Computational Lab in Physics:
Doing calculations in
C++ and using ROOT.
Derivatives
Steven Kornreich
www.beachlook.com
ROOT: http://root.cern.ch/
Using ROOT

Setting up ROOT in the Rm 106 linux
cluster
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By default, root is not in your path.
To add, add this line to your .bashrc:
. /usr/local/root/bin/thisroot.sh
You should have the following environment
variables:
ROOTSYS=/usr/local/root
 PATH should contain $ROOTSYS/bin

2
ROOT commands


Starting root, just type “root”
At the root prompt:
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.q = Exit from root
.ls = list the files loaded into root session
.! some-unix-command = execute some-unixcommand in the shell
Most c++ commands can also be
interpreted.
Executing a macro “myMacro.C”:

.x myMacro.C
3
ROOT Classes

Since it is C++, everything is represented by
classes:
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Windows (or canvases) : TCanvas
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Functions : TF1, TF2, TF3
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Class used to plot data on a canvas
Histograms: TH1, TH2, TH3
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Classes to manipulate mathematical functions, such
as sin(x), in order to draw, evaluate, and integrate
them.
Graphs : TGraph

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A window where we can draw data, functions, etc.
Classes to manipulate histograms. Can draw them
on a canvas, integrate them, obtain means and RMS
values, evaluate bin contents.
Tutorials (lots of code to try out ROOT):

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$ROOTSYS/tutorials/
For example: /hsimple.C, and /hist/h1draw.C
4
Derivatives. (Klein/Godunov Text, Ch. 5)
Forward difference. (5.2)




df
 f ( x  x)  f ( x) 
 lim 


x

0
dx
x


A derivative is defined as a limit.
Numerically, we need to use finite quantities
Note: needs to us small numbers in the computer.
 But watch out! Computer has finite representation!
 Must avoid machine precision problems.
We can define the Discrete Forward-difference operator:
 f ( x  x)  f ( x) 
D (f)

x



x


Introduce an error, typcially varies a polynomial in x.
O(xN)
5
Derivative:
What is the order of the error?

Use Taylor expansion.
df ( x)  h  d 2 f ( x)
f ( x  h)  f ( x )  h

 ...
2
dx
2! dx
Which implies:
df ( x)
 Dx ( f )  O(h)
dx


2
The error on the derivative will be of the
same order as the step size x.
This is akin to approximating the function
as a straight line between two x-values. 6
Homework, First part:
Example: Calculate d sin(x)/dx

Write a program that will do:




Print a statement that will say that the
program will calculate the derivative of
sin(x)
Ask us for the value of x
Calculate the derivative, using the finite
forward difference
Print it to the screen.

Note: Hwk: Chapter 5 Problem 1: Asks you
to do this, plus a similar program for cos(x).
7
5.3: Central difference and higher
order methods.


Improvement on the error behavior.
Use two versions of the Taylor expansion
ℎ ′
ℎ2 ′′
ℎ3 ′′′
𝑓 𝑥+ℎ =𝑓 𝑥 + 𝑓 𝑥 + 𝑓 𝑥 + 𝑓 𝑥 +⋯
1!
2!
3!
2
ℎ ′
ℎ ′′
ℎ3 ′′′
𝑓 𝑥−ℎ =𝑓 𝑥 − 𝑓 𝑥 + 𝑓 𝑥 − 𝑓 𝑥 +⋯
1!
2!
3!


Solve for f’(x) but first, subtract bottom
from top equation. What do we get?
𝑓 𝑥+ℎ −𝑓 𝑥−ℎ =
2ℎ𝑓 ′
𝑥
2ℎ 3 ′′′
+
𝑓
3!
𝑥 + odd terms
8
Central difference, and beyooond!

′
𝑓 𝑥 =



+ 𝒪(ℎ2 )
Key: Size of error we make is of order h2.
Improvement over forward difference
Can we keep going?


𝑓 𝑥+ℎ −𝑓(𝑥−ℎ)
2ℎ
Can we get to accuracy 𝒪 ℎ4 ? Yes!
1
𝑓 ′ 𝑥 = 12ℎ 𝑓 𝑥 − 2ℎ − 8𝑓 𝑥 − ℎ + 8𝑓 𝑥 + ℎ − 𝑓 𝑥 + 2ℎ
+ 𝒪(ℎ4 )
9
Homework, part I

Klein/Godunov Ch. 5

Problem 1 (25 points): Write a program to
calculate the derivatives of sin(x) and
cos(x).
Ask for value of x, calculate the derivative at
that value, print derivative.
 Use forward difference. h=10-5.


