Fourier series

Agnishom · 2014-02-22 02:34:43

Yes, that is what happened when I was on M. Why is that happening?

bobbym · 2014-02-22 02:36:45

This may be simplistic but you are obviously doing something wrong. Let's concentrate for a minute on my graph. What do you observe and how do you fix it?

Agnishom · 2014-02-22 02:39:00

I would subtract 0.3 frommy curve

Last edited by Agnishom (2014-02-22 02:40:02)

bobbym · 2014-02-22 02:40:25

That suggests that you computed a[0], the constant term wrong.

Agnishom · 2014-02-22 02:54:15

Hmm

(%i10) f(x) := if x>0 then sin(x) else 0$

(%i11) a(n) := (1/%pi)*(integrate(f(x)*cos(n*x),x,-%pi,%pi))$

(%i12) b(n) := (1/%pi)*(integrate(f(x)*sin(n*x),x,-%pi,%pi))$

(%i13) s(x,y) := a(0)/2 + sum((a(n)*cos(n*x)+b(n)*sin(n*x)),n,1,y)$

(%i14) plot2d([f(x),s(x,10)],[x,-%pi,%pi],[box, false]);

Why am I getting so many 'too many context' errors after that? Is it a bug?

bobbym · 2014-02-22 03:02:00

I do not know very much about Maxima and its error codes.

I do know that you are not computing a[0] right.

a0 = 1/(2 \[Pi]) Integrate[f[x], {x, -\[Pi], \[Pi]}]

Replace a[0] with a0.

Agnishom · 2014-02-22 03:47:25

Sorry, that is not possible right now.

f[x] is a piecewise function.

f[x_] := If [x>0, Sin[x],0]

How can M know what the following evaluates to without knowing if x is > 0?

a[0] + a[1] Cos[x] + a[2] Cos[2 x] + a[3] Cos[3 x] + 
 a[4] Cos[4 x] + a[5] Cos[5 x] + a[6] Cos[6 x] + a[7] Cos[7 x] + 
 b[0] + b[1] Sin[x] + b[2] Sin[2 x] + b[3] Sin[3 x] + b[4] Sin[4 x] + 
 b[5] Sin[5 x] + b[6] Sin[6 x] + b[7] Sin[7 x]

bobbym · 2014-02-22 03:51:35

Hi;

That is not the correct way to enter the piecewise function.

Agnishom · 2014-02-22 03:55:45

bobbym wrote:

Why didn't you tell me sooner?

bobbym · 2014-02-22 03:56:53

About the piecewise function? This was a question to you, not to me.

Agnishom · 2014-02-22 03:57:44

Hm, how does a normal M user enter a piecewise function?

bobbym · 2014-02-22 03:59:04

Piecewise[{{0, -\[Pi] <= x <= 0}, {Sin[x], \[Pi] >= x > 0}}]

Agnishom · 2014-02-22 04:00:01

How is it different from mine?

bobbym · 2014-02-22 04:12:40

1) You asked how an M programmer would do it.

2) Yours defines the function passed π, mine does not.

Agnishom · 2014-02-22 04:33:37

Agnishom wrote:

How can M know what the following evaluates to without knowing if x is > 0?

a[0] + a[1] Cos[x] + a[2] Cos[2 x] + a[3] Cos[3 x] + 
 a[4] Cos[4 x] + a[5] Cos[5 x] + a[6] Cos[6 x] + a[7] Cos[7 x] + 
 b[0] + b[1] Sin[x] + b[2] Sin[2 x] + b[3] Sin[3 x] + b[4] Sin[4 x] + 
 b[5] Sin[5 x] + b[6] Sin[6 x] + b[7] Sin[7 x]

bobbym · 2014-02-22 04:36:01

The piecewise function declares that x is sin(x) when x is between 0 and π.

Agnishom · 2014-02-22 05:12:04

Yes, so what?

bobbym · 2014-02-22 05:19:59

The formulas produce an expression that will closely approximate the given function between -π and π.

Look at the answer. Notice how closely the blue line ( Fourier series ) covers the red line ( original function ).

Agnishom · 2014-02-22 13:43:31

That is not my question

bobbym · 2014-02-22 13:57:25

I am not following you. Do you mean about the context error?

Agnishom · 2014-02-22 14:57:15

No, not that. I have now figured out the answer myself.

For now, I am integrating numerically.

