int m=1,n=2,*p=&m,*q=&n,*r;r=p;p=q;q=r;#includemain(){ int m=1,n=2,*p=&m,*q=&n,*r;r=p;p=q;q=r;printf("%d,%d,%d,%d\n",m,n,*p,*q);}值到底传不传递,都晕了.
传递.首先p指向m,q指向n；然后r=p,r指向p,p指向m,所以r指向mp=q,p指向q,q指向n,所以p指向nq=r,q指向r,r指向m,所以,q指向m所以输出结果是：m,n,n,m,即1,2,2,1
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扫描下载二维码Chapter 1 M-Files
Chapter 1 M-Files
Script File Index
Prints a short table of sine evaluations.
Displays a sequence of sin(x) plots.
Plots exp(x) and an approximation to exp(x).
Plots tan(x).
Superimposes plots of sin(x) and cos(x).
Displays nine regular polygons, one per window.
Displays the sum of four sine functions.
Displays a pair of sum-of-sine functions.
Sample core exploratory environment.
Framework for running UpDown.
Displays the distribution of rand and randn.
Displays 2-dimensional rand and randn.
Histogram of 1000 dice rolls.
Monte Carlo computation of pi.
Polygon smoothing.
Relative and absolute error in Stirling formula.
Plots relative error in Taylor approximation to exp(x).
Roundoff in the expansion of (x-1)^ 6.
Examines precision, overflow, and underflow.
Examines MyExp1 , MyExp2 , MyExp3 , and MyExp4.
Applies Derivative to Sin10.
Three-digit arithmetic sum of 1 + 1/2 + ... + 1/n.
Tests the function PadeArray.
Illustrates how to use fonts.
Shows how to generate math symbols.
Shows how to generate Greek letters.
Shows how to align with text.
Shows how vary line width in a plot.
Shows how to set tick marks on axes.
Shows how to add a legend to a plot.
Shows how to use built-in colors and user-defined colors.
Function File Index
For-loop Taylor approximation to exp(x).
Vectorized version of MyExpF.
Better vectorized version of MyExpF.
While-loop Taylor approximation to exp(x).
Vectorized version of MyExpW.
Better vectorized version of MyExpW.
Numerical differentiation.
Sets up 3-digit arithmetic representation.
Converts 3-digit representation to float.
Simulates 3-digit arithmetic.
Pretty prints a 3-digit representation.
Builds a cell array of Pade coefficients.
Generates Pade approximant coefficients
% Script File: SineTable
% Prints a short table of sine evaluations.
x = linspace(0,1,n);
y = sin(2*pi*x);
sin(x(k))')
disp('------------------------')
for k=1:21
degrees = (k-1)*360/(n-1);
disp(sprintf(' %2.0f
',k,degrees,y(k)));
disp( ' ');
disp('x(k) is given in degrees.')
disp(sprintf('One Degree = %5.3e Radians',pi/180))
% Script File: SinePlot
% Displays increasingly smooth plots of sin(2*pi*x).
for n = [4 8 12 16 20 50 100 200 400]
x = linspace(0,1,n);
y = sin(2*pi*x);
title(sprintf('Plot of sin(2*pi*x) based upon n = %3.0f points.',n))
% Script File: ExpPlot
% Examines the function
f(x) = ((1 + x/24)/(1 - x/24 + x^2/384))^8
% as an approximation to exp(z) across [0,1].
= linspace(0,1,200);
= 1 + x/24;
denom = 1 - x/12 + (x/384).*x;
quot = num./
y = quot.^8;
plot(x,y,x,exp(x))
% Script File: TangentPlot
% Plots the function tan(x), -pi/2 &= x &= 9pi/2
ymax = 10;
x = linspace(-pi/2,pi/2,40);
y = tan(x);
axis([-pi/2 9*pi/2 -ymax ymax])
title('The Tangent Function')
xlabel('x')
ylabel('tan(x)')
xnew = x+ k*
plot(xnew,y);
% Script File: SineAndCosPlot
% Plots the functions sin(2*pi*x) and cos(2*pi*x) across [0,1]
% and marks their intersection.
