Using nested FOR() loops to populate a matrix

Hello,

Having previously populated a matrix to demonstrated the MacLaurin Series, but entering all the data essentially manually, I'm having a go at using FOR() loops to do the same job again.

I'm trying to demonstrate how the MacLaurin series progressively and more accurately approximates e^x, by creating a vector series of X values from 0 to 1 in steps of 0.01 and then 25 Y series in a single matrix, each row being a progressively better approximation, i.e. row 1 being f(x)=f(0).x^0.1/0!, row 2 being f(x)=f(0).x^0*1/0! + f(0).x^1*1/1!, row 3 being f(x)=f(0).x^0*1/0! + f(0).x^1*1/1! + f(0).x^2*1/2! etc...

Here's my attempt at the code so far:

% MacLaurin Series of e^x

format longG

% define vector to hold series x values
X = 0 : 0.01 : 1;

% Create matrix to contain the results of each function in the series
% A FOR loop is used to populate the matrix without having to write out all
% the series iterations manually
for i = 0:25
    F(i+1,:) = [(X(1,:).^i)/factorial(i)]
end

% Create increasingly better approximations by summing increasing numbers
% of function results
for i = 1:26
    for j = 1:i
        Y(i,:) = [Y(i,:) + F(j,:)]
    end
end

I'm pretty sure vector F is created correctly, but trying the same method with matrix Y results in the error:

Unrecognized function or variable 'Y'.

Error in tutorial_6d (line 21)
        Y(i,:) = [Y(i,:) + F(j,:)]

The more I think about the error, though, the less certain I am that the entire concept of what I'm trying to do is correct...

If anybody has the time to help me wit the logic of this, I'd be very grateful,

Thanks,

Paul

[Stephen23]

Note that none of those square brackets do anything, so you might as well get rid of them. Why did you add them?

By the way, here is the very simple MATLAB approach without loops:

format longG
X = 0:0.01:1;
I = (0:25).';
F = (X.^I)./factorial(I);
Y = cumsum(F,1)

Y = 26×101

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1 1.01005 1.0202 1.03045 1.0408 1.05125 1.0618 1.07245 1.0832 1.09405 1.105 1.11605 1.1272 1.13845 1.1498 1.16125 1.1728 1.18445 1.1962 1.20805 1.22 1.23205 1.2442 1.25645 1.2688 1.28125 1.2938 1.30645 1.3192 1.33205 1 1.01005016666667 1.02020133333333 1.0304545 1.04081066666667 1.05127083333333 1.061836 1.07250716666667 1.08328533333333 1.0941715 1.10516666666667 1.11627183333333 1.127488 1.13881616666667 1.15025733333333 1.1618125 1.17348266666667 1.18526883333333 1.197172 1.20919316666667 1.22133333333333 1.2335935 1.24597466666667 1.25847783333333 1.271104 1.28385416666667 1.29672933333333 1.3097305 1.32285866666667 1.33611483333333 1 1.01005016708333 1.02020134 1.03045453375 1.04081077333333 1.05127109375 1.06183654 1.07250816708333 1.08328704 1.09417423375 1.10517083333333 1.11627793375 1.12749664 1.13882806708333 1.15027334 1.16183359375 1.17350997333333 1.18530363375 1.19721574 1.20924746708333 1.2214 1.23367453375 1.24607227333333 1.25859443375 1.27124224 1.28401692708333 1.29691974 1.30995193375 1.32311477333333 1.33640953375 1 1.01005016708417 1.02020134002667 1.0304545339525 1.04081077418667 1.05127109635417 1.06183654648 1.07250818108917 1.08328706730667 1.0941742829575 1.10517091666667 1.11627806795917 1.12749684736 1.13882837649417 1.15027378818667 1.1618342265625 1.17351084714667 1.18530481696417 1.19721731464 1.20924953049917 1.22140266666667 1.2336779371675 1.24607656802667 1.25859979736917 1.27124887552 1.28402506510417 1.29692964114667 1.3099638911725 1.32312911530667 1.33642662637417 1 1.01005016708417 1.02020134002676 1.03045453395351 1.04081077419236 1.05127109637587 1.0618365465448 1.07250818125257 1.08328706767076 1.09417428369561 1.10517091805556 1.11627807041967 1.1274968515072 1.13882838319807 1.15027379864436 1.16183424238281 1.17351087044836 1.18530485048857 1.1972173618792 1.20924959584067 1.22140275555556 1.23367805628711 1.24607672549876 1.25860000297457 1.2712491409408 1.28402540418837 1.29693007019636 1.30996442925651 1.32312978459876 1.33642745251767 1 1.01005016708417 1.02020134002676 1.03045453395352 1.04081077419239 1.05127109637602 1.06183654654536 1.0725081812542 1.08328706767492 1.0941742837051 1.1051709180754 1.11627807045833 1.1274968515783 1.13882838332257 1.15027379885351 1.16183424272182 1.17351087098097 1.18530485130273 1.19721736309392 1.20924959761422 1.22140275809524 1.2336780598607 1.24607673044788 1.25860000973017 1.27124915004094 1.28402541629852 1.29693008613249 1.30996445001118 1.32312981137044 1.33642748674361 1 1.01005016708417 1.02020134002676 1.03045453395352 1.04081077419239 1.05127109637602 1.06183654654536 1.07250818125422 1.08328706767496 1.09417428370521 1.10517091807564 1.11627807045886 1.12749685157936 1.13882838332459 1.15027379885717 1.16183424272818 1.17351087099162 1.18530485132003 1.19721736312125 1.20924959765634 1.22140275815873 1.23367805995451 1.24607673058398 1.2586000099244 1.27124915031395 1.28402541667696 1.29693008665041 1.30996445071165 1.32312981230745 1.3364274879843 1 1.01005016708417 1.02020134002676 1.03045453395352 1.04081077419239 1.05127109637602 1.06183654654536 1.07250818125422 1.08328706767496 1.09417428370521 1.10517091807565 1.11627807045887 1.12749685157938 1.13882838332462 1.15027379885723 1.16183424272828 1.17351087099181 1.18530485132036 1.1972173631218 1.20924959765723 1.22140275816014 1.2336780599567 1.24607673058731 1.25860000992936 1.27124915032123 1.28402541668747 1.29693008666537 1.30996445073267 1.3231298123366 1.33642748802428