Problem 2 (25 points)
Repeat problem 1, use central difference.
 Can you verify that the error has improved?
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Check derivative of sin at x=1.57079632679
What do you get the two cases? (Comment on this
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in your code)
Plotting the derivatives, use ROOT
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Homework part II, Problem 3 (25 points)
Use your programs to plot values of the
derivatives
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Simple way:
Calculate them between 0 – 2p, make 1000
points for each.
Store these points as 2 arrays of x and y
values.
Make a TGraph to draw these x and y values

See the TGraph example in the ROOT page.
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TGraph Example, from ROOT web
{
TCanvas *c1 = new TCanvas("c1","A Simple Graph
Example",200,10,700,500);
Double_t x[100], y[100];
Int_t n = 20;
for (Int_t i=0;i<n;i++) {
x[i] = i*0.1;
y[i] = 10*sin(x[i]+0.2);
}
gr = new TGraph(n,x,y);
gr->Draw("AC*");
return c1;
}
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Graph Draw Options
The various draw options for a graph are explained in
TGraph::PaintGraph. They are:
• "L" A simple poly-line between every points is drawn
• "F" A fill area is drawn
• “F1” Idem as "F" but fill area is no more repartee around X=0 or Y=0
• "F2" draw a fill area poly line connecting the center of bins
• "A" Axis are drawn around the graph
• "C" A smooth curve is drawn
• "*" A star is plotted at each point
• "P" The current marker of the graph is plotted at each point
• "B" A bar chart is drawn at each point
• "[]" Only the end vertical/horizontal lines of the error bars are drawn.
This option only
applies to the TGraphAsymmErrors.
• "1" ylow = rwymin
The options are not case sensitive and they can be concatenated in most
cases. Let us look at some examples
13
Homework, part III: Derivative
Operator

Write a program which will
calculate the first and second
derivatives for any function you
give
See Sec 5.4 for higher order
derivatives.
 Use central difference derivative
 To be specific,


make an example which has


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a linear function f(x)=x
a cubic function, f(x)=x3
Show that you can feed them into the same
derivative operator (one for first, one for
second), and obtain the derivatives.
14
Example code: derivative operator.
#include <cmath>
#include <iostream>
using namespace std;
double cube (double aD) {
return pow(aD,3);
}
double linear(double aD) {
return pow(aD,1);
}
int main() {
double del = 1.0e-1;
int choice;
cout << "Choose a function:" << endl;
cout << "1) cube " << endl;
cout << "2) linear " << endl;
cin >> choice;
switch (choice) {
case 1:
cout << derivOperator(cube,1.0,del) <<
endl;
break;
case 2:
cout << derivOperator(linear,1.0,del) <<
endl;
break;
default:
cout << "Incorrect input, exiting." <<
endl;
}
return 0;
}
15
Additional Material
16
Plotting simple functions in ROOT

Using TF1


write explicitly the function in the 2nd
argument
E.g. Aebt + Ce-dt :
TF1* myFunc = new
TF1(“myFunc",”[0]*exp([1]*x)+[2]*exp(-[3]*x)”,0,10);
Here:
 [0] is parameter A, [1] is b, etc.
 The parameters are set via, (e.g. if A=1):

myFunc->SetParameter(0,1);
17
Plotting a user defined function in
ROOT
double mysine(double* x, double* par) {
double Amplitude = par[0];
double wavelength = par[1];
double phase = par[2];
return Amplitude*sin(2*TMath::Pi()/wavelength*x[0]+phase);
}
void plotsine() {
TCanvas* sineCanvas = new TCanvas("sineCanvas","A*sin(2pi/lambda*x +
phi)",500,500);
}
TF1* sineFunc = new TF1("sineFunc",&mysine,0,2*TMath::Pi(),3);
sineFunc->SetParameters(2,TMath::Pi(),TMath::Pi()/2);
sineFunc->Draw();
return;
18
Resources for ROOT
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ROOT Web page:
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http://root.cern.ch/
User guides

http://root.cern.ch/root/doc/RootDoc.html
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Error dependence
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Note: the “approximation” is
actually exact for the linear case.

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Error estimation for linear case is
misleading.
When del is small, rounding errors
degrade the accuracy.

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double variables can do ~14 sig. figs.
what happens if del=1.0e-17?
20