x = linspace(0,1,200);
y1 = sin(2*pi*x);
y2 = cos(2*pi*x);
plot(x,y1,x,y2,'--',[1/8 5/8],[1/sqrt(2) -1/sqrt(2)],'*')
% Script File: Polygons
% Plots selected regular polygons.
theta = linspace(0,2*pi,361);
c = cos(theta);
s = sin(theta);
for sides = [3 4 5 6 8 10 12 18 24]
stride = 360/
subplot(3,3,k)
plot(c(1:stride:361),s(1:stride:361))
axis([-1.2 1.2 -1.2 1.2])
axis equal
% Script File: SumOfSines
% Plots f(x) = 2sin(x) + 3sin(2x) + 7sin(3x) + 5sin(4x)
% across the interval [-10,10].
x = linspace(-10,10,200)';
A = [sin(x) sin(2*x) sin(3*x) sin(4*x)];
y = A*[2;3;7;5];
title('f(x) = 2sin(x) + 3sin(2x) + 7sin(3x)
+ 5sin(4x)')
% Script File: SumOfSines2
% Plots the functions
f(x) = 2sin(x) + 3sin(2x) + 7sin(3x) + 5sin(4x)
g(x) = 8sin(x) + 2sin(2x) + 6sin(3x) + 9sin(4x)
% across the interval [-10,10].
x = linspace(-10,10,n)';
A = [sin(x) sin(2*x) sin(3*x) sin(4*x)];
y = A*[2 8;3 2;7 6;5 9];
% Script File: UpDown
% Generates a column vector x(1:n) of positive integers
% where x(1) is solicited and
x(k+1) = x(k)/2
if x(k) is even.
x(k+1) = 3x(k)+1
if x(k) is odd.
% The value of n is either 500 or the first index with the
% property that x(n) = 1, whichever comes first.
x = zeros(500,1);
x(1) = input('Enter initial positive integer:');
while ((x(k) ~= 1) & (k & 500))
if rem(x(k),2) == 0
x(k+1) = x(k)/2;
x(k+1) = 3*x(k)+1;
x = x(1:n);
disp(sprintf('x(1:%1.0f) = \n',n))
disp(sprintf('%-8.0f',x))
[xmax,imax] = max(x);
disp(sprintf('\n x(%1.0f) = %1.0f is the max.',imax,xmax))
density = sum(x&=x(1))/x(1);
disp(sprintf(' The density is %5.3f.',density))
title(sprintf('x(1) = %1.0f, n = %1.0f',x(1),n));
plot(-sort(-x))
title('Sequence values sorted.')
I = find(rem(x(1:n-1),2)==1);
if length(I)&1
plot((1:n),zeros(1,n),I+1,x(I+1),I+1,x(I+1),'*')
title('Local Maxima')
% Script File: RunUpDown
% Environment for studying the up/down sequence.
% Stores selected results in file UpDownOutput.
while(input('Another Example? (1=yes, 0=no)'))
diary UpDownOutput
if (input('Keep Output? (1=yes, 0=no)')~=1)
delete UpDownOutput
% Script File: Histograms
% Histograms of rand(1000,1) and randn(1000,1).
subplot(2,1,1)
x = rand(1000,1);
hist(x,30)
axis([-1 2 0 60])
title('Distribution of Values in rand(1000,1)')
xlabel(sprintf('Mean = %5.3f. Median = %5.3f.',mean(x),median(x)))
subplot(2,1,2)
x = randn(1000,1);
hist(x,linspace(-2.9,2.9,100));
title('Distribution of Values in randn(1000,1)')
xlabel(sprintf('Mean = %5.3f. Standard Deviation = %5.3f',mean(x),std(x)))
% Script File: Clouds
% 2-dimensional pictures of the uniform and normal distributions.
Points = rand(1000,2);
subplot(1,2,1)
plot(Points(:,1),Points(:,2),'.')
title('Uniform Distribution.')
axis([0 1 0 1])
axis square
Points = randn(1000,2);
subplot(1,2,2)
plot(Points(:,1),Points(:,2),'.')
title('Normal Distribution.')
axis([-3 3 -3 3])
axis square
% Script File: Dice
% Simulates 1000 rollings of a pair of dice.