[paul]

Hi Stephen,

thanks for your help with this.

I added the square brackets, as I was of the belief that I needed them to distinguish between creating a matrix and creating a vector - in this case, i thought I needed a matrix.

I'm rather new to all the array functions matlab is capable of - my background is more low-level, assembly and C type stuff. Your solution is definitely very much more succint than my laborious attempt!

Thanks very much for your time and help on this,

Paul

Answer 1

[VBBV]

% MacLaurin Series of e^x

format longG

% define vector to hold series x values
X = 0 : 0.01 : 1;

% Create matrix to contain the results of each function in the series
% A FOR loop is used to populate the matrix without having to write out all
% the series iterations manually
for i = 0:25
    F(i+1,:) = [(X(1,:).^i)/factorial(i)];
end
size(F)

ans = 1×2

26 101

% Create increasingly better approximations by summing increasing numbers
% of function results
Y = zeros(26,101);
for i = 1:26
    for j = 1:i
        Y(i,:) = [Y(i,:) + F(j,:)];
    end
end
Y

Y = 26×101

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1 1.01005 1.0202 1.03045 1.0408 1.05125 1.0618 1.07245 1.0832 1.09405 1.105 1.11605 1.1272 1.13845 1.1498 1.16125 1.1728 1.18445 1.1962 1.20805 1.22 1.23205 1.2442 1.25645 1.2688 1.28125 1.2938 1.30645 1.3192 1.33205 1 1.01005016666667 1.02020133333333 1.0304545 1.04081066666667 1.05127083333333 1.061836 1.07250716666667 1.08328533333333 1.0941715 1.10516666666667 1.11627183333333 1.127488 1.13881616666667 1.15025733333333 1.1618125 1.17348266666667 1.18526883333333 1.197172 1.20919316666667 1.22133333333333 1.2335935 1.24597466666667 1.25847783333333 1.271104 1.28385416666667 1.29672933333333 1.3097305 1.32285866666667 1.33611483333333 1 1.01005016708333 1.02020134 1.03045453375 1.04081077333333 1.05127109375 1.06183654 1.07250816708333 1.08328704 1.09417423375 1.10517083333333 1.11627793375 1.12749664 1.13882806708333 1.15027334 1.16183359375 1.17350997333333 1.18530363375 1.19721574 1.20924746708333 1.2214 1.23367453375 1.24607227333333 1.25859443375 1.27124224 1.28401692708333 1.29691974 1.30995193375 1.32311477333333 1.33640953375 1 1.01005016708417 1.02020134002667 1.0304545339525 1.04081077418667 1.05127109635417 1.06183654648 1.07250818108917 1.08328706730667 1.0941742829575 1.10517091666667 1.11627806795917 1.12749684736 1.13882837649417 1.15027378818667 1.1618342265625 1.17351084714667 1.18530481696417 1.19721731464 1.20924953049917 1.22140266666667 1.2336779371675 1.24607656802667 1.25859979736917 1.27124887552 1.28402506510417 1.29692964114667 1.3099638911725 1.32312911530667 1.33642662637417 1 1.01005016708417 1.02020134002676 1.03045453395351 1.04081077419236 1.05127109637587 1.0618365465448 1.07250818125257 1.08328706767076 1.09417428369561 1.10517091805556 1.11627807041967 1.1274968515072 1.13882838319807 1.15027379864436 1.16183424238281 1.17351087044836 1.18530485048857 1.1972173618792 1.20924959584067 1.22140275555556 1.23367805628711 1.24607672549876 1.25860000297457 1.2712491409408 1.28402540418837 1.29693007019636 1.30996442925651 1.32312978459876 1.33642745251767 1 1.01005016708417 1.02020134002676 1.03045453395352 1.04081077419239 1.05127109637602 1.06183654654536 1.0725081812542 1.08328706767492 1.0941742837051 1.1051709180754 1.11627807045833 1.1274968515783 1.13882838332257 1.15027379885351 1.16183424272182 1.17351087098097 1.18530485130273 1.19721736309392 1.20924959761422 1.22140275809524 1.2336780598607 1.24607673044788 1.25860000973017 1.27124915004094 1.28402541629852 1.29693008613249 1.30996445001118 1.32312981137044 1.33642748674361 1 1.01005016708417 1.02020134002676 1.03045453395352 1.04081077419239 1.05127109637602 1.06183654654536 1.07250818125422 1.08328706767496 1.09417428370521 1.10517091807564 1.11627807045886 1.12749685157936 1.13882838332459 1.15027379885717 1.16183424272818 1.17351087099162 1.18530485132003 1.19721736312125 1.20924959765634 1.22140275815873 1.23367805995451 1.24607673058398 1.2586000099244 1.27124915031395 1.28402541667696 1.29693008665041 1.30996445071165 1.32312981230745 1.3364274879843 1 1.01005016708417 1.02020134002676 1.03045453395352 1.04081077419239 1.05127109637602 1.06183654654536 1.07250818125422 1.08328706767496 1.09417428370521 1.10517091807565 1.11627807045887 1.12749685157938 1.13882838332462 1.15027379885723 1.16183424272828 1.17351087099181 1.18530485132036 1.1972173631218 1.20924959765723 1.22140275816014 1.2336780599567 1.24607673058731 1.25860000992936 1.27124915032123 1.28402541668747 1.29693008666537 1.30996445073267 1.3231298123366 1.33642748802428

