ASE 311 Notes
Chapter 4: Round-off and Truncation Errors
Sources of Error
●
1. Human Error
●
2. Model Error: you cannot consider everything that can happen
●
3. Data Uncertainty: noise, repeatability (possible to set up experiment exactly the same
everytime?), etc
●
4. Machine/Numerical Error: round-off error, Ex) pi is
digit numbers
Error Characterizations
if there exists a true value, that true value equals approximation + error
●
Accuracy: how closely a computed or measured value agrees with the true value
●
Precision: how closely individual computed values agree with each other
* we want everything to be both accurate and precise
Example
Case 1: true = 1, approx. = 1.1;
.1
Case 2: true = 10,000, approx. = 10,000.1
.1
* we would call Case 2 more accurate because relative to true value, the error is considerably
smaller
*Never have true value
Use
for the stopping condition for iterative methods
Numerical Error
1.
Round-Off:
exists because computers cannot represent numbers exactly
a) computers have size and precision limits on their ability to represent numbers
Computers represent numbers with a binary (base-2) system; this system has 0 or 1, or off/on
respectively

Example: 101.1 in base-2 =
in
base-10
Floating-Point Representation
,
MATLAB has 64 bits; 1 for the sign, 11 for the exponent, and 52 for the mantissa
More bits
if more bits are given to the mantissa, representation can be more accurate
if more bits are given to the exponent, representation can include larger numbers
Since numbers get normalized, in base-2, this implies that you get “53” bits in the mantissa
Range
:
Largest Value:
Smallest Value:
Precision
:
*We cannot control round-off error as much as Truncation error
2.
Truncation
: errors that result from using an approximation in place of an exact mathematical
procedure
Example
this is an approximation of a value that could be found using an exact mathematical
procedure
Taylor’s Theorem/Taylor’s Series
Taylor’s Series Expansion
As
, this is exact, but since you do not go to
, you have error (remainder term,
)
The term
is commonly represented in Lagrange form, where
The Lagrange form comes from the mean value theorem of integrals
In general, the error is
“Error is order
”
defines a convergence rate

Zeroth-Order Approximation
First-Order Approximation
Second-Order Approximation
...
Linear Systems
What is a linear system?
a system is linear if the system satisfies both the “addivity” and “homogeneity” properties
Let
be a mathematical operator (mapping), then
is called linear if
1)
“addivity”
2)
“homogeneity”
1) and 2) are called the principle of superposition
Example
This is linear
this is not linear
**in order for a line to be linear, a line must pass through the origin
**there is a distinction between a linear mapping and a “linear equation”
The inverse of matrix A

The inverse can be found as
, where the adjoint is the transpose of the
matrix of cofactors
The Determinant is a scalar measure of a square matrix,
where
where
Transpose - the transformation of rows to columns
* If a matrix is symmetric

#### You've reached the end of your free preview.

Want to read all 31 pages?

- Spring '12
- vasd
- Numerical Analysis, Linear Systems, Method, Convergence