= 1 + floor(6*rand(1000,1));
Second = 1 + floor(6*rand(1000,1));
Throws = First + S
hist(Throws, linspace(2,12,11));
title('Outcome of 1000 Dice Rolls.')
% Script File: Darts
% Estimates pi using random dart throws.
rand('seed',.123456)
NumberInside = 0;
PiEstimate = zeros(500,1);
for k=1:500
x = -1+2*rand(100,1);
y = -1+2*rand(100,1);
NumberInside = NumberInside + sum(x.^2 + y.^2 &= 1);
PiEstimate(k) = (NumberInside/(k*100))*4;
plot(PiEstimate)
title(sprintf('Monte Carlo Estimate of Pi = %5.3f',PiEstimate(500)));
xlabel('Hundreds of Trials')
% Script File: Smooth
% Solicits n, draws an n-gon, and then smooths it.
n = input('Enter the number of edges:');
axis([0 1 0 1])
axis square
x = zeros(n,1);
y = zeros(n,1);
title(sprintf('Click in %2.0f more points.',n-k+1))
[x(k) y(k)] = ginput(1);
plot(x(1:k),y(1:k), x(1:k),y(1:k),'*')
x = [x;x(1)];
y = [y;y(1)];
plot(x,y,x,y,'*')
title('The Original Polygon')
xlabel('Click inside window to smooth, outside window to quit.')
[a,b] = ginput(1);
while (v(1)&=a) & (a&=v(2)) & (v(3)&=b) & (b&=v(4));
x = [(x(1:n)+x(2:n+1))/2;(x(1)+x(2))/2];
y = [(y(1:n)+y(2:n+1))/2;(y(1)+y(2))/2];
m = max(abs([x;y]));
plot(x,y,x,y,'*')
axis square
title(sprintf('Number of Smoothings = %1.0f',k))
xlabel('Click inside window to smooth, outside window to quit.')
[a,b] = ginput(1);
% Script File: Stirling
% Prints a table showing error in Stirling's formula for n!
Relative')
Approximation
disp('----------------------------------------------------------------')
for n=1:13
nfact = n*
s = sqrt(2*pi*n)*((n/e)^n);
abserror = abs(nfact - s);
relerror = abserror/
s1 = sprintf('
%13.2f',n,nfact,s);
s2 = sprintf('
%5.2e',abserror,relerror);
disp([s1 s2])
% Script File: ExpTaylor
% Plots, as a function of n, the relative error in the
% Taylor approximation
1 + x + x^2/2! +...+ x^n/n!
% to exp(x).
nTerms = 50;
for x=[10 5 1 -1 -5 -10]
f = exp(x)*ones(nTerms,1);
for k=1:50
term = x.*term/k;
err(k) = abs(f(k) - s);
relerr = err/exp(x);
semilogy(1:nTerms,relerr)
ylabel('Relative Error in Partial Sum.')
xlabel('Order of Partial Sum.')
title(sprintf('x = %5.2f',x))
% Script File: Zoom
% Plots (x-1)^6 near x=1 with increasingly refined scale.
% Evaluation via x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x +1
% leads to severe cancellation.
for delta = [.1 .01 .008 .007 .005
x = linspace(1-delta,1+delta,n)';
y = x.^6 - 6*x.^5 + 15*x.^4 - 20*x.^3 + 15*x.^2
- 6*x + ones(n,1);
subplot(2,3,k)
plot(x,y,x,zeros(1,n))
axis([1-delta 1+delta -max(abs(y)) max(abs(y))])
% Script File: FpFacts
% Prints some facts about the underlying floating point system.
