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Numerical Analysis and Simulation
The study of algorithms for problems related to continuous mathematics
818 Questions
What is an example of an enumerated item?
The word enumerated refers to counting. The most counted in countries are people, along with statistics about them. Census population is the #1 example of enumeration.
What is the largest number on Earth?
There is no largest number. One simple way to consider this is to look at the function:
f(x) = x + 1
Pick any number, and plug it into that function. No matter what number you plug in, the function will always give you a number that is one larger than the previous one. It doesn't matter how big "x" is, there will always be an "x + 1" value, which will always be larger than x. This is a concept we refer to as infinity, which is usually represented with the symbol ∞. It is important to note that infinity is not in itself a number, but the concept of a set of numbers that has no limit.
There is however a large finite named number. That would be a googolplexian, which is a one followed by a googolplex of zeroes. A googolplex itself is a one followed by a googol of zeroes, and a googol is a one followed by a hundred zeroes. In other words:
1 googol = 1 × 10100
1 googolplex = 1 × 10googol or 1 x 10^(10^100)
1 googolplexian = 1 × 10googolplex or 1 x 10^(10^(10^100))
There are also large numbers that can only be expressed as complex exponentials, such as Graham's Number. Once again though, none of these are the largest number that exists, as there is no such thing. You need merely to add another value on top to reach a larger number.
999,999,999,999
To solve the LCM of 180 and 252, factor them in their prime factors...
180 = 2 * 2 * 3 * 3 * 5 = 22 * 32 * 51
252 = 2 * 2 * 3 * 3 * 7 = 22 * 32 * 71
... then multiply the prime factors with the highest power together...
22 * 32 * 51 * 71 = 1260
180 and 252 is a simple example which does not completely illustrate the priniciple involved. A better example might be 48 and 180...
48 = 2 * 2 * 2 * 2 * 3 = 24 * 31
180 = 2 * 2 * 3 * 3 * 5 = 22 * 32 * 51
... so the LCM would be 24 * 32 * 51 = 720
they are letters to crack an ancient code (The Shape Of Them)
What is the process of finding the solution of an equation?
A system of equations simply means, what number can you plug in for X and Y that will make both equations work?
For example,
Solve the following system of equations:
x + y = 11
3x - y = 5
In order to solve a system of equations, you have 2 ways.
Method 1: Substitution
Choose 1 equation. It doesn't matter which!
To make it easier, pick an equation without alot of numbers multiplied to X or Y.
ie: x + y = 11.
Now, solve for X or Y. You pick! OK, I'll pick. (but it doesn't matter which. just pick either!)
I'll solve for X.
x + y = 11
Subtract y from both sides, because I want my equation to look like x = something.
x = 11 - y
Now that we've solved for a variable (x), plug what x is equal to (11-y) into the equation you didn't solve, wherever you see x. Make sure to use parenthesis!
3x - 5 = y
3(11-y) - 5 = y
Now use your algebra to simplify the equation.
33 - 3y -5 = y (distributive property -- this is why we use parenthesis....)
28 - 3y = y
28 = 4y
7 = y
Now that we have a number for Y, plug 7 in wherever you see Y into one of your original equations. We're trying to solve for X now. In fact, to make it easier, you can plug y = 7 into the equation we solved x for!
Remember, x = 11-y ?
x = 11 - (7)
x = 4
So we know y = 7 and x = 4. These numbers should work in BOTH original equations!! If they don't, you made a mistake and you should double check your algebra.
x + y = 11
(4) + (7) = 11
11 = 11 (check)
3x - y = 5
3(4) - (7) = 5
12 - 7 = 5
5 = 5 (check)
Method 2: Elimination
As this method suggests, we need to "eliminate" or get rid of 1 variable.
Line both equations up so the x's and y's are stacked on top of each other, and the = signs as well. Like you were going to do an addition, or subtraction problem.
x + y = 11
3x - y = 5
-------------
What our goal is, is to eliminate one variable. Just like in method 1, you choose which variable. The easiest way is to pick a variable with no numbers in front.
Those y's look pretty good!
When using the elimination method we want 2 things to happen.
1) The number in front of the variable being eliminated is the same in both equations
2) The variable being eliminated must have opposite signs in both equations.
Luckily, our y variable has no coefficient (number in front) AND already has an opposite sign! (+y on top, -y on bottom). All we need to do now is add downward, like a regular addition problem.
