Preliminaries

Numbers are represented in binaries, thus creating errors.
Numerical procedures also introduce errors.
Numerical analysis is the study of the behavior of errors in computation.

• Suppose that $\widehat{p}$ is an approximation to $p$. The (absolute) error is $E_p = \vert p - \widehat{p}\vert$, and the relative error is $R_p = \frac{E_p}{\vert p\vert}$, provided that $p\neq 0$.
  • Let $x = 3.141592$ (approx. $\pi$?) and $\widehat{x} = 3.14$; then the error is
    
    
    $$E_x = \vert\widehat{x} - x\vert = \vert 3.14 - 3.141592\vert = 0.001592$$
    
    
    and the relative error is
    
    
    $$R_x =\frac{\vert\widehat{x} - x\vert}{\vert x\vert}=\frac{0.001592}{3.141592}= 0.00507$$
    
    

  • Let $y = 1,000,000$ and $\widehat{y} = 999,996$; then the error is (large?)
    
    
    and the relative error is (small?)
    
    
    
  • Let $z = 0.000012$ and $\widehat{z} = 0.000009$; then the error is (small?)
    
    and the relative error is (large?)
    
    The relative error $R_p$ is a better indicator of accuracy and is preferred for floating-point representations since it deals directly with the mantissa.
• The number $\widehat{p}$ is said to approximate $p$ to $d$ significant digits if $d$ is the largest positive integer for which
  
  $$\frac{\vert\widehat{p} - p\vert}{\vert p\vert} < 0.5 × 10^{-d}$$
  
  
  • If $x = 3.141592$ and $\widehat{x} = 3.14$, then $\frac{\vert\widehat{x}-x\vert}{\vert x\vert} = 0.000507 < 0.5×10^{-2}$. Therefore, $\widehat{x}$ approximates $x$ to 2 significant digits.
  • If $y = 1000000$ and $\widehat{y} = 999996$, then $\frac{\vert\widehat{y}-y\vert}{\vert y\vert} = 0.000004 < 0.5×10^{-2}$. Therefore, $\widehat{y}$ approximates $y$ to ?? significant digits.
  • If $z = 0.000012$ and $\widehat{z} = 0.000009$, then $\frac{\vert\widehat{z}-z\vert}{\vert z\vert} = 0.25 < 0.5×10^{-2}$. Therefore, $\widehat{z}$ approximates $z$ to ?? significant digits.
• Given that
  
  $$p = \int_0 ^{1/2}e^{x^2}dx = 0.544987104184$$
  
  
  and is approximated by using Taylor series as
  
  $$\widehat{p} = \int_0^{1/2}P_8(x)dx =$$
  
  
  Since $0.5*10^{-5} > R_p = 7.03442×10^{-7} > 10^{-6}/2$, the approximation $\widehat{p}$ agrees with the true answer $p$ to 5 significant figures.
• Calculate $f(500)$ and $g(500)$ using 6 digits and rounding, with
  
  $$f(x) = x(\sqrt{x+1} - \sqrt{x}) ,~ g(x) = \frac{x}{\sqrt{x+1}+\sqrt{x}}$$
  
  
  
  
  Note that $g(x)$ is algebraically equivalent to $f(x)$, but $g(500) = 11.1748$ is more accurate than $f(500)$ to the true answer $11.174755300747198\ldots$ to six digits.
• Let $P(x) = x^3-3x^2+3x-1$ , $Q(x) = ((x-3)x+3)x-1$. Use 3-digit rounding arithmetic to compute $P(2.19) = Q(2.19) = 1.685159$:
  
  The errors are 0.015159 and -0.004841, respectively. Thus the approximation $Q(2.19)\approx 1.69$ has less error.
• Consider the Taylor polynomial expansions
  
  $$e^h = 1+h+\frac{h^2}{2!} + \frac{h^3}{3!} +O(h^4)$$
  
  
  
  
  $$cos h = 1 -\frac{h^2}{2!} +\frac{h^4}{4!}+O(h^6)$$
  
  
  With $O(h^4)+O(h^6) = O(h^4) = O(h^4)+\frac{h^4 }{4!}$, we have the sum
  
  $$e^h +cos h = 2+h+ \frac{h^3}{3!}+\frac{h^4}{4!}+O(h^4)+O(h^6)= 2+h+\frac{h^3}{3!}+O(h^4)$$
  
  
  The difference behaves similarly.
  The product
  
  $$e^h * cos h =$$
  
  
  
  
  $$= 1+h -\frac{h^3}{3}+O(h^4)$$
  
  
  and the order of approximation is $O(h^4)$.
• $x_n =\frac{1}{3^n}$, approximated by (for n = 1, 2, · · ·)
  
  $$r_0 = 1 , r_n =\frac{1}{3}r_{n-1}\left(=\frac{A}{3^n}\right)$$
  
  
  
  
  $$p_0 = 1 , p_1 =\frac{1}{3}, p_n =\frac{4}{3}p_{n-1} -\frac{1}{3}p_{n-2}\left(A\frac{1}{3^n}+B\right)$$
  