% p = smallest positive integer so 1+1/2^p = 1.
x=1; p=0; y=1; z=x+y;
while x~=z
disp(sprintf('p = %2.0f
is the smallest positive integer so 1+1/2^p = 1.',p))
% q = smallest positive integer so 1/2^q = 0.
disp(sprintf('q = %2.0f is the smallest positive integer so 1/2^q = 0.',q))
% r = smallest positive integer so 2^r = inf.
while x~=inf
disp(sprintf('r = %2.0f is the smallest positive integer so 2^r = inf.',r))
% The Kahan example. In exact arithmetic f/e = 0/0:
h = 1./2.;
x = 2./3. -
y = 3./5. -
e = (x+x+x) -
f = (y+y+y+y+y)-h;
disp(sprintf('\nz = %10.5f',z))
% Script File: TestMyExp
% Checks MyExpF, MyExp1, MyExp2, MyExp3, and MyExp4.
x = linspace(-1,1,m);
exact = exp(x);
y = zeros(1,m);
for n = [4 8 16 20]
y(i) = MyExpF(x(i),n);
RelErr = abs(exact - y)./
subplot(2,2,k)
plot(x,RelErr)
title(sprintf('n = %2.0f',n))
nRepeat = 1;
% Note: The following fragment may take a long time to execute
% with this value of nRepeat.
disp('T1 = MyExp1 benchmark.')
disp('T2 = MyExp2 benchmark.')
disp(' Length(x)
disp('-----------------------')
xL = linspace(-1,1,L);
for k=1:nRepeat
y = MyExp1(xL);
for k=1:nRepeat
y = MyExp2(xL);
disp(sprintf('%6.0f
',L,T2/T1))
disp(' Length(x)
MyExp4(x) ')
disp('---------------------------------------')
for L = 50:50:300
xL = linspace(-1,1,L);
y = MyExp3(xL);
y = MyExp4(xL);
disp(sprintf('%6.0f
',L,f3,f4))
% Script File: TestDerivative
% Numerical differentiation of f(x) = sin(10x)
% Look at errors across [0,1].
% err1 = error with no derivative information.
% err2 = error with 2nd derivative information.
x=linspace(0,1,m);
exactDer = a*cos(a*x(i));
err1(i) = abs(exactDer - Derivative('sin10',x(i)));
err2(i) = abs(exactDer - Derivative('sin10',x(i),eps,M2));
% Plot err1(1:m)
plot(x,err1);
title('Derivative(''sin10'',a)')
xlabel('a')
ylabel('Der Error')
% Plot err2(1:m)
plot(x,err2);
title('Derivative(''sin10'',a,eps,100)')
xlabel('a')
ylabel('Der Error')
% Script File: Euler
% Sums the series 1 + 1/2 + 1/3 + .. in 3-digit floating point arithmetic.
% Terminates when the addition of the next term does not change
% the value of the running sum.
oldsum = represent(0);
one = represent(1);
while convert(sum) ~= convert(oldsum)
= represent(k);
term = float(one,kay,'/');
= float(sum,term,'+');
disp(['The sum for ' num2str(k) ' or more terms is ' pretty(sum)])
% Script File: ShowPadeArray
% Illustrates the function PadeArray that creates a cell array.
P = PadeArray(5,5);
disp(sprintf('\n (%1d,%1d) Pade Coefficients:\n',i-1,j-1))
' sprintf('%7.4f
',P{i,j}.num)])
' sprintf('%7.4f
',P{i,j}.den)])
% Script File: ShowFonts
HA = 'HorizontalAlign';
fonts = {'Times-Roman' 'Helvetica' 'AvantGarde' 'Bookman' 'Palatino'...