1x + 1y = 11
3x - 1y = 5
-----------------
4x = 16
Notice the y's cancelled each other out? They "eliminated" themselves, which was our goal! Remember to add the numbers on the other side of the equals sign as well.
Now just solve for X.
4x = 16 -> x = 4
Once we find a value for x, substitute x=4 into either original equations, to solve for y.
x + y = 11
(4) + y = 11
y = 11 - 4
y = 7
So x =4 and y=7, just like in the first method. Use whichever makes you feel comfortable. I find substitution is much easier, but you might prefer elimination. Good luck!
What is a negation of a statement?
Negation says you should write the opposite.
So let's take the statement, Today is Monday. The negation is today is NOT monday. Sometimes it is harder.
Say we have the statement, EVERY WikiAnswers.com question is a great questions. The negation is Some questions on WikiAnswers.com are not great questions.
What is absolute and relative error?
An absolute measurement is based on first principle measurements. Most measurements are comparison. An absolute measurement doesn't rely on calibration of the instrument. For example wavelength measurements can be made without calibration by looking at the number of beats per seconds (Hertz).
Absolute error is the magnitude of the difference between the exaxt value of the value measured. It can be expressed as a number, e.g. the molecular weight measured is 27 000 grams per moles while the known molecular weight of the structure is 27 500, the absolute error is 500 grams per mole.
34.872 to 1 significant figure?
34.872 rounded and chopped to 1 sig fig is 30. The previous answer 34.8 is wrong, 34.8 is 34.872 chopped to 3 sig figs.
Any digit in a number that is not zero is classed as a significant figure e.g. your number above has 5 sig figs but if it was 30.072 it would only have 3 sig figs: 3, 7 and 2. When someone asks something to be taken to n sig figs they mean the first n significant figures of the number i.e. 103.42 to 2 sig figs would be 103, not 100.
Also, incase you don't know what I mean when i used the terms rounded and chopped above, rounding is when you round the number either up or down to the nearest sig fig so your number rounded to 2 sig figs would be 35. Chopping however is when you take however sig figs you need and chop the rest of regardless of how close the n+1th digit is to a higher sig fig. e.g. 34.372 is 34 chopped to 2 s.f. but 34.872 is also 34 chopped to 2 s.f. in fact if a number is virtually the same as a higher number it will get chopped to a lower one e.g. 34.999999999999999 chopped to 2 s.f. is 34, not 35!
What are the advantages of the finite element method?
Models Bodies of Complex Shape
- Can Handle General Loading/Boundary Conditions
- Models Bodies Composed of Composite and Multiphase Materials
- Model is Easily Refined for Improved Accuracy by Varying
Element Size and Type
C program to solve equation in runge kutta method?
PROGRAM :-
/* Runge Kutta for a set of first order differential equations */
#include
#include
#define N 2 /* number of first order equations */
#define dist 0.1 /* stepsize in t*/
#define MAX 30.0 /* max for t */
FILE *output; /* internal filename */
void runge4(double x, double y[], double step); /* Runge-Kutta function */
double f(double x, double y[], int i); /* function for derivatives */
void main()
{
double t, y[N];
int j;
output=fopen("osc.dat", "w"); /* external filename */
y[0]=1.0; /* initial position */
y[1]=0.0; /* initial velocity */
fprintf(output, "0\t%f\n", y[0]);
for (j=1; j*dist<=MAX ;j++) /* time loop */
{
t=j*dist;
runge4(t, y, dist);
fprintf(output, "%f\t%f\n", t, y[0]);
}
fclose(output);
}
void runge4(double x, double y[], double step)
{
double h=step/2.0, /* the midpoint */
t1[N], t2[N], t3[N], /* temporary storage arrays */
k1[N], k2[N], k3[N],k4[N]; /* for Runge-Kutta */
int i;
for (i=0;i
{
t1[i]=y[i]+0.5*(k1[i]=step*f(x,y,i));
}
for (i=0;i
{
t2[i]=y[i]+0.5*(k2[i]=step*f(x+h, t1, i));
}
for (i=0;i
{
t3[i]=y[i]+ (k3[i]=step*f(x+h, t2, i));
}
for (i=0;i
{
k4[i]= step*f(x+step, t3, i);
}
for (i=0;i
{
y[i]+=(k1[i]+2*k2[i]+2*k3[i]+k4[i])/6.0;
}
}
double f(double x, double y[], int i)
{
if (i==0)
x=y[1]; /* derivative of first equation */
if (i==1)
x= -0.2*y[1]-y[0]; /* derivative of second equation */
return x;
}
An object traveling at an average speed of 35 miles per hour, for 3 hours and 24 minutes, will travel 35 x (3 + 24/60) miles, or 119 miles.