[VBBV]

Y = zeros(26,101);

Preallocate the size for Y before using it on RHS side, Matlab assumes everything on RHS as defined (either predefined or iteratively // successively ) values

[paul]

Thanks very much for that!

If I use the instruction:

Y = zeros(size(F));

...is that acceptable? It seems to work...

Thanks very much for your help,

Paul

[Steven Lord]

Alternately take a look at the cumsum function.

Answer 2

[John D'Errico]

You have a solution. It works. Do you need for loops, or worse, nested for loops? Of course not. The entire point of this answer is to show that you don't need to use loops to solve a problem like this. Use MATLAB as it was designed to be used.

Nmax = 10; % maximum number of terms in the expansion
Nx = 20; % number of points to be sampled in [0,1]
N = 0:Nmax; % explicitly a row vector
X = linspace(0,1,Nx)'; % see that A is explicitly a column vector
M = X.^N./factorial(0:Nmax); % % using .^ and ,/ to expand the vectors automatically
M = cumsum(M,2); % accumulation, instead of a loop
err = M - exp(X); % the error of approximation

Note that several of the lines above require R2016b or later to operate correctly, but by now, most people should have that much.

[XX,NN] = meshgrid(X,N);
surf(X,N,err')
xlabel X
ylabel N
zlabel 'error of approximation'

Next, looking at the error in terms of a logged z-axis so we can better see the behavior near zero...

surf(X,N,abs(err'));
xlabel X
ylabel 'Number of terms'
zlabel 'Absolute error'
set(gca,'zscale','log')

So as expected, the error is maximal near x==1. As x grows larger, you need more terms in the Taylor series.

Near x==0 we see almost full double precision accuracy for only or one or two terms. Again, this is expected. You need more terms as x moves away from zero.

[paul]

Hello John,

Thanks very much for your time on this and for such a complete answer - this is extremely helpful!

My knowledge of matlab syntax is unfortunately quite rudimentary - I'm picking it up as I go along. I was unsure whether looping was a good solution to this problem (coming from a programming background, it seemed intuitive), but reading through your solution, I can see looping is very certainly not the bext solution.

The cumalitive sum function is very useful, although I have to admit, understanding quite how the output is structured is a bit tricky. I shall most certainly go and have a play with this and inspect the matrices and vectors it creates - thanks very much.

Many thanks,

Paul

[Stephen23]

"coming from a programming background, it seemed intuitive"

The name MATLAB comes from MATrix LABoratory. While FOR/WHILE loops are certainly useful, the simple and intuitive way to solve many problems in MATLAB is to work with entire matrices (vectors/arrays) at once.

[John D'Errico]

Looping is A solution. In fact, it is often a good solution. As long as your code is not taking too much time to run, then all that matters is that you wrote it, and it works. Get on to solving the next problem, rather than optimize the curent one. :)

At the same time, the reason for writing this answer was to give you a target, something to shoot for. I want to teach you to write code that uses MATLAB efficiently, and that uses the capabilities of MATLAB. The goal is to get people so they automatically write code that only uses loops when that is the only solution.

[paul]

Thanks John - I'm very grateful for the time you've taken to write your answer and the explanations you've given - to my mind, your responses exemplify the purpose of this forum 👍.

Thanks,

Paul