'ZapfChancery' 'Courier' 'NewCenturySchlbk' 'Helvetica-Narrow'};
plot([-10 100 100 -10 -10],[0
axis([-10 100 0 60])
F = fonts{k};
text(45,55,F,'FontName',F,'FontSize',24,HA,'center')
text(10,47,'Plain','FontName',F,'FontSize',22,HA,'center')
text(45,47,'Bold','FontName',F,'Fontweight','bold','FontSize',22,HA,'center')
text(82,47,'Oblique','FontName',F,'FontAngle','oblique','FontSize',22,HA,'center')
for size=[22 18 14 12 11 10 9]
text(10,v,'Matlab','FontName',F,'FontSize',size,HA,'center')
text(45,v,'Matlab','FontName',F,'FontSize',size,HA,'center','FontWeight','bold')
text(82,v,'Matlab','FontName',F,'FontSize',size,HA,'center','FontAngle','oblique')
% Script File: ShowSymbols
plot([0 12 12 0 0],[0 0 12 12 0])
text(6,10.5,'Math Symbols','FontSize',18,'HorizontalAlign','center')
x = 1; x1 = x+.7;
y = 4.6; y1 = y+.7;
z = 9; z1 = z+.7;
text(y,9,'\leftarrow','Fontsize',n);
text(y,8,'\rightarrow','Fontsize',n);
text(y,7,'\uparrow','Fontsize',n)
text(y,6,'\downarrow','Fontsize',n)
text(y,5,'\Leftarrow','FontSize',n)
text(y,4,'\Rightarrow','FontSize',n)
text(y,3,'\Leftrightarrow','FontSize',n)
text(y,2,'\partial','FontSize',n)
text(x,9,'\neq','FontSize',n)
text(x,8,'\geq','FontSize',n)
text(x,7,'\approx','FontSize',n)
text(x,6,'\equiv','Fontsize',n)
text(x,5,'\cong','Fontsize',n)
text(x,4,'\pm','Fontsize',n)
text(x,3,'\nabla','FontSize',n)
text(x,2,'\angle','FontSize',n)
text(z,9,'\in','FontSize',n)
text(z,8,'\subset','Fontsize',n)
text(z,7,'\cup','Fontsize',n)
text(z,6,'\cap','Fontsize',n)
text(z,5,'\perp','FontSize',n)
text(z,4,'\infty','FontSize',n)
text(z,3,'\int','Fontsize',n)
text(z,2,'\times','FontSize',n)
text(y1,9,'\\leftarrow','Fontsize',n);
text(y1,8,'\\rightarrow','Fontsize',n);
text(y1,7,'\\uparrow','Fontsize',n)
text(y1,6,'\\downarrow','Fontsize',n)
text(y1,5,'\\Leftarrow','FontSize',n)
text(y1,4,'\\Rightarrow','FontSize',n)
text(y1,3,'\\Leftrightarrow','FontSize',n)
text(y1,2,'\\partial','FontSize',n)
text(x1,9,'\\neq','FontSize',n)
text(x1,8,'\\geq','FontSize',n)
text(x1,7,'\\approx','FontSize',n)
text(x1,6,'\\equiv','Fontsize',n)
text(x1,5,'\\cong','Fontsize',n)
text(x1,4,'\\pm','Fontsize',n)
text(x1,3,'\\nabla','FontSize',n)
text(x1,2,'\\angle','FontSize',n)
text(z1,9,'\\in','FontSize',n)
text(z1,8,'\\subset','Fontsize',n)
text(z1,7,'\\cup','Fontsize',n)
text(z1,6,'\\cap','Fontsize',n)
text(z1,5,'\\perp','FontSize',n)
text(z1,4,'\\infty','FontSize',n)
text(z1,3,'\\int','Fontsize',n)
text(z1,2,'\\times','FontSize',n)
% Script File: ShowGreek
plot([-1 12 12 -1 -1],[-1 -1 12 12 -1])
text(4,10,'Greek Symbols','FontSize',18)
x = 0; x1 = x+.7;
y = 4; y1 = y+.7;
z = 8; z1 = z+.7;
text(x,8,'\alpha','Fontsize',n);
text(x,7,'\beta','Fontsize',n);
text(x,6,'\gamma','Fontsize',n)
text(x,5,'\delta','Fontsize',n)
text(x,4,'\epsilon','FontSize',n)
text(x,3,'\kappa','FontSize',n)
text(x,2,'\lambda','Fontsize',n)
text(x,1,'\mu','Fontsize',n)
text(x,0,'\nu','Fontsize',n)
text(x1,8,'\\alpha','Fontsize',n);
text(x1,7,'\\beta','Fontsize',n);
text(x1,6,'\\gamma','Fontsize',n)
