bimbo36

isn't this all about understanding how atoms works at its deepest levels ??

i am just a computer science student , i am not a physicist ...

i still don't know what holds an electron to its place ...

does this resemble atoms ?

but after watching a lot of documentaries on quantum physics ...

i ended up watching this documentary ...

its not bad as it sounds ...

just a bit of an update of the last post ...

i am adding one more text and some algorithm notes too , to it ...

 

Peter Selby, Steve Slavin Practical Algebra_ A Self-Teaching Guide
Tom M. Apostol Calculus, Volume I_ One-Variable Calculus, with an Introduction to Linear Algebra
Tom M. Apostol Calculus, Volume II_ Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability
Numerical Analysis - Richard L. Burden (Author), J. Douglas Faires (Author), Annette M. Burden (Author)

Numerical recipes in C -The Art of Scientific Computing
Computer oriented numerical methods - N Datta
Computer Fundamentals and Programming in C - J.B. Dixit

 

An analytical solution to a problem is one that has a "proof": a series of logical steps that can be followed and verified as correct. If you use the quadratic formula to solve for x in a quadratic equation, this is an analytical solution to the problem.

In contrast to this, some problems are solved via other means. If, instead of using the quadratic formula, you try a lot of values for x and y, this is a numerical solution. Some problems simply do not have analytical solutions and must be approximated using numerical methods -- for example, many complicated integrals.

for example ...

this is an instantaneous rate of change , find the function ...

and the answer is a function that has the property of ,this instantaneous rate of change ...

example 2 :

this is an instantaneous rate of change ...

find the function ,the solution , the answer which is a set of points that has certain underlying qualitative property which produced, this instantaneous rate of change ...

 

Say we were to solve the initial value problem:

y' = 2x

y(0) = 0

It's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (Yes, I know we don't know how to do this yet, but go with me on this for a second!) The resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.


Using Numerical Solutions

So what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as Mathematica, do do with these solutions. (Mathematica produces a join-the-dots function that it calls InterpolatingFunction.)

When actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: to find the values of the solution at specific
points. This is exactly what we can still do with a numerical solution

 

Solving an ordinary differential equation (ODE) or initial value problem means finding a clear expression for y in terms of a finite number of elementary functions of x. Euler’s method is one of the simplest method for the numerical solution of such equation or problem. This C program for Euler’s method considers an ordinary differential equations, and the initial values of x and y are known.

Mathematically, here, the curve of solution is approximated by a sequence of short lines i.e. by the tangent line in each interval. (Derivation) Using these information, the value of the value of ‘y_n’ corresponding to the value of ‘x_n‘ is to determined by dividing the length (x_n – x) into n strips.
Therefore, strip width= (x_n – x)/n and x_n=x₀+ nh. Again, if m be the slope of the curve at point, y₁₌ y_{0 + m}(x₀ , y_o)h.
Similarly, values of all the intermediate y can be found out.

Below is a source code forEuler’s method in C to solve the ordinary differential equation dy/dx = x+y. It asks for the value of of x_{0 ,} y_{0 ,}x_n and h. The value of slope at different points is calculated using the function ‘fun’.
The values of y are calculated in while loop which runs till the initial value of x is not equal to the final value. All the values of ‘y’ at corresponding ‘x’ are shown in the output screen. dy/dx = x+y

#include<stdio.h>
#include<math.h>
  
  
main()
{
                            
      float x;                                                                  /*defining variables*/
      float y;
      float h;
      float targetx;
        
       
        
      puts("This program will solve the differential equation y' = y - x \nusing Euler's Method with y(0)=1/2 \n\n");
      puts("Please enter the desired constant step size. (h-value)\n\n");
      scanf("%f", &h);                                                           /* Defining step size*/
      puts("\n\nNow enter the desired x-value to solve for y.\n\n");
      scanf("%f", &targetx);
        
      y = 0.5;
      x = 0.0;
        
      puts("\n\nX                Y");
        
      while ( x != targetx )
      {
        
      printf("\n\n%f     %f", x, y);
        
      y = y + ((y - x)*h);
        
      x= x+h;
      }
         
      printf("\n\n%f     %f\n", x, y);
        
      printf("\nThe value of y at the given x is %f.\n\n", y, h);
             
      system("pause");
        
}

here the solution is a set of points or a table of values that has certain underlying qualitative property which produced, this instantaneous rate of change ...

