**Programming Team: Practice Contest,
12-Feb-05 (Matrices)**

*Mt Lebanon HS 2004-5
David Kosbie*

**Link to the Programming
Team Home Page.**

**Q1: Gaussian Elimination (n linear equations in n
unknowns)
**Read in an integer n followed by n linear equations in n unknowns and solve
for the n unknowns. Consider the following system of equations:

2a + 2b - 5c = -18

4a + 3b - 4c = -4

5a + 4b - 3c = 8

These would be entered by the following numbers (one per line):

3 2 2 -5 -18 4 3 -4 -4 5 4 -3 8

Print out the values of the unknowns rounded to nearest 0.01.

For the given data, your program should output:

2.0

4.0

6.0

**Q2: Polynomial Fitting
**Read in an integer n followed by n points (x1,y1,x2,y2,...), one value per
line, and print out a polynomial of degree (n-1) that fits those points. Round
coefficients to the nearest 0.01.

Sample input (entered one number per line): 2 1 1.5 2 2.5

Sample output: 1.0x^1 + 0.5x^0

Sample input: 3 -1.5 5 2 -2 4 5

Sample output: 1.0x^2 + -2.5x^1 + -1.0x^0

**Q3: Sum of Powers (1 ^{k} + 2^{k}
+ ... + x^{k})**Read in a positive integer k and print out the polynomial which represents
the sum of the first x integers raised to the kth power. That is, 1^k + 2^k +
... + x^k. Round coefficients to the nearest 0.01.

Sample input: 1

Sample output: 0.5x^2 + 0.5x^1 + 0.0x^0

Sample input: 2

Sample output: 0.33x^3 + 0.5x^2 + 0.17x^1 + 0.0x^0

Sample input: 4

Sample output: 0.2x^5 + 0.5x^4 + 0.33x^3 + 0.0x^2 + -0.03x^1 + 0.0x^0

To see that this is true, notice that:

1^{4} + 2^{4} + 3^{4} + 4^{4}
= 1 + 16 + 81 + 256 = 354

and

0.2*4^{5} + 0.5*4^{4} + 0.33*4^{3} + 0.02*4^{2}
+ -0.03*4^{1} + 0.0*4^{0
} = 204.8 + 128 + 21.12 + 0.32 + -0.12 + 0

= 354.12 (the extra 0.12 is due to roundoff error)