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Mathematical Preliminaries and Error Analysis C HA PT E R 1
Mathematical Preliminariesand Error Analysis
C HA PT E R 1
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Limit
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Continuity
Set X: C(X) Interval : C[a,b]
limit of a sequence
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Differentiability
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Integration
function f to be continuousthe points xi to be equally spacedzi = xi
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
When g(x) ≡ 1 Mean Value Theorem for Integrals
the average value of the function f over the interval [a, b]
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Taylor Polynomials and Series
(nth Taylor polynomial)
(remainder term or truncation error)
n→∞ : Pn(x) is called the Taylor series for f about x0.x0 = 0 : Maclaurin polynomial.
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
→ cos 0.01 ≅ 0.99995
trunca on error →
Determining a bound for the accuracy of the approximation:
More accurate approximation:
→ cos 0.01 ≅ 0.99995
trunca on error →
1.1 Review of CalculusInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
trunca on error →
Actual error → which is within the error bound.
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Round-off error:
Decimal Machine Numbers
k-digit decimal machine numbers:
• Machine arithmetic involves numbers with only a finite number of digits.• The calculations are performed with approximate representation.• The error that results from replacing a number with its floating-point form (rounding or
chopping)
3= 1.732050808 3 = (1.732050808)2=3.000000001
Any positive real number:
The floating-point form of y: ChoppingRounding
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Absolute error VS. Relative error
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Significant digits
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
relative error of f l(y)
k-digit chopping
k-digit rounding
Finite-Digit Arithmetic
In addition to inaccurate representation of numbers, the arithmetic performed in a computeris not exact.
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Nested Arithmetic: reducing the round-off error by rearranging calculations
1.2 Round-off Errors and Computer ArithmeticInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Writing f(x) in a nested manner:
Three-digit rounding answer is: −14.3
Polynomials should always be expressed in nested form before performing an evaluation!
1.2 Algorithms and ConvergenceInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
AlgorithmUnambiguousFinite sequenceSpecified order
1.2 Algorithms and ConvergenceInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Algorithms and ConvergenceInstructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Algorithms and Convergence Instructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
AlgorithmStableUnstableConditionally stable
Algorithm Stability:
small changes in the initial data
small changes in the final results
1.2 Algorithms and Convergence Instructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Algorithms and Convergence Instructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Algorithms and Convergence Instructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Algorithms and Convergence Instructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
Rates of Convergence:
1.2 Algorithms and Convergence Instructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
1.2 Algorithms and Convergence Instructor: Dr. Ali AmiriMathematical Preliminaries and Error Analysis