??

blue89 ,

no problems ...i hope you will find what you are looking for ...

good luck ...

blue89 ,

have you used Gnuplot ?

i have not tried this program before .. and i don't even know if this is the sort of application you are looking for ...

anyway i was a bit curious ... so i followed this tutorial here ...

http://people.duke.edu/~hpgavin/gnuplot.html

does this help ??

i am still confused about ceratin things ...

i have few more doubts ...

i was trying to learn "computer oriented numerical methods in c programming language "

we had a messed up syllabus ...

it was supposed to start with a program for a polynomial factorization first ... but i cannot even find one example program of a polynomial factorization in c in any texts or online...

so i have been trying to re arrange the whole syllabus , so that it becomes something nice to look at and easy to learn ...

i have been following at least this much amount of books to narrow it down ...

 

Peter Selby, Steve Slavin Practical Algebra_ A Self-Teaching Guide
Tom M. Apostol Calculus, Volume I_ One-Variable Calculus, with an Introduction to Linear Algebra
Tom M. Apostol Calculus, Volume II_ Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability
Numerical Analysis - Richard L. Burden (Author), J. Douglas Faires (Author), Annette M. Burden (Author)
Computer oriented numerical methods - N Datta
Computer Fundamentals and Programming in C - J.B. Dixit

 

An analytical solution to a problem is one that has a "proof": a series of logical steps that can be followed and verified as correct. If you use the quadratic formula to solve for x in a quadratic equation, this is an analytical solution to the problem.

In contrast to this, some problems are solved via other means. If, instead of using the quadratic formula, you try a lot of values for x and y, this is a numerical solution. Some problems simply do not have analytical solutions and must be approximated using numerical methods -- for example, many complicated integrals.

for example ...

this is an instantaneous rate of change , find the function ...

and the answer is a function that has the property of ,this instantaneous rate of change ...

example 2 :

this is an instantaneous rate of change ...

find the function ,the solution , the answer which is a set of points that has certain underlying qualitative property which produced, this instantaneous rate of change ...

 

Say we were to solve the initial value problem:

y' = 2x

y(0) = 0

It's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (Yes, I know we don't know how to do this yet, but go with me on this for a second!) The resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.


Using Numerical Solutions

So what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as Mathematica, do do with these solutions. (Mathematica produces a join-the-dots function that it calls InterpolatingFunction.)

When actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: to find the values of the solution at specific
points. This is exactly what we can still do with a numerical solution

#include<stdio.h>
#include<math.h>
  
  
main()
{
                            
      float x;                                                                  /*defining variables*/
      float y;
      float h;
      float targetx;
        
       
        
      puts("This program will solve the differential equation y' = y - x \nusing Euler's Method with y(0)=1/2 \n\n");
      puts("Please enter the desired constant step size. (h-value)\n\n");
      scanf("%f", &h);                                                           /* Defining step size*/
      puts("\n\nNow enter the desired x-value to solve for y.\n\n");
      scanf("%f", &targetx);
        
      y = 0.5;
      x = 0.0;
        
      puts("\n\nX                Y");
        
      while ( x != targetx )
      {
        
      printf("\n\n%f     %f", x, y);
        
      y = y + ((y - x)*h);
        
      x= x+h;
      }
         
      printf("\n\n%f     %f\n", x, y);
        
      printf("\nThe value of y at the given x is %f.\n\n", y, h);
             
      system("pause");
        
}

here the solution is a set of points or a table of values that has certain underlying qualitative property which produced, this instantaneous rate of change ...

??

blue89,

thanks a lot for all that helpful posts ...

i have been trying to follow this book ..for sometime now ...

i have managed it this far ...

