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Approximation and Errors
Numerical Methods © Wen-Chieh Lin 2
Well-Posed Problems
Problem is well-posed if solution exits is unique depends continuously on problem data
Otherwise, problem is ill-posed
Numerical Methods © Wen-Chieh Lin 3
Remedy to ill-posed problem
Replace difficult problem by easier one havingsame or close enough solution nonlinear linear infinite finite differential algebraic complicate simple
Solution obtained may only approximate thatof the original problem
Numerical Methods © Wen-Chieh Lin 4
Sources of Approximation Before computation
Modeling Empirical measurements Previous computations
During computation Truncation or discretization Rounding
Accuracy of final result reflects all these Uncertainty in input may be amplified by problem Perturbations during computation may be amplified
by algorithmFrom Michael T. Heath’s slide
Numerical Methods © Wen-Chieh Lin 5
Example: ApproximationsComputing surface area of Earth using formula
involves several approximations
Earth is modeled as sphere, idealizing its trueshape
Value for radius is based on empiricalmeasurements and previous computations
Value forπrequires truncating infinite processValues for input data and results of arithmetic
operations are rounded in computer
24 rV
From Michael T. Heath’s slide
Numerical Methods © Wen-Chieh Lin 6
Error Definition
Absolute error: |approximate value –true value| Relative error: (absolute error)/(true value) True value is usually unknown, so we estimate or
bound error than compute it exactly Relative error often taken relative to approximate
value rather than (unknown) true value
From Michael T. Heath’s slide
ionapproxiamtcurrentionapproximatprevious-ionapproximatcurrent
Numerical Methods © Wen-Chieh Lin 7
Truncation ErrorDifference between true result and result
produced by given algorithm using exactarithmetic
Due to approximations such as truncatinginfinite series or terminating iterative sequencebefore convergence
!3!2!11
32 xxxex
4
32
!!3!2!11
n
nx
nxxxx
e
Truncation error
Numerical Methods © Wen-Chieh Lin 8
!11
xex
15.0 e1xe
5.15.0 e
%3.33%1005.1
15.1
ionapproxiamtcurrentionapproximatprevious-ionapproximatcurrent
4
32
!!3!2!11
n
nx
nxxxx
e
Numerical Methods © Wen-Chieh Lin 9
!11
xex
15.0 e1xe
5.15.0 e
%69.7%100625.1
1625.1
4
32
!!3!2!11
n
nx
nxxxx
e
!2!11
2xxex 625.15.0 e
%3.33
Numerical Methods © Wen-Chieh Lin 10
!11
xex
15.0 e1xe
5.15.0 e
%27.1
4
32
!!3!2!11
n
nx
nxxxx
e
!2!11
2xxex 625.15.0 e
%3.33
%69.7
!3!2!11
32 xxxex 645833333.15.0 e
Numerical Methods © Wen-Chieh Lin 11
Round-off Error
Difference between result produced by givenalgorithm using exact arithmetic and resultproduced by same algorithm using limitedprecision arithmetic
Due to inexact representation of real numbersand arithmetic operations upon them
Computational error = truncation error +round-off error
Numerical Methods © Wen-Chieh Lin 12
Propagated Error
An error in the succeeding steps of a processdue to an occurrence of an earlier error
Numerical Methods © Wen-Chieh Lin 13calcx
Forward and Backward Error
Suppose we want to compute y=f(x), where f is a realvalue function, but obtain approximate value
Forward error
Backward error
calcy
)(xfyexact
x
)(1calccalc yfx calcy
exacty
exactcalcfwd yyE
xxE calcbackw
Numerical Methods © Wen-Chieh Lin 14x
Forward and Backward Error
Suppose we want to compute y=f(x), where f is a realvalue function, but obtain approximate value
Forward error
Backward error
y
)(xfy
x
)(ˆ 1 yfx y
y
yyy ˆ
xxx ˆ
Numerical Methods © Wen-Chieh Lin 15
Backward Error Analysis Idea: approximate solution is exact solution to
modified problemHow much must original problem change to
give result actually obtained?How much data error in input would explain
all error in computed result?Approximate solution is good if it is exact
solution to near problem
From Michael T. Heath’s slide
Numerical Methods © Wen-Chieh Lin 16
Example: Backward Error Analysis
Approximating cosine function by truncatingTaylor series after two terms gives
Forward error is given by
Backward error is given by , where
21)(ˆ 2xxfy
)cos(21)()(ˆ 2 xxxfxfyy
)arccos()(ˆ 1 yyfx
xx
Numerical Methods © Wen-Chieh Lin 17
Example (cont.)
