**3. Algorithms**

In this section, we give two algorithms for finding the determinant and inverse of a periodic tridiagonal Toeplitz matrix with perturbed corners of type 1, which is called A. Besides, we analyze these algorithms to illustrate our theoretical results.

Firstly, based on Theorem 1, we give an algorithm for computing determinant of A as following: Based on Algorithm 1, we make a comparison of the total number operations for determinant of A between LU decomposition and Algorithm 1 in Table 1. Specifically, we get that the total number operation for the determinant of A is 2*n* + 11, which can be reduced to *O*(*logn*) (see, [41] pp. 226–227).

**Table 1.** Comparison of the total number operations for determinant of A.

**Algorithm 1:** The determinant of a periodic tridiagonal Toeplitz matrix with perturbed corners of type 1

Step 1: Input *α*1, *αn*, *γ*1, *γn*, ¯*h*, order *n* and generate Mersenne numbers *Mi*(*i*=*n*−3,*n*−2,*n*−1)by(1).

 Step 2: Calculate and output the determinant of A by (4).

Next, based on Theorem 2, we give an algorithm for computing inverse of A as following:

**Algorithm 2:** The inverse of a periodic tridiagonal Toeplitz matrix with perturbed corners of type 1

Step 1: Input *α*1, *αn*, *γ*1, *γn*, ¯*h*, order *n* and generate Mersenne numbers *Mi* (*i* = *n* − 3, *n* − 2, *n* − 1) by (1).

Step 2: Calculate *ψ* by (13) and six initial values *a*˘1,1, *a*˘1,2, *a*˘2,1, *a*˘2,2, *a*˘3,1, *a*˘3,2 by (12). Step 3: Calculate the remaining elements of the inverse:

$$\begin{split} \mathfrak{u}\_{2,3} &= 3\mathfrak{u}\_{2,2} - 2\mathfrak{u}\_{2,1} + \frac{1}{\hbar}, \\ \mathfrak{u}\_{3,4} &= 3\mathfrak{u}\_{3,2} - 2\mathfrak{u}\_{3,1} + \frac{1}{\hbar}, \\ \breve{\mathfrak{u}}\_{i,j} &= 3\breve{\mathfrak{u}}\_{i,j-1} - 2\breve{\mathfrak{u}}\_{i,j-2}, \ i \in \{1,2,3\}, i+2 \le j \le n\_{i}, \\ \breve{\mathfrak{u}}\_{i,j} &= 3\breve{\mathfrak{u}}\_{i,j-1} - 2\breve{\mathfrak{u}}\_{i,j-2}, \ i \in \{1,2,3\}, 3 \le j \le i \le n\_{i}, \\ \mathfrak{u}\_{i,j} &= \frac{3}{2}\mathfrak{u}\_{i-1,j} - \frac{1}{2}\mathfrak{u}\_{i-2,j}, \ j \in \{1,2\}, 4 \le i \le n\_{i} \\ \breve{\mathfrak{u}}\_{i,j} &= \frac{3}{2}\breve{\mathfrak{u}}\_{i-1,j} - \frac{1}{2}\breve{\mathfrak{u}}\_{i-2,j}, \ 4 \le i < j \le n. \end{split}$$

Step 4: Output the inverse A−1 = (*a*˘*i*,*j*)*ni*,*j*=1.

To test the effectiveness of Algorithm 2, we compare the total number of operations for the inverse of A between LU decomposition and Algorithm 2 in Table 2. The total number operation of LU decomposition is 5*n*3 6 + 3*n*2 + 91*n*6 − 21, whereas that of Algorithm 2 is 7*n*2 2 − 3*n*2 + 30.

**Table 2.** Comparison of the total number operations for inverse of A.

## **4. Discussion**

In this paper, explicit determinants and inverses of periodic tridiagonal Toeplitz matrices with perturbed corners are represented by the famous Mersenne numbers. This helps to reduce the total number of operations during the calculation process. Some recent research related to our present work can be found in [42–48]. Among them, Qi et al. presented some closed formulas for the Horadam polynomials in terms of a tridiagonal determinant and derived closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell-Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.


**Understanding the Genetics of Clubroot Resistance for Effectively Controlling this Disease in Brassica Species**

Arvind H. Hirani and Genyi Li

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60936

#### **Abstract**

Clubroot disease is one of the most serious diseases of Brassica species, which is caused by soil-borne pathogen *Plasmodiophora brassicae* Woronin. Clubroot disease has a long history on vegetable crops belonging to the Brassica species; most recently, this disease is also invading rapeseed/canola crop around the globe. The clubroot disease causes significant yield and quality losses in highly infected fields. Clubroot pathogens invade into the host plant roots and infect root tissues with the formation of abnormal clubs, named as galls, which results in incompetent plant roots to intake water and nutrients and eventually dead plants. As it is a soil-borne disease and accomplishes its disease cycle in two different phases and both phases are highly efficient to damage root system as well as to release more inoculum, there are many challenges to control this disease through chemical and other cultural practices. In general, clubroot disease can be effectively managed by developing resistant cultivars. In this chapter, various resistance sources of clubroot disease in different Brassica species have been discussed with potential applications in canola/rapeseed breeding programs worldwide. Importance of gene mapping and molecular marker development efforts by different research studies for clubroot in *B. rapa*, *B. oleracea*, and *B. napus* has been stressed. Transcriptomic and metabolomic changes occurring during host–pathogen interac‐ tions are also covered in this chapter, which would enhance our understanding and utilization of clubroot resistance in Brassica species.

**Keywords:** brassica species, clubroot resistance, molecular marker development, marker-assisted selection

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.