Applied Numerical Methods Using MATLAB. Won Y. Yang

Figure 1.5 Distribution (histogram) of noise generated by the rand()/ randn() command.

      %nm01f05a.m N=1e4; x=rand(N,1); % An Nx1 noise vector with U(0,1) mx=mean(x); sgm2=sum((x-mx).̂2)/(N-1), var(x) subplot(221), M=20; hist(x,M) % Histogram having M=20 bins a=-1; b=1; y=(b-a)*x+a; %Eq.(1.1.4): 1000x1 noise vector with U(-1,1) subplot(222), hist(y,M) % Histogram

       1.1.8.2 Random Number with Normal (Gaussian) Distribution

      The numbers in a matrix generated by the MATLAB function ‘ randn(M,N)’ have normal (Gaussian) distribution with average m = 0 and variance σ² = 1, as described by N(0,1). The random number x generated by ‘ rand()’ has the probability density function

(1.1.5)

If you want another Gaussian number y with a general normal distribution N(m,σ²), transform the standard Gaussian number x as follows:

      (1.1.6)

      The probability density function of the new Gaussian number generated by this transformation is obtained by substituting x = (y − m)/σ into Eq. (1.1.5) and dividing the result by the scale factor σ (which can be seen in dx = dy/σ) so that the integral of the density function over the whole interval ( ‐∞,+∞) amounts to 1.

      (1.1.7)

      %nm01f05b.m N=1e4; x=randn(N,1); % An Nx1 noise vector with N(0,1) subplot(223), M=20; hist(x,M) % Histogram having M=20 bins f=@(x,m,sgm)exp(-(x-m).̂2/2/sgm̂2)/sqrt(2*pi)/sgm; %Gaussian density ftn [Ns,Cs]=hist(x,20); dx=Cs(2)-Cs(1); % Bin width x_=[-5:0.01:5]; fx=f(x_,0,1); hold on, plot(x_,fx*dx*N,'r:') sgm=1/2; m=1; y=sgm*x+m; % An Nx1 noise vector with N(m,sgm̂2) subplot(224), hist(y,M) % Histogram [Ns,Cs]=hist(y,M); dy=Cs(2)-Cs(1); % Bin width y_=[-5:0.01:5]; fy=f(y_,m,sgm); hold on, plot(y_,fy*dy*N,'r:')

      For practice, we make a vector consisting of 1000 standard Gaussian numbers, transform it to make a vector of numbers having normal distribution N(1,1/4), with mean m = 1 and variance σ² = 1/4, and then draw the histograms for the two Gaussian number vectors (Figure 1.5b1,b2) by running the above block of MATLAB statements.

1.1.9 Flow Control

      1.1.9.1 if‐end and switch‐case‐end Statements

An if‐ end block basically consists of an if statement, a sequel part and an end statement categorizing the block. An if statement, having a condition usually based on the relational/logical operator (Table 1.4), is used to control the program flow, i.e. to adjust the order in which statements are executed according to whether or not the condition is met, mostly depending on unpredictable situations. The sequel part consisting of one or more statements may contain else or elseif statements, possibly in a nested structure containing another if statement inside it. The switch‐case‐end block might replace a multiple if‐elseif‐..‐end statement in a neat manner.

Table 1.4 Relational operators and logical operators.

Let us see the following examples:

      1.1.9.2 for index=i_0:increment:i_last‐end Loop

      A for loop makes a block of statements executed repeatedly for a specified number of times, with its loop index increasing from i_ 0 to a number not greater than i_ last by a specified step ( increment) or by 1 if not specified. The loop iteration normally ends when the loop index reaches i_ last, but it can be stopped by a break statement inside the for loop. The for loop with a positive/negative increment will never be iterated if the last value ( i_ last) of the index is smaller/greater than the starting value ( i_0).