Numerical Methods: Errors, Significant Figures, and Floating-Point Arithmetic

What are significant figures?

Digits in a number that carry meaningful information about its precision.

What is the definition of accuracy in measurements?

Accuracy refers to how closely a measured value compares with the true value.

What does precision refer to in measurements?

Precision indicates how closely individual measured values agree with each other.

What is truncation error?

Error made by using a numerical method to represent an exact mathematical procedure.

What is roundoff error?

Error introduced by a digital computer due to a finite number of significant figures stored.

What is absolute error?

The difference between the true value and the approximate value.

How is relative error calculated?

Relative error is calculated as the absolute error( which is just true value - approx value) divided by the true value.

What is an iterative method?

A numerical procedure that is repeated to get closer to the true value.

What is the stopping criterion in an iterative method?

Stop when the absolute error is less than a specified threshold.

What is machine epsilon?

The largest relative error that can occur when representing a number in a computer.

What is the condition number of a function?

A measure of how the relative change in the input of a function affects the relative change in its output.

What happens when a function is ill-conditioned?

Small changes in input can result in large changes in output.

What is the significance of significant figures in floating point representation?

They determine the precision of the number that can be stored.

What is subtractive cancellation?

Loss of significant figures that occurs when subtracting two nearly equal numbers.

What is the range of a 16-bit integer representation?

From -32,768 to +32,767.

What is the effect of rounding on numerical computations?

Rounding can introduce errors, especially in iterative calculations.

What is the formula for calculating the condition number?

Condition Number = xo * f'(xo) / f(xo)

What is the significance of the factor 0.5 in relation to significant figures?

It is used to avoid loss of the last digit due to roundoff error.

What is the impact of large values of f'(x) on condition number?

Large values of f'(x) can indicate that the function is ill-conditioned.

What is the relationship between significant figures and machine epsilon?

Machine epsilon is determined by the number of significant figures used in the mantissa. machine epsilon = b^1-t (t is the number of sig figs).

How does a computer represent floating point numbers?

In the form of mantissa and exponent.

What is the effect of overflow and underflow in numerical computations?

Overflow occurs when a number exceeds the maximum representable value; underflow occurs when a number is too small to be represented.