text(x1,5,'\\delta','Fontsize',n)
text(x1,4,'\\epsilon','FontSize',n)
text(x1,3,'\\kappa','FontSize',n)
text(x1,2,'\\lambda','Fontsize',n)
text(x1,1,'\\mu','Fontsize',n)
text(x1,0,'\\nu','Fontsize',n)
text(y,8,'\omega','Fontsize',n)
text(y,7,'\phi','Fontsize',n)
text(y,6,'\pi','FontSize',n)
text(y,5,'\chi','Fontsize',n)
text(y,4,'\psi','Fontsize',n)
text(y,3,'\rho','FontSize',n)
text(y,2,'\sigma','Fontsize',n)
text(y,1,'\tau','FontSize',n)
text(y,0,'\upsilon','FontSize',n)
text(y1,8,'\\omega','Fontsize',n)
text(y1,7,'\\phi','Fontsize',n)
text(y1,6,'\\pi','FontSize',n)
text(y1,5,'\\chi','Fontsize',n)
text(y1,4,'\\psi','Fontsize',n)
text(y1,3,'\\rho','FontSize',n)
text(y1,2,'\\sigma','Fontsize',n)
text(y1,1,'\\tau','FontSize',n)
text(y1,0,'\\upsilon','FontSize',n)
text(z,8,'\Sigma','FontSize',n)
text(z,7,'\Pi','FontSize',n)
text(z,6,'\Lambda','FontSize',n)
text(z,5,'\Omega','FontSize',n)
text(z,4,'\Gamma','FontSize',n)
text(z1,8,'\\Sigma','FontSize',n)
text(z1,7,'\\Pi','FontSize',n)
text(z1,6,'\\Lambda','FontSize',n)
text(z1,5,'\\Omega','FontSize',n)
text(z1,4,'\\Gamma','FontSize',n)
% Script File: ShowText
t = pi/6 + linspace(0,2*pi,7);
x = r*cos(t);
y = r*sin(t);
plot(x,y);
axis equal off
HA = 'HorizontalAlignment';
VA = 'VerticalAlignment';
text(x(1),y(1),'\leftarrow {\itP}_{1}',
HA,'left','FontSize',14)
text(x(2),y(2),'\downarrow',
HA,'center',VA,'baseline','FontSize',14)
text(x(2),y(2),'{
\itP}_{2}',
HA,'left',VA,'bottom','FontSize',14)
text(x(3),y(3),'{\itP}_{3} \rightarrow', HA,'right','FontSize',14)
text(x(4),y(4),'{\itP}_{4} \rightarrow', HA,'right','FontSize',14)
text(x(5),y(5),'\uparrow',
HA,'center',VA,'top','FontSize',14)
text(x(5),y(5),'{\itP}_{5}
HA,'right',VA,'top','FontSize',14)
text(x(6),y(6),'\leftarrow {\itP}_{6}',
HA,'left','FontSize',14)
text(0,1.4*r,'A Labeled Hexagon^{1}',HA,'center','FontSize',14)
text(0,-1.4*r,'^{1} A hexagon has six sides.',HA,'center','FontSize',10)
% Script File: ShowLineWidth
plot([0 14 14 0 0],[0 0 14 14 0])
text(7,13,'LineWidth','FontSize',18,'HorizontalAlign','center')
for width=0:10
h = plot([3 12],[11-width 11-width]);
if width&0
set(h,'LineWidth',width)
text(1,11-width,sprintf('%3d',width),'FontSize',14)
text(1,11-width,'default','FontSize',14)
% Script File: ShowAxes
F = 'Times-Roman'; n = 12;
t = linspace(0,2*pi); c = cos(t); s = sin(t);
axis([-1.3 1.3,-1.3 1.3])
axis equal
title('The Circle ({\itx-a})^{2} + ({\ity-b})^{2} = {\itr}^{2}','FontName',F,'FontSize',n)
xlabel('x','FontName',F,'FontSize',n)
ylabel('y','FontName',F,'FontSize',n)
set(gca,'XTick',[-.5 0 .5])
set(gca,'YTick',[-.5 0 .5])
% Script File: ShowLegend
t = linspace(0,2);
axis([0 2 -1.5 1.5])
y1 = sin(t*pi);
y2 = cos(t*pi);
plot(t,y1,t,y2,'--',[0 .5 1 1.5 2],[0 0 0 0 0],'o')
set(gca,'XTick',[])
set(gca,'YTick',[0])
legend('sin(\pi t)','cos(\pi t)','roots',0)
% Script File: ShowColor
plot([-1 16 16 -1 -1],[-3 -3 18 18 -3])
text(0.5,16.5,'Built-In Colors','FontSize',14)
text(9.5,16.5,'A Gradient','FontSize',14)
% The predefined colors.