(%i1)  f(x) := if x>0 then sin(x) else 0$

(%i2) a(n) := (1/%pi)*(first(quad_qags(f(x)*cos(n*x),x,-%pi,%pi)))$

(%i3) b(n) := (1/%pi)*(first(quad_qags(f(x)*sin(n*x),x,-%pi,%pi)))$

(%i4) s(x,y) := a(0)/2 + sum((a(n)*cos(n*x)+b(n)*sin(n*x)),n,1,y)$

(%i5) plot2d([f(x),s(x,10)],[x,-%pi,%pi],[box, false]);
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
 ***MESSAGE FROM ROUTINE DQAGS IN LIBRARY SLATEC.
 ***INFORMATIVE MESSAGE, PROG CONTINUES, TRACEBACK REQUESTED
 *  ABNORMAL RETURN
 *  ERROR NUMBER = 2
 *   
 ***END OF MESSAGE
 
(%o5)

Yields the attached image

The polynomial is the following

(%i6) s(x,10);
      1.3877787807814457E-16 sin(10 x)   0.02020202020202 cos(10 x)
(%o6) -------------------------------- - --------------------------
                    %pi                             %pi
   3.3219954564955856E-16 sin(9 x)   1.3877787807814457E-17 cos(9 x)
 + ------------------------------- - -------------------------------
                 %pi                               %pi
   4.163336342344337E-17 sin(8 x)   0.031746031746032 cos(8 x)
 + ------------------------------ - --------------------------
                %pi                            %pi
   3.2265856653168612E-16 sin(7 x)   5.7245874707234634E-17 cos(7 x)
 - ------------------------------- - -------------------------------
                 %pi                               %pi
   3.9091435841719085E-16 sin(6 x)   0.057142857142857 cos(6 x)
 - ------------------------------- - --------------------------
                 %pi                            %pi
   4.8572257327350599E-17 sin(5 x)   1.6653345369377348E-16 cos(5 x)
 - ------------------------------- + -------------------------------
                 %pi                               %pi
   2.0816681711721685E-16 sin(4 x)   0.13333333333333 cos(4 x)
 - ------------------------------- - -------------------------
                 %pi                            %pi
   3.2612801348363973E-16 sin(3 x)   1.9081958235744878E-17 cos(3 x)
 + ------------------------------- + -------------------------------
                 %pi                               %pi
   1.6653345369377348E-16 sin(2 x)   0.66666666666667 cos(2 x)
 - ------------------------------- - -------------------------
                 %pi                            %pi
   1.570796326794897 sin(x)   1.1102230246251565E-16 cos(x)   1.0
 + ------------------------ - ----------------------------- + ---
             %pi                           %pi                %pi

bobbym · 2014-02-22 15:04:34

There are a lot of error messages in that calculation. Also, many of those coefficients are just fungi.

But that is the correct answer. Very good.

Agnishom · 2014-02-22 15:06:08

fungi?

bobbym · 2014-02-22 15:20:32

A number that is very, very tiny so it is obviously 0.

Agnishom · 2014-02-22 15:43:36

All tiny numbers are essentially zero? Why?

There are a lot of error messages in that calculation.

Not the calculation, its just the plotting part.

Last edited by Agnishom (2014-02-22 15:50:39)

Math Is Fun Forum

#51 2014-02-22 02:34:43

Re: Fourier series

#52 2014-02-22 02:36:45

Re: Fourier series

#53 2014-02-22 02:39:00

Re: Fourier series

#54 2014-02-22 02:40:25

Re: Fourier series

#55 2014-02-22 02:54:15

Re: Fourier series

#56 2014-02-22 03:02:00

Re: Fourier series

#57 2014-02-22 03:47:25

Re: Fourier series

#58 2014-02-22 03:51:35

Re: Fourier series

#59 2014-02-22 03:55:45

Re: Fourier series

#60 2014-02-22 03:56:53

Re: Fourier series

#61 2014-02-22 03:57:44

Re: Fourier series

#62 2014-02-22 03:59:04

Re: Fourier series

#63 2014-02-22 04:00:01

Re: Fourier series

#64 2014-02-22 04:12:40

Re: Fourier series

#65 2014-02-22 04:33:37

Re: Fourier series

#66 2014-02-22 04:36:01

Re: Fourier series

#67 2014-02-22 05:12:04

Re: Fourier series

#68 2014-02-22 05:19:59

Re: Fourier series

#69 2014-02-22 13:43:31

Re: Fourier series

#70 2014-02-22 13:57:25

Re: Fourier series

#71 2014-02-22 14:57:15

Re: Fourier series

#72 2014-02-22 15:04:34

Re: Fourier series

#73 2014-02-22 15:06:08

Re: Fourier series

#74 2014-02-22 15:20:32

Re: Fourier series

#75 2014-02-22 15:43:36

Re: Fourier series

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