#include<stdio.h>
#include<conio.h>
#include<math.h>
void main()
{
float x[10],y[10],temp=1,f[10],sum,p;
int i,n,j,k=0,c;
clrscr();
printf("\nhow many record you will be enter: ");
scanf("%d",&n);
for(i=0; i<n; i++)
{
printf("\n\nenter the value of x%d: ",i);
scanf("%f",&x[i]);
printf("\n\nenter the value of f(x%d): ",i);
scanf("%f",&y[i]);
}
printf("\n\nEnter X for finding f(x): ");
scanf("%f",&p);

for(i=0;i<n;i++)
{
temp = 1;
k = i;
for(j=0;j<n;j++)
{
if(k==j)
{
continue;
}
else
{
temp = temp * ((p-x[j])/(x[k]-x[j]));
}
}
f[i]=y[i]*temp;
}

for(i=0;i<n;i++)
{
sum = sum + f[i];
}
printf("\n\n f(%.1f) = %f ",p,sum);
getch();
}

/*
______________________________________

OUT PUT
______________________________________

how many record you will be enter: 4


enter the value of x0: 0


enter the value of f(x0): 0


enter the value of x1: 1


enter the value of f(x1): 2


enter the value of x2: 2


enter the value of f(x2): 8


enter the value of x3: 3


enter the value of f(x3): 27


Enter X for finding f(x): 2.5


f(2.5) = 15.312500


*/

#include<stdio.h>
#include<math.h>
  
  
main()
{
                            
      float x;                                                                  /*defining variables*/
      float y;
      float h;
      float targetx;
        
       
        
      puts("This program will solve the differential equation y' = y - x \nusing Euler's Method with y(0)=1/2 \n\n");
      puts("Please enter the desired constant step size. (h-value)\n\n");
      scanf("%f", &h);                                                           /* Defining step size*/
      puts("\n\nNow enter the desired x-value to solve for y.\n\n");
      scanf("%f", &targetx);
        
      y = 0.5;
      x = 0.0;
        
      puts("\n\nX                Y");
        
      while ( x != targetx )
      {
        
      printf("\n\n%f     %f", x, y);
        
      y = y + ((y - x)*h);
        
      x= x+h;
      }
         
      printf("\n\n%f     %f\n", x, y);
        
      printf("\nThe value of y at the given x is %f.\n\n", y, h);
             
      system("pause");
        
}

i think these root finding algorithms for polynomial equations does not involve factorization ...

i think i should find more examples of equations to solve by numerical techniques involving polynomials and differential equations ...

??

i might be able to program all those 7 programs properly in Dev c++ ...

 

#include<stdio.h>
#include<conio.h>
#include<math.h>
void main()
{
float x[10],y[10],temp=1,f[10],sum,p;
int i,n,j,k=0,c;
clrscr();
printf("\nhow many record you will be enter: ");
scanf("%d",&n);
for(i=0; i<n; i++)
{
printf("\n\nenter the value of x%d: ",i);
scanf("%f",&x);
printf("\n\nenter the value of f(x%d): ",i);
scanf("%f",&y);
}
printf("\n\nEnter X for finding f(x): ");
scanf("%f",&p);

for(i=0;i<n;i++)
{
temp = 1;
k = i;
for(j=0;j<n;j++)
{
if(k==j)
{
continue;
}
else
{
temp = temp * ((p-x[j])/(x[k]-x[j]));
}
}
f=y*temp;
}

for(i=0;i<n;i++)
{
sum = sum + f;
}
printf("\n\n f(%.1f) = %f ",p,sum);
getch();
}

 

/*
______________________________________

OUT PUT
______________________________________

how many record you will be enter: 4


enter the value of x0: 0


enter the value of f(x0): 0


enter the value of x1: 1


enter the value of f(x1): 2


enter the value of x2: 2


enter the value of f(x2): 8


enter the value of x3: 3


enter the value of f(x3): 27


Enter X for finding f(x): 2.5


f(2.5) = 15.312500


*/

i have few more doubts ...

in certain equations why do we change the unit of x to delta x , and the unit of y to delta y ??

blue89,

i have been trying hard to keep it as appropriate as possible ...

using c language isn't exactly that easy either ...

thanks for the replies ...

blue89,

this thread is a bit of a mess ...

i was trying to narrow it down with the help of some pictures ...

we had this subject called "computer oriented numerical methods in c programming language " in the second semester ...

that was the only maths related subject we had ...

the rest was 12 programming languages ...

the "computer oriented numerical methods in c programming language " looked a bit like this ...

