• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Fixed-Point Iteration

Extracts from this document...

Introduction

Carrera Falk Mr. Moore IB HL Math Methods January 3, 2003 IB HL Portfolio Assignment: Fixed-Point Iteration 1. i. A. Equation: f(x) = 0.25+6 Fixed point: x = 8 First "guess": xn-1 = 1 f(1) = 0.25(1) + 6 = 6.25 Second iteration: f(6.25) = 0.25(6.25) +6 = 7.5625 Third iteration: f(7.5625) = 0.25(7.5625) + 6 = 7.890625 Fourth iteration: <my iteration> f(7.890625) = 0.25(7.890625) + 6 = 7.97265625 B. - Copied axes located on page 6 Staircase diagram from right side of graph: <Same equation and fixed point as above> First "guess": 20 f(20) = 0.25(20) + 6 = 11 Second iteration: f(11) = 0.25(11) + 6 = 8.75 Third iteration: f(8.75) = 0.25(8.75) + 6 = 8.1875 - New "staircase" diagram shown on axes - "Staircases" graphed on TI-83+ 2. A. Solve x = 1.25x - 2 to find a value of x at the fixed point: x= 1.25x -2 -.25x = -2 x = 8 B. ...read more.

Middle

/3 or 1/3x2 - 4/3 ii. Iterations on page 8 It is an attracting point, especially due to the fact that I used the number 1 as my first "guess" which made every other answer to the iteration "-1" because it is one of the numbers that solve the problem evenly. iii. Iterations on page 8 It is a repelling point, and maybe this is because the first "guess" number was a decimal, but also, the staircase here is found to be on the left side of the graph, whereas the above iteration was a right staircase. B. Consider the equation x = -0.5x + 2 Iterations on page 8 It is an attracting point, staircasing left of the fixed point. Again, here I noticed that I used a whole number as a first "guess" to be multiplied by a fraction of one in the equation. ...read more.

Conclusion

It is a repelling fixed point when examining the last graph where x = 1 and the first "guess" is x = 1.1. Once looking over these last equations, I am beginning to think that a greater meaning lies behind why some equations have repelling and some have attracting fixed points. E. Table of Attracting Fixed Point and Repelling Fixed Points Tables on page 10 i. Conjecture: The condition, which must be present for a fixed point to be attracting, is apparent to me that every one has a fractional slope of the tangent, or f ' (x). The repelling points also have a few fractional tangential slopes, but all of the attracting ones do as a trend. ii. Testing the conjecture: Unfortunately, this conjecture did not hold true for the equation f(x) = x^5 -2 when viewing its staircase from the left, as its fixed point seems to be centered around -3, and the tangential slope of that becomes a positive term, -405. (Iterations on page 10.) iii. Iterations on page 11 ...read more.

The above preview is unformatted text

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related AS and A Level Core & Pure Mathematics essays

1. 8 27 64 I am using the Increment Method, because this graph has a very awkward curve which is hard to find the gradient, so I will need to be a specific as possible and use a method which will calculate the gradient accurately.

2. ## Math Portfolio Type II - Applications of Sinusoidal Functions

R | 5.516 ? y ? 7.208}. The period of the function representing the time of sunrise for Miami is , which results in the answer 0.017. The domain represents the number of days, starting from day 1 to day 365, since day 0 does not exist.

1. ## The open box problem

X 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 V 242 244.944 247.192 248.768 249.696 250 249.704 248.832 247.408 245.456 243 This table now suggests that the maximum volume is 250 and that x is 2.5. To prove this again I will construct two graphs.

2. ## Mathematics portfolio - Translations.

If the number is positive, the curve will shift to the left. If the number is negative, the curve will shift to the right. = sin (x-90)2 is the effect of translation vector of = sinx. It moves right 90 units. This has the same effect with the previous examples. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 