EGR 511



EGR 511 NUMERICAL METHODS __________________

LAST NAME, FIRST

PROBLEM SET #3

1. (P. 14.5 Chapra) Find the minimum value of f(x, y) = (x ( 2)2 + (y ( 3)2 starting at x = 1 and y = 1, using one iteration of the steepest descent method.

Ans: x = 2 and y = 3 ( f(x, y) = 0

2. (P. 14.6 Chapra) Perform one iteration of the steepest ascent method to locate the maximum of f(x, y) = 3.5x + 2y + x2 ( x4 ( 2xy ( y2 using initial guesses x = 0 and y = 0.

Ans: x = 0.9766 and y = 0.5581

3. Let f(x) = [pic]and P2(x) be the interpolation polynomial on x0 = 0, x1 and x2 = 1. Find the largest value of x1 in (0,1) for which f(0.5) – P2(0.5) = -0.25.

Ans: x1 = 0.872675

4. Let P3(x) be the interpolation polynomial for the data (0,0), (0.5, y), (1,3), and (2,2). Find y if the coefficient of x3 in P3(x) is 6.

Ans: y = 4.25

5. Show that the following sequences {pn} converge linearly to p = 0. How large must n be before |pn – p| [pic] 5(10-2?

a) pn = [pic], n ( 1 b) pn = [pic], n ( 1

Ans: a) n = 20 and b) n = 5

6. For the given functions f(x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct the Lagrange interpolating polynomials of degree 1 and 2 to approximate f(0.45) and find the error bound and the actual error.

a) f(x) = cos x b) f(x) = [pic] c) f(x) = tan x

Ans:

|f(x) |cos x |[pic] |tan x |

|P1(x=0.45) |0.86900 |1.19868 |0.5131 |

|error bound |0.03375 |4.65(10-2 |0.220 |

|actual error |3.1445(10-2 |5.47616(10-2 |0.03005 |

|P2(x=0.45) |0.89810 |1.20343 |0.454614 |

|error bound |5.0625(10-3 |6.98(10-3 |0.151 |

|actual error |2.357(10-3 |7.3573(10-4 |0.02844 |

7. Construct a divided-difference table from:

|x |0.5 |-0.2 |0.7 |0.1 |0.0 |

|f(x) |-1.1518 |0.7028 |-1.4845 |-0.14943 |0.13534 |

Use the divided-difference table to estimate f(0.15), using

a) a polynomial of degree 2 through the first three points.

b) a polynomial of degree 2 through the last three points.

c) a polynomial of degree 3 through the first four points.

d) a polynomial of degree 3 through the first four points.

e) a polynomial of degree 4.

Which three points are best to use for constructing the quadratic if we want f) f(0.15)? g) f(-0.1)? h) f(1.2)?

Ans:

a) ( 0.35871

b) ( 0.28514

c) ( 0.28941

d) ( 0.28938

e) ( 0.28939

f) f(0.15): use 0, 0.1, 0.5

g) f(-0.1): use ( 0.2, 0, 0.1

h) f(1.2): use 0.1, 0.5, 0.7

8. (P. 14.10 Chapra) Develop a one-dimensional equation in the pressure gradient direction at the point (4, 2). The pressure function is

f(x, y) = 5x2y ( 8y2 ( 7x2

Ans:

g(h) = 16 + 2880h + 29376h2 + 138240h3

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