©THEBRIGHT EXAM STUDY SOLUTIONS 8/26/2024 11:32 AM
Numerical methods - numerical analysis
Exam Questions And Correct Answers
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
, ©THEBRIGHT EXAM STUDY SOLUTIONS 8/26/2024 11:32 AM
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
Numerical methods - numerical analysis
Exam Questions And Correct Answers
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
, ©THEBRIGHT EXAM STUDY SOLUTIONS 8/26/2024 11:32 AM
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
