How to plot Taylors approximation using pre-calculated generalized summation formula

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Stefan 2024-3-3

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https://ww2.mathworks.cn/matlabcentral/answers/2089786-how-to-plot-taylors-approximation-using-pre-calculated-generalized-summation-formula

评论： Stefan 2024-3-11

Hi All,

I have came accross the function taylor() exampl. T = taylor(log(x), x, 'ExpansionPoint', 2); by using it I get perfect result

but I'd like to plot results of my own pre-calculated Taylors aproximation of x Order. I started with f(x)=ln(x) for 0<=x<5 when approximation is around x=1 (therefor a=1) at n points [0,2,4,6]

This is what I got.

BTW: I am total newbie, and any hint more then appriciated

clear

clc

close all

%taylors series of ln(x)

x=0:5;

a=1;

SumN=0; % initialize SumN

sign=-1; % variable that assigns a sign to a term

timepoints = 0:0.1:6;

y= zeros(1,length(timepoints));

%adding bells and whistles

fig = figure();

set(fig,'color','white')

grid on

xlabel('x')

ylabel('y')

for n=1:length(y)

SumN= @(x) ((sign).^(n+1)*((x-a).^n))/n;

y(n) = SumN(timepoints(n));

end

plot(timepoints,y,'r-','LineWidth',2);

legend('Taylor series')

Thank you

7 个评论
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VBBV 2024-3-8

编辑：VBBV 2024-3-8

在 MATLAB Online 中打开

@Stefan, In your code, the difference i noticed is you were evaluating the function at each timepoint instead of x and not summing the terms, By summing all the timepoints and evaluate the function at each value of x, you would still arrive at the same result as shown below

clear

clc

close all

%taylors series of ln(x)

x=0:0.1:5;

a=1;

SumN=0; % initialize SumN

sign=-1; % variable that assigns a sign to a term

timepoints = 1:6; % Note the zero is excluded !!! in his function too

y= zeros(1,length(timepoints));

%adding bells and whistles

figure

plot(x,log(x))

hold on

for n=1:length(x) % evaluate the function at every x

y = (sign).^(timepoints+1).*((x(n)-a).^timepoints)./timepoints;

ySum(n) = sum(y);

end

ySum

ySum = 1x51

-2.4500 -1.9187 -1.5023 -1.1726 -0.9077 -0.6911 -0.5105 -0.3566 -0.2231 -0.1054 0 0.0953 0.1823 0.2623 0.3363 0.4047 0.4674 0.5233 0.5701 0.6035 0.6167 0.5995 0.5376 0.4114 0.1950 -0.1453 -0.6521 -1.3786 -2.3900 -3.7655

plot(x,ySum,'r-','LineWidth',1);ylim([-2 2])

legend('ln(1)','Taylor series'); grid

Stefan 2024-3-8

在 MATLAB Online 中打开

Thank you for hints, and making the code work, much appriciate

I was messing around and Here is what I got

I wanted to implement the section of code below, but when I assigne my plot to variable "p", set function has an issue to accep "p" element

p = plot(rand(4));
NameArray = {'LineStyle'};
ValueArray = transpose({'-','--',':','-.'});
set(p,NameArray,ValueArray)

Anyhow, I learned a lot with your help, research and , trial and error experiments, Thanks

clear
clc
close all
%%taylors series of ln(1)
%variables
x=0:0.1:5;
a=1;
ySum = 0; % initialize ySum
sign =-1; % variable that assigns a sign to a term
y= zeros(1,length(6)); %initialize y variable
%adding bells and whistles
figure;
ylim([-2 2])
grid on
xlabel('x')
ylabel('y')
legend({"f(x)=ln(1)","Taylor Deg0","Taylor Deg2","Taylor Deg4","Taylor Deg6"}, ...
        'location','northwest')
hold on;
plot(x,log(x),'LineWidth',1.5);
for i=0:2:6
    timepoints = 1:i;
    for n=1:length(x)
        y = (sign).^(timepoints+1).*((x(n)-a).^timepoints)./timepoints;     
        ySum(n) =sum(y);
    end
hold on    
plot(x,ySum,'LineWidth',1.5);
end

VBBV 2024-3-9

编辑：VBBV 2024-3-9

在 MATLAB Online 中打开

p = plot(rand(4)); % plot returns 4 x 1 line array

NameArray1 = {'LineStyle'}; %

NameArray2 = {'LineWidth'};

ValueArray1 = {'-','--',':','-.'};

ValueArray2 = [1.5, 1.5 2 2];

for k = 1:numel(p) % use a loop to set the individual line styles

set(p(k),NameArray1{1},ValueArray1{k},NameArray2{1},ValueArray2(k)); %

end

Stefan 2024-3-11

在 MATLAB Online 中打开

Hi VBBV,

Thanks for contribution to this project, this is what I ended up with

%% Paylors series of ln(1)
%% initialize workspace
clear
clc
close all
%% declare variables
x=0:0.1:5; 
a=1; %constant variable
sign =-1; % constant variable 
y= zeros(1,length(6)); %initialize y array
ySum = zeros(1,length(6)); % initialize ySum dynamic array
%% Plot bells and whistles
markers = {'o','hexagram','*','diamond','d','v','*','h'};
colors = {'#e81416','#79c314','#ffa500','#FFA500','#70369d','#487de7','#79c314'};
linestyle = {':','--','-.',':','-.','--'};
getFirst = @(v)v{1}; 
getprop = @(options, idx)getFirst(circshift(options,-idx+2));
%% Plot of ln(1)
figure()
plot(x,log(x),'Marker',getprop(markers,5),...        
    'Color',getprop(colors,3),...
    'linestyle',getprop(linestyle,6),...
    'DisplayName', ['Line ', num2str(6)],...
    'MarkerIndices',1:10:length(log(x)),...
    'LineWidth',1.5);
hold on
%% Plot Taylors aproximation of ln(1) in 0,2,4,6 order
for i=0:2:6
    timepoints = 1:i;
    for n=1:length(x)                       
        y = (sign).^(timepoints+1).*((x(n)-a).^timepoints)./timepoints;     
        ySum(n) =sum(y);                  
    end           
    plot(x,ySum ,'Marker',getprop(markers,i),...
    'Color',getprop(colors,i),...
    'linestyle',getprop(linestyle,i),...
    'DisplayName', ['Line ', num2str(i)],...
    'LineWidth',1.5,....
    'MarkerIndices',1:5:length(ySum)); 
    hold on               
end    
%% Figure set up        
ylim([-2 2])
grid on
xlabel('x')
ylabel('y')
title('Tayylors series of ln(1)');
set(gca,'fontSize',10);
legend({"  f(x) = ln(1)","Taylor Deg0",...
    "Taylor Deg2","Taylor Deg4",...
    "Taylor Deg6"},'location','northwest')
%% End