  
  
  
  $$q_0 = 1 , q_1 =\frac{1}{3}, q_n =\frac{10}{3}q_{n-1} -q_{n-2}\left(A\frac{1}{3^n}+B3^n\right)$$
  
  
  Generate a table for $x_n-r_n,x_n-p_n,x_n-q_n$, with errors introduced in the starting values:
  
  $$r_0 = 0.99996 , p_0 = q_0 = 1 , p_1 = q_1 = 0.33332$$
  
  
  The error for ${r_n}$ is stable and decreases exponentially.
  The error for ${p_n}$ is stable, but eventually dominates as $p_n \rightarrow 0$.
  The error for ${q_n}$ is unstable and grows exponentially.
• Write the following code and study the response.

  %      Determines effective machine precision for MATLAB
    a = 1.0 ;
    while  ( (1. + a) ~= 1)
       a     = a/2.  ;
    end
    delta = 2.0*a ;
    sprintf(' Machine Precision of MATLAB  is  %9.2e', delta )
  

• Write the following code and study the response.

  % uses the MATLAB   chop.m   function to find simulated  machine
  % precision for a NDIGITS decimal ( base 10 ) machine.
  data = [] ;
  for NDIGITS = 2: 20 ;
     a = 1.0 ;
     while  ( chop( (1.+a), NDIGITS ) ~= chop( (1.+a/2.), NDIGITS) )
         a  = chop( a/2. , NDIGITS) ;
     end
     theoret = 0.5*10^(1-NDIGITS) ;
     data = [ data ; NDIGITS (1.5)*a theoret ] ;
  end
  %  Note the use of (semi)logarithmic plots is usually preferable
  %  for displaying error behavior.
  semilogy( data(:,1) , data(:,2) , '*',  ...
              data(:,1) , data(:,3)  ) ;
  xlabel('NDIGITS');
  ylabel('Machine Precision')
  legend('Observed','Theoretical');
  title('Dependence of Machine Precision on Machine "Size"');
  

• Write the following code and study the response.

  % Determines the accuracy of a computed expression which is potentially 
  % subject to cancellation errors, using the MATLAB chop.m  function.
    clear ;
    data = [] ;
    NDIGITS    = 8 ;
    mu_NDIGITS = 0.5*10^(1-NDIGITS) ;
    mu_calc    = 50*mu_NDIGITS      ;
    for n = 1: 30 ;
       x = 2^n ;
       xsing      = chop( x , NDIGITS ) ;
       xm1_sing   = chop( xsing - 1 , NDIGITS ) ;
       xsq_sing   = chop( xsing*xsing , NDIGITS) ;
       xsqp4_sing = chop( xsq_sing + 4 , NDIGITS ) ;
       sroot_sing = chop( sqrt( xsqp4_sing ) , NDIGITS ) ;
       fval_sing  = chop( sroot_sing - xm1_sing , NDIGITS ) ;
       f_double   = sqrt( x^2 + 4 ) - ( x - 1 ) ;
       rel_err    = abs( f_double - fval_sing )/abs(f_double + eps ) + eps ;
       data       = [ data ; x rel_err f_double fval_sing]  ;
    end
    xmin =  min(data(:,1)) ;   xmax =  max(data(:,1)) ;
    loglog( data(:,1) , data(:,2) , '-.' , ...
                 [ xmin xmax ] , [ mu_calc  mu_calc ] , ':' ) ;
    axis( [ xmin  10*xmax   10^(-10) 10^3 ] );
    xlabel( 'x' ) ; ylabel( 'Relative Difference') ;
    legend('Observed','"Acceptable"');
    title('Variation of the Accuracy of a Computed Function with x');
    figure(2);
    semilogx( data(:,1), data(:,3), data(:,1), data(:,4),':');
    xlabel('x') ; ylabel('Computed Value of f(x)')
    axis([min(data(:,1)), 10*max(data(:,1)),-.25, 2.25])
    legend('Double Precision','Single Precision');
    title('Effect of Machine Precision on the Accuracy of a Computed Function')
  

• Write the code and analysis output

  a=123*2*pi*/360
  L=inline('9/sin(pi-2.1468-c)+7/sin(c)')
  fplot(L,[0.4,0.5]); grid on
  fminbnd(L,0.4,0.5)
  L(0.4677)
  fminbnd(L,0.4,0.5,optimset('Display','iter'))
  

• Write a code that adds 0.0001 one thousand times. The result should equal 1.0 exactly but this is not true for single precision.
• Write a code that computes values of this expression
  
  $$z=\frac{(x+y)^2-2xy-y^2}{x^2}$$
  
  
  with different values of $x$ and $y$. (Hint: use $y=10000$ and change the x-value as $0.01,0.001,0.0001,,\ldots$)