 

Numerical Methods and errors

Interpolation

Numerical Differentiation

Numerical Integration

Solution of Algebraic and Transcendental Equations

Numerical Solution of a system of Linear Equations

Numerical Solution of Ordinary differential equations

Curve fitting

Numerical Solution of problems associated with Partial Differential Equations

 

An Example

Say we were to solve the initial value problem:

y′ = 2x

y(0) = 0

It's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (Yes, I know we don't know how to do this yet, but go with me on this for a second!) The resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.


Using Numerical Solutions

So what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as Mathematica, do do with these solutions. (Mathematica produces a join-the-dots function that it calls InterpolatingFunction.)

When actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: to find the values of the solution at specific points. This is exactly what we can still do with a numerical solution

terms like "Solution of equations by iterative algorithms" were very confusing in the beginning ... now i am a little bit more familiar with all these terms ...

thanks a lot for the replies ...

programming is a lot more harder than that ...

on a usual case ...

we are given problems like these only...

 

An Example

Say we were to solve the initial value problem:

y′ = 2x

y(0) = 0

It's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (Yes, I know we don't know how to do this yet, but go with me on this for a second!) The resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.


Using Numerical Solutions

So what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as Mathematica, do do with these solutions. (Mathematica produces a join-the-dots function that it calls InterpolatingFunction.)

When actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: to find the values of the solution at specific points. This is exactly what we can still do with a numerical solution

i think it would be a good idea to focus on those 7 programs instead of going after the things outside the syllabus ...

thanks to you ..

from now on i can focus a bit more on programming ... rather than spent all my time in front of mathematics ...

sorry for the long posts ...

i was able to narrow it down to these ...

An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation

they are mainly classified into two ..
ordinary differential equation
partial differential equation ..

then comes first order differential equations to nth order differential equations ...

order is the highest number of the differentiations appearing
degree is the power of the highest order derivative in the equation ...

then there are types of differential equations , depending on their order

separable equations
homogeneous equations
linear equations
exact equations

A partial differential equation is an equation involving functions and their partial derivatives ...

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant

i was very confused when i began studying this ...

because i was mostly aiming for the numerical methods ...

blue89 ,

Thanks a lot for all the explanations ...

blue89 ,

thanks a lot for the replies ...

your maths looks a bit too advanced to me ...

i am a bsc computer science student ... and we had this "computer oriented numerical methods in c language " in college ...

our practical lab , involved solving 7 questions with numerical methods...

at that time , i was not aware of the depth of this subject ...

since then , i have been trying to understand this ...

my understanding so far is , that we are dealing with .... iterations of the "instantaneous rate of change " in the question ...

 

 

 

1-->> "I wonder whether iteration's properties affects the results" or ""...how it affect the results."

at this point of time , i am not exactly sure how the algorithms , or the iteration properties affects the results ...

finding delta y / delta x for an instantaneous rate of change is like already a step for approximating or finding function values ...

i guess the euler's method is one such algorithm to take smaller intervals of the instantaneous rate of change in the question ...

i have learned from your post that , the smaller the interval .. the much clearer the properties of the function we are trying to find ...

??

i like this sentence actually ...

could you determine that which expression is the best suitable for your question below..

 

1-->> "I wonder whether iteration's properties affects the results" or ""...how it affect the results."

 

 

is iteration like another word for numerical methods ??

and we are trying to iterate the instantaneous rate of change ... in the question right ?

and the Euler's method is one such iteration for ... the instantaneous rate of change in the question ...

after these iterations with the help of these algorithms ...

we get function values ...

then if you increment x ... you get properties of that function again at certain points or places

if you add all the function values ... you get the overall properties of a function ...

and that is the function with all those properties that has given that instantaneous rate of change ...

??

i don't know if i am understanding this properly ...

this thread , post # 8 ...

picture one ..

that question is about finding the "function" that has that instantaneous rate of change ... right ??

we found the function ... that has that rate of change ...

same thread picture two ,

we are given a function and we are trying to solve it numerically ...

if you take dy/dx of that instantaneous rate of change you get certain properties of that function at certain points or places ...

then if you increment x ... you get properties of that function again at certain points or places

if you add all the function values ... you get the overall properties of a function ...

and that is the function with all those properties that has given that instantaneous rate of change ... ??