x = [3 6 6 3 3];
y = [-.5 -.5 .5 .5 -.5];
fill(x,y,'c')
text(0,0,'cyan','FontSize',12)
fill(x,y+2,'m')
text(0,2,'magenta','FontSize',12)
fill(x,y+4,'y')
text(0,4,'yellow','FontSize',12)
fill(x,y+6,'r')
text(0,6,'red','FontSize',12)
fill(x,y+8,'g')
text(0,8,'green','FontSize',12)
fill(x,y+10,'b')
text(0,10,'blue','FontSize',12)
fill(x,y+12,'k')
text(0,12,'black','FontSize',12)
fill(x,y+14,'w')
text(0,14,'white','FontSize',12)
% Making your own colors.
for f = [.4 .6 .7 .8 .85 .90 .95 1]
color = [f 1 1];
text(x(1)+5,y(1)+(2*(k-1)),sprintf('[ %4.2f , %1d , %1d ]',f,1,1),'VerticalAlign','bottom')
fill(x+9,y+(2*(k-1)),color)
function y = MyExpF(x,n)
% y = MyExpF(x,n)
% x is a scalar, n is a positive integer
% and y = n-th order Taylor approximation to exp(x).
for k = 1:n
term = x*term/k;
function y = MyExp1(x)
% y = MyExp1(x)
% x is a column n-vector and y is a column n-vector with the property that
% y(i) is a Taylor approximation to exp(x(i)) for i=1:n.
nTerms = 17;
n = length(x);
y = ones(n,1);
y(i) = MyExpF(x(i),nTerms);
function y = MyExp2(x)
% y = MyExp2(x)
% x is an n-vector and y is an n-vector with with the same shape
% and the property that y(i) is a Taylor approximation to
% exp(x(i)) for i=1:n.
y = ones(size(x));
nTerms = 17;
term = ones(size(x));
for k=1:nTerms
term = x.*term/k;
function y = MyExpW(x)
% y = MyExpW(x)
% x is a scalar and y is a Taylor approximation to exp(x).
while abs(term) & eps*abs(y)
k = k + 1;
term = x*term/k;
function y = MyExp3(x)
% y = MyExp3(x)
% x is a column n-vector and y is a column n-vector with the property that
% y(i) is a Taylor approximation to exp(x(i)) for i=1:n.
n = length(x);
y = ones(n,1);
y(i) = MyExpW(x(i));
function y = MyExp4(x)
% y = MyExp4(x)
% x is an n-vector and y is an n-vector with
the same shape and the
% property that y(i) is a Taylor approximation to exp(x(i)) for i=1:n.
y = zeros(size(x));
term = ones(size(x));
while any(abs(term) & eps*abs(y))
term = x.*term/k;
function [d,err] = Derivative(fname,a,delta,M2)
% d = Derivative(fname,a,delta,M2);
a string that names a function f(x) whose derivative
% at x = a is sought. delta is the absolute error associated with
% an f-evaluation and M2 is an estimate of the second derivative
% magnitude near a. d is an approximation to f'(a) and err is an estimate
% of its absolute error.
Derivative(fname,a)
Derivative(fname,a,delta)
Derivative(fname,a,delta,M2)
if nargin &= 3
% No derivative bound supplied, so assume the
% second derivative bound is 1.
if nargin == 2
% No function evaluation error supplied, so
% set delta to eps.
% Compute optimum h and divided difference
hopt = 2*sqrt(delta/M2);
= (feval(fname,a+hopt) - feval(fname,a))/
err = 2*sqrt(delta*M2);
function y = sin10(x)
y = sin(10*x);
function f = Represent(x)
% f = Represent(x)
% Yields a 3-digit mantissa floating point representation of f:
f.mSignBit
mantissa sign bit (0 if x&=0, 1 otherwise)
mantissa (= f.m(1)/10 + f.m(2)/100 + f.m(3)/1000)
f.eSignBit
the exponent sign bit (0 if exponent nonnegative, 1 otherwise)
the exponent (-9&=f.e&=9)
% If x is out side of [-.999*10^9,.999*10^9], f.m is set to inf.