Why is the mathematics is the queen of all science?
math is the quen of sience becaus without math there would be no sience
If you have to pick one hand and are then asked to change to the other hand should you change?
No. Always trust your gut.
There are 2.54 centimeters in one inch.
19.0 inches (2.54 centimeters/1 inch)
= 48.3 centimeters
Simeon Fatunla was a leader of the scientific community in Nigeria and by any standards one of the outstanding men of his generation. His death in a car accident, aged 51 and at the height of his powers, deprives Nigeria of a talent that combined vision with technical prowess, and of personal qualities that put both at the service of his country.
Fatunla was born in 1943 near Igede-Ekiti in what is now Ondo State, where he attended the Baptist High School from 1956 to 1961. After a year at college in Ijebu-Ode he took a degree in Mathematics at the University of Ibadan in 1965-68. There followed three years with Mobil Oil in Lagos which laid the foundations of his formidably professional computing skills; then his first period of overseas research at Loughborough University of Technology under Professor David Evans, completing a PhD in Computer Studies in 1975.
Fatunla's career as teacher had already begun: by 1969 he had had two spells schoolteaching in his home town and one in Lagos and he combined his PhD with a year as a lecturer at the university in Ile-Ife.
Fatunla knew his own powers and the value of developing them by international contacts. His charm, energy and determination to be based in Nigeria and contribute to the academic growth of his country won him friends world- wide. He travelled extensively to conferences, and to research visits in Illinois, 1977, Dublin, 1981 and 1988, Simon Fraser University, British Columbia, 1990. From 1984 he was Head of Mathematics at Benin University (Uniben). His widening reputation was recognised by a 1993 Distinguished Leadership Award of the American Biographical Institute and brought responsibilities in other developing countries: external examiner, Nairobi; guest conference speaker in Delhi and Tehran; setting up an Industrial mathematics MSc in Mauritius.
His main personal research was in numerical methods for stiff and oscillatory ordinary differential equations, in which many students at Benin followed him. But his view of priorities in a developing country led him to teaching and exposition. In 1983, and biennially from 1986, he led the now famous Uniben Scientific Computing Conferences. At first almost single-handed, then with a growing support group at Uniben, he attracted funding and impressive teams of overseas speakers, and edited the Proceedings. Since the 1988 conference these have been typeset and printed in Nigeria, latterly by the Ada/Jane Press which he founded. He found time to write two fine textbooks, Numerical Methods for Initial Value Problems in Ordinary Differential Equations (1988), a best-seller on Academic Press's list, and Fundamentals of FORTRAN Programming (1993). Widely adopted by Nigerian universities, the latter is both up-to-date and adapted to African needs.
His death occurred the week before he was due to address the Computer Association of Nigeria on "What Parallel Computing is All About" at their annual international conference, when it was widely expected that he would be elected president of the association.
As well as immense appetite for work - like many high achievers he needed little sleep - Fatunla had great zest for enjoyment, and many friends will remember sitting down to beer, political discussion and television after a day of conference, or enjoying pepper soup and bushmeat at one of his clubs.
Simeon Olujuyigbe Fatunla, mathematician: born near Igede-Ekiti, Nigeria 5 September 1943; Head of Mathematics Department, Benin University 1984- 95; married 1971 Grace Itiola (five sons, one daughter); died near Auchi, Nigeria 19 May 1995.
Why is lower percentage error better?
I would have thought this blindingly obvious but no matter, a lower percentage error is better because it means your approximation to a solution is closer to the real answer than an approximation with a higher error.
What is the the volume of a container that has 10 centimeters height and 10 centimeters width?
You need to know another dimension to find the volume. The volume is the length times height times width. You can only findthe area with two dimensions... the area would be 100cm2
See below for more info on area and volume.
How fast would a 75 kilo male fall from 12000 feet?
The acceleration of an object that falls from a certain height does not depend on its mass, in an ideal condition with no air resistance.
The value of acceleration is the acceleration due to gravity, which is 9.81 m s-2.
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However, in this case, air resistance is going to matter. 12000 feet is high enough for the person to accelerate to what we call terminal velocity. Terminal velocity is the velocity where the force of acceleration due to gravity (9.81 m s-2) is matched by the air resistance. That velocity varies, depending on the outline shape of the person, and is typically around 200 km/h or 125 mph. That will be the velocity of the fall.