??

unfortunately i only understand this part clearly ...

i don't understand the importance of function values or what they actually mean ...

??

i have been trying to follow at least this much amount of books ...

 

Peter Selby, Steve Slavin Practical Algebra_ A Self-Teaching Guide

Tom M. Apostol Calculus, Volume I_ One-Variable Calculus, with an Introduction to Linear Algebra

Tom M. Apostol Calculus, Volume II_ Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability
Numerical Analysis by Richard L. Burden (Author), J.Douglas Faires (Author), Annette M. Burden
Computer oriented numerical methods - N Datta

Computer Fundamentals and Programming in C - J.B. Dixit

i have few doubts ...

how to find an initial guess ... ??

 

Nonlinear optimization routines enable you to find the values of variables that optimize an objective function of those variables. When you use a numerical optimization routine, you need to provide an initial guess, often called a "starting point" for the algorithm. Optimization routines iteratively improve the initial guess in an attempt to converge to an optimal solution. Consequently, the choice of a starting point determines how quickly the algorithm converges to a solution and—for functions with multiple local extrema—to which optimum the algorithm converges.

But how can you produce a "good" starting point for multivariate functions? Some analysts use a constant vector, such as a vector of all 0s or a vector of all 1s. Although this practice is widespread, this default initial guess might not converge to a global extremum. You can usually do better by choosing a starting point that is based on a few easy evaluations of the objective function.

i am also trying to understand the steps involved , in these two questions ...

first we are given a gradient of the curve or an instantaneous rate of change

then we are trying to find a function that is giving this gradient of the curve or this instantaneous rate of change

then we change the dy/dx , the gradient of the curve or an instantaneous rate of change to ... delta y /delta x which is an even smaller gradient of the curve or a smaller instantaneous rate of change ...

we start with x = 0 , we get y =0

when we increment x by 0.1 , we get the value of y around -0.299

which is another gradient of the curve for delta y / delta x , the smaller instantaneous rate of change

why are we adding up these gradients of the curve or the smaller instantaneous rate of change for these small small change in delta y / delta x?

this means that these smaller gradients of the curve or these smaller instantaneous rate of change has certain property at each value of x ..., the gradient of the curve with respect to x which is indeed "the rate" at which y is changing as x is changing ....

so the function has the property of this "rate" ???

usually , when we try to solve ...

we get a function that has this instantaneous rate of change ...

but when we try to solve it with numerical methods , we only get a " rate " which is the property of that function ???

i am sorry for the long posts ... as i learn more , the size of the post seems to be getting smaller ...

i really need to understand how these works ...



lagrange's interpolation formula
newton raphson method
bisection method
eulers method
newtons forward
runge kutta method
trapezoidal rule


and i dont have one text that explains it all ...

 

A numerical iteration method or simply iteration method is a mathematical

procedure that generates a sequence of improving approximate solutions for a class of

problems. A specific way of implementation of an iteration method, including the

termination criteria, is called an algorithm of the iteration method. In the problems of

finding the solution of an equation an iteration method uses an initial guess to generate

successive approximations to the solution.

i am so confused ...

what is this initial guess ??

the initial guess of what ??

 

#include<stdio.h>

#include<math.h>

main()

{

float x; /*defining variables*/

float y;

float h;

float targetx;

puts("This program will solve the differential equation y' = y - x \nusing Euler's Method with y(0)=1/2 \n\n");

puts("Please enter the desired constant step size. (h-value)\n\n");

scanf("%f", &h); /* Defining step size*/

puts("\n\nNow enter the desired x-value to solve for y.\n\n");

scanf("%f", &targetx);

y = 0.5;

x = 0.0;

puts("\n\nX Y");

while ( x != targetx )

{

printf("\n\n%f %f", x, y);

y = y + ((y - x)*h);

x= x+h;

}

printf("\n\n%f %f\n", x, y);

printf("\nThe value of y at the given x is %f.\n\n", y, h);

system("pause");

}

??

like this ??

i am a computer science student ... not an expert programmer ... this is not an easy thing to do ...

all those things requires a lot of programming...

i don't know if these pictures might help you ...

that was my own attempt to program something useful , with html , php , mysql , javascrip and ajax ...

i had to play around with some scripts i downloaded from the internet ...

the first picture is supposed to be the client side ... and the last picture is supposed to be a linux machine where you host your files...