% If x is in the range [-.100*10^-9,.100*10^-9] f is the representation of zero
% in which both sign bits are 0, e is zero, and m = [0 0 0].
f = struct('mSignBit',0,'m',[0 0 0],'eSignBit',0,'e',0);
if abs(x)&.100*10^-9
% Underflow. Return 0
if x&.999*10^9
% Overflow. Return inf
if x&-.999*10^9
% Overflow. Return -inf
f.mSignBit = 1;
% Set the mantissa sign bit
f.mSignBit = 0;
f.mSignBit = 1;
% Determine m and e so .1 &= m & 1 and abs(x) = m*10^e
m = abs(x); e = 0;
while m &= 1,
m = m/10; e = e+1; end
while m & 1/10,m = 10*m; e = e-1; end
% Determine nearest integer to 1000m
z = round(1000*m);
% Set the mantissa
f.m(1) = floor(z/100);
f.m(2) = floor((z - f.m(1)*100)/10);
f.m(3) = z - f.m(1)*100 - f.m(2)*10;
% Set the exponent and its sign bit.
f.eSignBit = 0;
f.eSignBit = 1;
function x = Convert(f)
% x = Convert(f)
% f is a is a representation of a 3-digit floating point number. (For details
% type help represent. x is the value of v.
% Overflow situations
if (f.m == inf) & (f.mSignBit==0)
if (f.m == inf) & (f.mSignBit==1)
% Mantissa value
mValue = (100*f.m(1) + 10*f.m(2) + f.m(3))/1000;
if f.mSignBit==1
mValue = -mV
% Exponent value
eValue = f.e;
if f.eSignBit==1
eValue = -eV
x = mValue * 10^eV
function z = Float(x,y,op)
% z = Float(x,y,op)
% x and y are representations of a 3-digit floating point number. (For details
% type help represent.
% op is one of the strings '+', '-', '*', or '/'.
% z is the 3-digit floating point representation of x op y.
sx = num2str(convert(x));
sy = num2str(convert(y));
z = represent(eval(['(' sx ')' op '(' sy ')' ]));
function s = Pretty(f)
% s = Pretty(f)
% f is a is a representation of a 3-digit floating point number. (For details
% type help represent.
% s is a string so that disp(s) &pretty prints& the value of f.
if (f.m == inf) & (f.mSignBit==0)
s = 'inf';
elseif (f.m==inf) & (f.mSignBit==1)
s = '-inf';
% Convert the mantissa.
m = ['.' num2str(f.m(1)) num2str(f.m(2)) num2str(f.m(3))];
if f.mSignBit == 1
m = ['-' m];
% Convert the exponent.
e = num2str(f.e);
if f.eSignBit == 0
e = ['+' num2str(f.e)];
e = ['-' num2str(f.e)];
s = [m ' x 10^' e ];
function P = PadeArray(m,n)
% P = PadeArray(m,n)
% m and n are nonnegative integers.
% P is an (m+1)-by-(n+1) cell array.
% P{i,j} represents the (i-1,j-1) Pade approximation N(x)/D(x) to exp(x).
P = cell(m+1,n+1);
for i=1:m+1
for j=1:n+1
P{i,j} = PadeCoeff(i-1,j-1);
function R = PadeCoeff(p,q)
% R = PadeCoeff(p,q)
% p and q are nonnegative integers and R is a representation of the
% (p,q)-Pade approximation N(x)/D(x) to exp(x):
is a row (p+1)-vector whose entries are the coefficients of the
p-degree numerator polynomial N(x).
is a row (q+1)-vector whose entries are the coefficients of the
q-degree denominator polynomial D(x).
R.num(1) + R.num(2)x + R.num(3)x^2
------------------------------------
R.den(1) + R.den(2)x
% is the (2,1) Pade approximation.
a(1) = 1; for k=1:p, a(k+1) =
a(k)*(p-k+1)/(k*(p+q-k+1)); end
b(1) = 1; for k=1:q, b(k+1) = -b(k)*(q-k+1)/(k*(p+q-k+1)); end
R = struct('num',a,'den',b);}