Revision sheet 1

Question 1

What is the difference between computational and propagated error

Answer 1

Computional error is the result of the model and algorithm approximations including truncation and rounding (computer problem).

Propagated data error is the error caused by inputing data

Question 2

What is Truncation error?

Answer 2

difference between true result and result that would’ve been produced by given algorithm

Question 3

What is rounding error?

Answer 3

when a numerical value is approximated to a certain number of decimal places.

The discrepancy between exact and rounding value

Question 4

What is a condition number of computational problem?

Answer 4

the measure of how sensitive an output of a problem is of an input

Question 5

If a conditional number is 10^6 is this good or bad?

Answer 5

Bad, problem is sensitive to small changes in input

Question 6

estimate the condition number of evaluating function f(x)

Answer 6

cond ≈ |(x* f’(x))/f(x)|

Question 7

describe the structure of a floating point system

Answer 7

(-1)^sign x fraction x base^exponent

a floating-point number xˆ has the form ±d0.d1d2 ···dp−1 × βE, where d0.d1d2 · · · dp−1 is called mantissa (or significand) and E is called exponent.

x^= +/- (d0+d1/β+d2/β^2+ d3/β^3 +…. d(p-1)/β^(p-1))xβ^E

Question 8

what is machine precision?

Answer 8

determines the maximum possible relative error in representing an non-zero real number x in a floating point system.

Question 9

what is a floating point system?

Answer 9

a way to represent a number on a computer

Question 10

Explain why a divergent series (sum 1/n from n to infinity) can have a finite sum in floating point arithmetic.

At what point will the partial sum cease to change when computed using floating-point arithmetic with machine precision.

Answer 10

If we sum this series directly, then at some point the partial sum, Sn = sum(1/k) from k = 1 to n , will become bigger than 1/(n+1) by more than E^-1 mach, where E mach is machine prescision.

Question 11

when is a matrix non singular?

Answer 11

ANY of the following properties applies

• inverse of matrix exists
• non zero determinate exists
• the rank is n for n x n matrix (number of LI rows/columns)
• fi=or any vector not equal to zero, the product of the matrix by the vector is not equal to 0

Question 12

Define a transpose matrix

Answer 12

transpose of a mxn matrix is a mxn matrix with entries (A^T)ij = Aji

Question 13

show that (AB)^T = B^T A^T

Answer 13

C = AB and Cij = sum AikBkj
(C^T)ij = Cij = sum AjkBkj = sum Akj^T Bik^T = sum Bik^T Akj^T = (b^tA^T)ij

Question 14

Let A be a nonsingular matrix. Show that C = A^T A is symmetric and positive definite

Answer 14

symmetric (A^T = A)

C^T = (A^TA)^T = A^TA^TT = A^TA = C

PD = symmetric all eigenvalues positive
For all x not equal to 0, x^T C x >0,
X^T A^T A X = (AX)^T (AX) = ||AX||2^2 >= 0

since A is nonsingular, Ax =/ 0 for any x =/ 0 for ||AX||2^2 = 0

Question 15

prove that ||Ax||<= ||A||||x||

Answer 15

look at notes

Question 16

prove that ||AB||<= ||A||||B||

Answer 16

look at notes