 

An Example

 

Say we were to solve the initial value problem:

 

y′ = 2x

 

y(0) = 0

 

It's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (Yes, I know we don't know how to do this yet, but go with me on this for a second!) The resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

 

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.

 

 

Using Numerical Solutions

 

So what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

 

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

 

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as Mathematica, do do with these solutions. (Mathematica produces a join-the-dots function that it calls InterpolatingFunction.)

 

When actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: to find the values of the solution at specific points. This is exactly what we can still do with a numerical solution

 

Numerical Methods and errors

Interpolation

Numerical Differentiation

Numerical Integration

Solution of Algebraic and Transcendental Equations

Numerical Solution of a system of Linear Equations

Numerical Solution of Ordinary differential equations

Curve fitting

Numerical Solution of problems associated with Partial Differential Equations

 

Fixed Point Iteration Method

Bisection and Regula False Methods

Newton Raphson Method etc.

Finite Differences Operators

Numerical Interpolation

Newton’s and Lagrangian Formulae

Part I

Newton’s and Lagrangian Formulae

Part II

Interpolation by Iteration

Numerical Differentiaton

Numerical Integration

Solution of System of Linear

Equations

Solution by Iterations

Eigen Values

Taylor Series Method

Picard’s Iteration Method

Euler Methods

Runge – Kutta Methods

Predictor and Corrector Methods

my questions sort of looks a bit like this ...

??

i am sorry .. this is seriously one last time ... i collected notes like this from many different places and it took a lot of time to arrange it like this ...

can i please keep it for one last time ??

i wont repeat this ...

from next time onward i promise i would keep it short and neat ...

let me try to refresh this thread with few more things ...

i am trying to put all the stuff from my syllabus in this thread , so that i don't have to look anywhere else ...

my syllabus was called , "computer oriented numerical methods in c programming language "

it had lot of mathematics ...

it was a bit of a mess ...

we really lacked proper texts and materials to really understand that subject ....

none of the texts i have read had the proper mathematics or the examples of numerical related programming in c languages properly ...

i was trying to follow these two instead ...

for some understanding of the overall mathematics , before trying to do programming ...

http://www.universityofcalicut.info/SDE/BSc_maths_numerical_methods.pdf

for programming some maths related things , i would recommend this book ...

Computer Fundamentals and Programming in C - J.B. Dixit

even those books are still lacking many needed information's ...

which is why i am using pictures nowadays to learn it properly ...

let me arrange some things properly in this thread ...

so you too know , how it looked like .. and where it is heading to ...


well it begins with something like this ...

 

mathematical expressions
equations in one variable
equations in two variables
system of 2 equations containing 2 variables
functions in one variable
functions in two variables

differential equations

first order differential equations
second order differential equations
higher order differential equations ...

linear differential equations
separable differential equations
exact differential equations
homogeneous differential equations
non homogeneous differential equations
using the method of undetermined coefficients ...



partial differential equations ...

http://www.mathsisfun.com/algebra/systems-linear-equations.html

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

http://www.sosmath.com/soe/SE211105/SE211105.html

 

A linear equation is always a polynomial of degree 1 (for example x+2y+3=0). In the two dimensional case, they always form lines; in other dimensions, they might also form planes, points, or hyperplanes. Their "shape" is always perfectly straight, with no curves of any kind. This is why we call them linear equations.

Every other equation is nonlinear. Higher degree polynomials are nonlinear. Trigonometric functions (like sin or cos) are nonlinear. Square roots are nonlinear. The main exception is if the nonlinear piece can evaluate to a constant--for example, sqrt(4)*x is linear because sqrt(4) is just 2, and 2x is linear.

Linear equations have some useful properties, mostly in that they are very easy to manipulate and solve. Although they are quite limited in what they can represent, it is often useful to try and approximate complicated systems using linear equations so that they will be easier to think about and deal with.

