Manual for Applied Numerical Methods with Python for
Engineers and Scientists 1st Edition By Steven Chapra,
David Clough
Accuracy - ANSWER How closely a computed value agrees with the true value
Precision - ANSWER How closely individual values from different numerical
analyses of same problem agree with each other
True error - ANSWER Difference between the true value and the approximation
Relative error - ANSWER True error divided by the true value
Tolerance - ANSWER Usually want to know if true error is lower than pre-specified
tolerance - computation is repeated until true error less than tolerance - when this
happens is called the stopping criterion
Round-off errors - ANSWER Arise as computers can not represent quantities
exactly
1) size and precision limits computers ability to represent numbers
2) certain numerical manipulations highly sensitive to round off errors
How can round off errors be reduced - ANSWER Use of large steps but need to
balance as precision achieved with small steps
Truncation error - ANSWER Result from using an approximation in place of an
exact mathematical procedure - in numerical analysis use approximate mathematical
function to represent physical properties e.g. Taylor series
Why is select of step size important - ANSWER Results depend on step size used -
usually use of small steps give good results - however in some cases small steps
increase error
Propagation of errors - ANSWER Error though initially small will grow significantly in
subsequent arithmetic operations
A special case of propagation of errors - ANSWER Catastrophic cancellation - error
will tend to infinity
Regression analysis - ANSWER Development of mathematical model used to full
use of collections of experimental data - also known as curve fitting
Why is regression analysis useful? - ANSWER Raw test data alone can not be
used in practice - once appropriate curve has been established further analysis can
be carried out
, Applications of regression analysis (4) - ANSWER 1) analysis of astronomical
observations
2) economists used regression analysis to estimate key economic statistics
3) machine learning - analyse data from various sensors/ databases - structural
health monitoring
4) civil engineering - model extreme loads earthquakes, traffic management analysis,
rainfall data, ground settlement
Theory of regression analysis - ANSWER A statistical technique for estimating
relationship among variables
Piecewise linear interpolation - ANSWER Is the simplest form of curve fitting - need
to find equations to represent the piecewise linear functions. Do this by finding linear
function between the two data points and then also need to find linear function
between each two adjacent data points
Limitations of piecewise linear interpolation (4) - ANSWER 1) simple model - 1st
order polynomial
2) practically less useable since most data can not be represented using linear
functions
3) sharp change in the first derivative at data points
4) higher order polynomials are often used in practical problems so can't use this for
them
Polynomial fits - ANSWER Polynomials used to model data more accurately
Theory of polynomial fit - ANSWER Use a matrix to represent data set
Advantages of using polynomial fit to represent and analyse data (4) - ANSWER 1)
simple model
2) polynomials are smooth functions
3) polynomial of degree n-1 can be represented exactly with a set of n coefficients
4) if f(x) is any continuous function defined over any finite interval [a,b] then for each
e, ther the exists a polynomial fit such that |f(x) - F(x)| < e for all a < x < b
Limitations of polynomial fit - ANSWER When higher order polynomials are used
the output can be very sensitive to the changes in a single or a few terms only - eg.
x^5 is too significant and will decide the value. May then oscillate widely between
adjacent data points - known as ill condition f polynomials
Lagrange polynomials - ANSWER Instead of using general form of polynomials can
use alternative polynomial system >> Lagrange polynomials - is a way to avoid effect
of ill conditioned polynomials
Cubic splines theory - ANSWER High order functions required to fit data since
otherwise can not have smooth lower order functions which pass through point, use
of high order polynomials may not be accurate. Alternative can define a set of low
order polynomials (piecewise) to pass through all data points instead of single high
order function. Choice of low order piecewise polynomials allows us to add additional
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