For x=1,
Forward error:Backward error:
5.0211)1(ˆ 2 fy
5403.0)1cos()1( fy
0472.1)5.0arccos()arccos(ˆ yx
0403.05403.05.0ˆ yyy
0472.010472.1ˆ xxx
Numerical Methods © Wen-Chieh Lin 18
Sensitivity and Conditioning Problem is insensitive or well-conditioned, if relative
change in input causes similar change in solution Problem is sensitive or ill-conditioned, if relative
change in solution can be much larger than that ininput
Condition number
xxxxfxfxf
)ˆ()()]()([
|datainputinchagerelative||solutioninchangerelative|
cond
Numerical Methods © Wen-Chieh Lin 19
Sensitivity and Conditioning (cont.)
Problem is sensitive or ill-conditioned, ifcond >> 1
A well-posed problem can be ill-conditioned
Computational algorithm should not makesensitivity worse
Numerical Methods © Wen-Chieh Lin 20
Condition Number Condition number is amplification factor relating
relative forward error to relative backward error
Condition number is usually not known exactly andmay vary with input, so rough estimate or upperbound is used
xxyy
xxxxfxfxf
)ˆ()()]()([
cond
Numerical Methods © Wen-Chieh Lin 21
)()(')()('
)()()]()([
)ˆ()()]()([
|datainputinchagerelative||solutioninchangerelative|
cond
xfxxf
xhxfxhf
xxhxxfxfhxf
xxxxfxfxf
Example: Evaluating Function Evaluating function f for approximate input
)(')()( xhfxfhxf
hxx
hxx
Numerical Methods © Wen-Chieh Lin 22
Example: Sensitivity Tangent function is sensitive for arguments
near π/2
For
51058058.1)57079.1tan(
41058058.6)57078.1tan(
51048275.2cond,57079.1 x
Numerical Methods © Wen-Chieh Lin 23
Stability Algorithm is stable if result produced is relatively
insensitive to perturbations during computation
Stability of algorithm is analogous to conditioning ofproblems
From viewpoint of backward error analysis,algorithm is stable if result produced is exact solutionto nearby problem
For stable algorithm, effect of computational error isworse than effect of small data error in input
From Michael T. Heath’s slide
Numerical Methods © Wen-Chieh Lin 24
Accuracy Accuracy: closeness of computed solution to true
solution of problem
Stability alone does not guarantee accurate results
Accuracy depends on conditioning of problem as wellas the stability of algorithm
Inaccuracy can result from applying stable algorithm to ill-conditioned problem unstable algorithm to well-conditioned problem
Applying stable algorithm to well-conditionedproblem yields accurate solution
From Michael T. Heath’s slide
Numerical Methods © Wen-Chieh Lin 25
Floating-Point Arithmetic
Floating-point number is represented by sign fraction (mantissa) exponent
Eppddd
x 2)222
1( 11
221
UELpidi and,1,,1,10where
Numerical Methods © Wen-Chieh Lin 26
Machine Numbers
Not all real numbers exactly representable;those that are are called machine numbers
Machine numbers are unequally spaced
If a real number is not exactly repersentable,then it is approximated by a nearby machinenumber causing round-off error
Numerical Methods © Wen-Chieh Lin 27
Machine Precision
Accuracy of floating-point systemcharacterized by unit round-off (or machineprecision or machine epsilon)
Smallest number such that
If1)1( epsfloat
)1(
)(1
eps
Numerical Methods © Wen-Chieh Lin 28
Machine Precision
Accuracy of floating-point systemcharacterized by unit round-off (or machineprecision or machine epsilon)
Smallest number such that
If1)1( epsfloat
1)1(
)(1
eps
Numerical Methods © Wen-Chieh Lin 29
Machine Precision
Accuracy of floating-point systemcharacterized by unit round-off (or machineprecision or machine epsilon)
Smallest number such that
If1)1( epsfloat
1)1(
1)(1
eps
Numerical Methods © Wen-Chieh Lin 30
Summary
Well-posed problemComputational error
truncation error round-off error
Sensitivity (conditioning) of problemCondition number
Stability of algorithm