Nonlinear equations, for the most part, are much harder to solve and manipulate. Sometimes you need them--nature doesn't always work in straight lines, and nor do mathematicians--but generally speaking, you can only solve nonlinear equations if the systems are fairly small and simple. Solving a linear system with a million interacting variables is very doable with a computer, and most nonlinear solvers aren't going to get even close to that

 

The (standard) calculus is broken into two pieces.

i) Differential calculus - which is looking at the instantaneous rates of change of objects with respect to some variables. We have the notion of the derivative of a function.
ii) Integral calculus - which is calculating the area under curves, calculating volumes and so on. This is all given in terms if the (indefinite or definite) integral of a function.

The two notions are tied together via the fundamental theorem of calculus. This says that the derivative and indefinite integral are basically mutual inverses (but not quite).

Differential equations ...

 

An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation

 

Numerical Methods and errors
Interpolation
Numerical Differentiation
Numerical Integration
Solution of Algebraic and Transcendental Equations
Numerical Solution of a system of Linear Equations
Numerical Solution of Ordinary differential equations
Curve fitting
Numerical Solution of problems associated with Partial Differential Equations

 

Solution of Algebraic and Transcendental Equation
2.1 Introduction
2.2 Bisection Method
2.3 Method of false position
2.4 Iteration method
2.5 Newton-Raphson Method
2.6 Ramanujan's method
2.7 The Secant Method Finite Differences


3.1 Introduction
3.3.1 Forward differences
3.3.2 Backward differences
3.3.3 Central differences
3.3.4 Symbolic relations and separation of symbols
3.5 Differences of a polynomial Interpolation
3.6 Newton's formulae for intrapolation
3.7 Central difference interpolation formulae
3.7.1 Gauss' Central Difference Formulae
3.9 Interpolation with unevenly spaced points
3.9.1 Langrange's interpolation formula
3.10 Divided differences and their properties
3.10.1 Newton's General interpolation formula
3.11 Inverse interpolation Numerical Differentiation and Integration


5.1 Introduction
5.2 Numerical differentiation (using Newton's forward and backward formulae)
5.4 Numerical Integration
5.4.1 Trapizaoidal Rule
5.4.2 Simpson's 1/3-Rule
5.4.3 Simpson's 3/8-Rule Matrices and Linear Systems of equations
6.3 Solution of Linear Systems – Direct Methods
6.3.2 Gauss elimination
6.3.3 Gauss-Jordan Method
6.3.4 Modification of Gauss method to compute the inverse
6.3.6 LU Decomposition
6.3.7 LU Decomposition from Gauss elimination
6.4 Solution of Linear Systems – Iterative methods
6.5 The eigen value problem
6.5.1 Eigen values of Symmetric Tridiazonal matrix Numerical Solutions of Ordinary Differential Equations


7.1 Introduction
7.2 Solution by Taylor's series
7.3 Picard's method of successive approximations
7.4 Euler's method
7.4.2 Modified Euler's Method
7.5 Runge-Kutta method
7.6 Predictor-Corrector Methods
7.6.1 Adams-Moulton Method
7.6.2 Milne's method

 

for example , for a polynomial ... a solution of a polynomial equation is also called a root of the polynomial ... a value for the variable that makes the polynomial zero if you can't find an exact expression, then you can use numerical methods to get approximations ... with numerical methods you can choose how close to zero you want, and it will give you a value that's at least that close ...

at this point of time i am not sure how the alphabets , variables , arrays ... etc ...
f(x) ... theta , d^2y/dx^2 , (dy/dx)^3 , integral symbol... are declared ...

again if you use the numerical methods or the appropriate algorithm for the differential equations properly ...

you should be able to find the solution of the problem with a programming language like c ..

my question is , what does numerical method achieve for each one these types of numerical method questions??

because the questions ranges from ...

polynomials

trigonometric

transcendental

differentiation

integration

differential equations

i know for a polynomial ...

the numerical method means ...

 

for example , for a polynomial ... a solution of a polynomial equation is also called a root of the polynomial ... a value for the variable that makes the polynomial zero if you can't find an exact expression, then you can use numerical methods to get approximations ... with numerical methods you can choose how close to zero you want, and it will give you a value that's at least that close ...

for differentiation and differential equations ...

its about the instantaneous rate of change of the variable x with respect to y

then what happens , when you apply numerical methods on their solutions ??

what does this approximate values represent ??

again is it about the instantaneous rate of change of the variable x with respect to y ...

is it always about how close to zero you want the solutions to be ??