NUMERICAL METHODS -
NUMERICAL ANALYSIS QUESTIONS
& ANSWERS(GRADED A+)
Accuracy - ANSWERHow closely a computed value agrees with the true value
Precision - ANSWERHow closely individual values from different numerical analyses
of same problem agree with each other
True error - ANSWERDifference between the true value and the approximation
Relative error - ANSWERTrue error divided by the true value
Tolerance - ANSWERUsually 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 - ANSWERArise 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 - ANSWERUse of large steps but need to
balance as precision achieved with small steps
Truncation error - ANSWERResult 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 - ANSWERResults depend on step size used -
usually use of small steps give good results - however in some cases small steps
increase error
Propagation of errors - ANSWERError though initially small will grow significantly in
subsequent arithmetic operations
A special case of propagation of errors - ANSWERCatastrophic cancellation - error
will tend to infinity
Regression analysis - ANSWERDevelopment of mathematical model used to full use
of collections of experimental data - also known as curve fitting
Why is regression analysis useful? - ANSWERRaw 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) - ANSWER1) 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 - ANSWERA statistical technique for estimating
relationship among variables
Piecewise linear interpolation - ANSWERIs 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
Least squares - ANSWERDraw curve of bat fit which minimises the associated error
of different data points
What does least squares allow? (3) - ANSWER1) ignore obviously wrong data
2) give a weight for each data - to take into account the uncertainty in each data
point
3) define a best fit curve by considering all data
In least squares what is the error between the predicted value and the actual data
point called - ANSWERResidue
What is the total error in least squares - ANSWERSum of square residues
Least squares using polynomials - ANSWERIs the same as method used for a
straight line
Limitations of piecewise linear interpolation (4) - ANSWER1) 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 - ANSWERPolynomials used to model data more accurately
Theory of polynomial fit - ANSWERUse a matrix to represent data set
Advantages of using polynomial fit to represent and analyse data (4) - ANSWER1)
simple model
2) polynomials are smooth functions
3) polynomial of degree n-1 can be represented exactly with a set of n coefficients
NUMERICAL ANALYSIS QUESTIONS
& ANSWERS(GRADED A+)
Accuracy - ANSWERHow closely a computed value agrees with the true value
Precision - ANSWERHow closely individual values from different numerical analyses
of same problem agree with each other
True error - ANSWERDifference between the true value and the approximation
Relative error - ANSWERTrue error divided by the true value
Tolerance - ANSWERUsually 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 - ANSWERArise 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 - ANSWERUse of large steps but need to
balance as precision achieved with small steps
Truncation error - ANSWERResult 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 - ANSWERResults depend on step size used -
usually use of small steps give good results - however in some cases small steps
increase error
Propagation of errors - ANSWERError though initially small will grow significantly in
subsequent arithmetic operations
A special case of propagation of errors - ANSWERCatastrophic cancellation - error
will tend to infinity
Regression analysis - ANSWERDevelopment of mathematical model used to full use
of collections of experimental data - also known as curve fitting
Why is regression analysis useful? - ANSWERRaw 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) - ANSWER1) 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 - ANSWERA statistical technique for estimating
relationship among variables
Piecewise linear interpolation - ANSWERIs 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
Least squares - ANSWERDraw curve of bat fit which minimises the associated error
of different data points
What does least squares allow? (3) - ANSWER1) ignore obviously wrong data
2) give a weight for each data - to take into account the uncertainty in each data
point
3) define a best fit curve by considering all data
In least squares what is the error between the predicted value and the actual data
point called - ANSWERResidue
What is the total error in least squares - ANSWERSum of square residues
Least squares using polynomials - ANSWERIs the same as method used for a
straight line
Limitations of piecewise linear interpolation (4) - ANSWER1) 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 - ANSWERPolynomials used to model data more accurately
Theory of polynomial fit - ANSWERUse a matrix to represent data set
Advantages of using polynomial fit to represent and analyse data (4) - ANSWER1)
simple model
2) polynomials are smooth functions
3) polynomial of degree n-1 can be represented exactly with a set of n coefficients
