinant of a
square matrix will be presented briefly in this section. The
determinant of a matrix of arbitrary size can be evaluated
using Leibnitz or Laplace formulations [37,38]. If ‘A’ is a
square matrix:
(aij)i,j=1,….,n
(1)
The determinant of A is defined as:
(2)
where Sn is the permutation group and sgn returns +1 or -1 for
even and odd permutations respectively. Laplace formula can
be written as:
(8)
(3)
And simply the determinant of the main matrix [X] is acquired
by:
Where the i,j cofactor of B is the scalar Cij defined by
(4)
(9)
The determinant can also be expanded using Laplace
formulas by complementary minors.
It is worth mentioning that there are several notations and
presentations of the above basic formulas which do not carry
significant difference in fundamental.
Other ways of computing the determinant of a matrix,
includes: applying Gaussian Elimination, Decomposition
Methods which computes the determinant of a given matrix A
as a product of matrices which their determinants can be
simply evaluated, LU decomposition, QR decomposition,
Cholesky decomposition, matrix determinant lemma, Cramer
and some other methods [see references section (VII) for
detailed information].
In a simpler form of formulation, consider [Y] to be a 2×2
matrix, build up of the elements which are composed by the
determinant of the cofactors of the corner elements of the main
matrix. That is:
III.
THE NOVEL ALGORITHM (TABE)
In this method, we build a 2×2 matrix which its
elements are only four determinant of order (n-1) × (n-1)
[instead of n * [n-1] × [n-1] in most methods]. It is notable
that the method of picking the elements in our algorithm is
totally different with Dodgson method [1]. In the Dodgson
method, an [n-1] × [n-1] matrix is created which its elements
are all the determinants of order 2×2, calculated in an especial
manner. Now let us introduce the new method.
Consider the main matrix [x]:
(5)
And we create a 2×2 matrix [Y]:
(10)
In the next step, we create a (n-2) × (n-2) matrix which is
composed by eliminating the adjacent, top and bottom rows
and columns of the [ ]. At the end, only a division of the
determinants is needed to evaluate the determinant of the main
matrix by (7):
(11)
Now, as you may guess, the machine code of the
proposed method is considerably easier than the Dodgson
method (and nearly most other algorithms) because:
1.
The number of the chosen matrices at first is always
4 (independent of the order of the main matrix),
2.
The algorithm reduces the order of the matrix in a
complete systematic manner,
3.
The picking of the
is really easy
in mathematical programming as well as for humans,
and
4.
The machine code of this method can even be written
by low-level programming languages with no predefined mathematical functions, therefore it is nearly
as easy as high-level languages with its fast topology
(few numbers of loops only).
(6)
Where
(7)
The (n-2) × (n-2) matrix [Z] is configured as
So the machines or humans alike can calculate the
determinant of high order matrices with minimum or no
mistakes.
Let’s now consider a numerical example to illustrate
the concepts.
At first sight, we may see a drawback in the algorithm:
division by zero when |Z| = 0. This can be addressed by the
following suggestions:
As can be seen, for n≥5 the usefulness and rapidness of
the TaBe method increases quickly. For instance, assume
[X] be a 10×10 matrix. In general method,
1.
We can prevent division by zero (| | = 0) by using
simple permutation of rows and columns, and other
elementary operations (which do not change | |).
2.
We can substitute some elements of
with ʓ such
that the | | ≠ 0, and then continue calculating the
determinant of . At the end, we may have a
statement in terms of ʓ. Now we can reversely
replace ʓ by the element which was already
substituted with.
10×10
10 * (9×9)
By applying the above two solutions, the algorithm would
always work. In(a) next section, we will illustrate the
mathematical demonstration of the proposed method.
9*(8×8)
8*(7×7)
7*(6×6)
IV.
MATHEMATICAL PROOF
In this section a proof the statements will be presented. As
is known, the determinant of a matrix can be defined by:
6*(5×5)
(12)
5*(4×4)
where xuv are the elements of the square matrix, and
ε_(k,r,t,..,z) is the Levi-Civita symbol.
4*(3×3)
3*(2×2)
First, we need to know about cofactor, minor, Laplace and
Cauchy expressions as preliminaries.
Let xuv to be the element of the square matrix X_(n×n).
The minor of X is the matrix which is obtained by eliminating
the Uth row and Vth coloumn of X and is defined by Xuv. The
cofactor of Xuv is defined and presented as follows:
(a)
Now, by using the TaBe method,
10×10
(13)
4 * (9×9)
4*(8×8)
4*(7×7)
4*(6×6)
Laplace theory (b)
for determining the determinant of X states
that the summation of the multiplication of a row (or
columns)'s elements by its cofactor gives the determinant of X
. That is
(14)
4*(5×5)
In a similar way, the Cauchy theory is presented as follows:
4*(4×4)
4*(3×3)
(15)
4*(2×2)
(b)
Also
is defined as:
(16)
So:
(27)
Therefore it can be shown that:
(17)
Now we can begin demonstrating the proposed algorithm. For
demonstration, we need to prove that for
(18)
We can find Y
(19)
Where:
(20)
And we know:
Now, let us define matrix α as:
(28)
(29)
With simple calculation, it can be showed that:
(30)
As we have already seen, the order of both X and α matrices
are equal; hence we can multiply them by each other.
(21)
(31)
(23)
By multiplying the 2nd, 3rd, …., nth column of the above
matrices by x12, x13, …., x1 n-1 respectively, and then adding the
results to the first coloumn, and repeatedly multiplying the 2nd,
3rd, …., nth column by xn2, xn3, …., xn n-1 and adding the results to
the last column, we acquire [γ] which can be written as follows
by using Laplace and Cauchy theorems:
(24)
(32)
(22)
and Z
where we used Laplace theorem for γ expansion so
Such that
(25)
and as we have derived γ by elementary operations on β, it is
obviously clear that
For proof, we can use the following substitutions:
(34)
,
,
(33)
(26)
So we can conclude:
(35)
,
Thus we have reached to the proof
(36)
V.
ADVANTAGES
As mentioned before, several methods have been presented
to evaluate the determinant of a square matrix. The new
proposed method, TaBe, has some especial advantages in
comparison with other algorithms. Some of the relative
advantages are as follows:
1.
2.
3.
As it can be seen in (a) and (b), as examples, the
mass of the calculation is quite less than the general
methods (almost all methods). The main point here is
that the number of the chosen determinants for the
2×2 matrix at first is independent of the order of the
main matrix and is always four. So the reduction in
calculation volume is obvious.
As mentioned before, the simplicity of the machine
code for this algorithm is really considerable
(detailed explanations presented in section II). As a
result, the speed of the code using this algorithm is
relatively higher than the other algorithms especially
for parametric and symbolic matrices.
The other benefit of this algorithm is its systematical
topology. To calculate the determinant of a high
order matrix (even for n=4) by humans correctly,
high level of accuracy and concentration is needed;
otherwise mistakes will be made during the
calculation and mixing up is very common. But in
this method we can reduce the order of the matrix
absolutely systematically, by glance of view and
continue using this method repeatedly. So the
possibility of making mistakes by humans can be
decreased considerably.
VI.
REFERENCES
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CONCLUSION
In this paper we propose a novel method for calculating the
determinant of a square matrix. We showed that is correct for
any square matrix of order n, and is different from all the
previous methods. We showed that the TaBe algorithm has
lots of advantages which can be used in appropriate
applications and situations; For instance, the reduction of
calculations, and minimizing the possibility of making
mistakes by humans. It has easier machine code, and faster
result acquisition for parametric and symbolic matrices can be
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