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I - Introduction
- Modeling real systems involves converting the physical system into mathematical equations
that capture the embedded physics.
- The mathematical model often comprises complex and complicated equations that are
unmanageable analytically and lack closed-form and exact solutions. Thus, numerical
methods are essential for handling and solving such systems.
- Each problem can be approached using various numerical methods, each with its own set of
advantages, disadvantages, and levels of simplicity, complexity, and accuracy. Selecting the
appropriate method is crucial for effectively solving such problems.
- Computational tools like computers and calculators are commonly used for fast and accurate
numerical calculations. However, their accuracy is limited due to finite precision, rounding
errors, and algorithm stability.
Floating-Point Numbers and Rounding
- Fundamental concepts in numerical computing, deeply influencing the accuracy and
reliability of numerical calculations.
- Floating points: a way to represent real numbers in computations that can accommodate a
very wide range of values. It represents numbers using three components,
o Sign : indicating positive or negative.
o Exponent : scales the number.
o Mantissa : provides the precision of the number in the binary system.
๐๐ข๐๐๐๐ = ๐๐๐๐ ร ๐๐๐๐ก๐๐ ๐ ๐ ร 2
๐ธ๐ฅ๐๐๐๐๐๐ก
- Rounding errors occur because of the need to round off numbers that cannot be exactly
represented with the available level of precision. Rounding can lead to several types of errors
in calculations.
o Rounding Error : this is the difference between the mathematically exact value of a
number and its rounded floating-point approximation. Even simple arithmetic operations
can accumulate noticeable rounding errors over many iterations or complex calculations.
o Precision Loss : when operating on numbers of vastly different magnitudes, precision
loss can occur, where significant digits of the smaller number are lost. For example, when
a very large number and a very small number are added, the small number may not affect
the sum due to the limited precision of the floating-point format.
o Cancellation : significant digits can be lost when subtracting two close numbers, leading
to a result with less precision than the original numbers.
o Accumulated Errors : in iterative processes or algorithms involving many steps
(numerical integration, solving differential equations, โฆetc.), small rounding errors can
accumulate, potentially leading to significant deviation from the true solution.
Definitions
- Degree of condition : a measure of the uncertainties in the initial or boundary conditions. It
influences the uncertainty and sensitivity of the solution. The degree of condition is a
function of the mathematical problem itself, independent of the numerical method
implemented.
o Ill-conditioned problem : large uncertainty in the initial or boundary conditions, leading
to very sensitive solution to small changes or errors in the initial data.
o Well-conditioned problem : small uncertainty in the initial or boundary conditions,
leading to very insensitive solution to small changes or errors in the initial data.
โฒ
2 ๐ฅ
2 ๐ฅ
The solution is highly sensitive to changes or errors in the initial conditions, particularly
for large values of ๐ฅ, resulting in ill-conditioned problem. Conversely, for large negative
values of ๐ฅ, it appears as a well-conditioned problem.
- Stability : refers to the degree of certainty in ensuring that the method delivers reliable and
meaningful results. Unstable methods accumulate and propagate errors to levels that are
inaccurate and unreliable.
Error Analysis
Numerical methods are computational techniques used to approximate solutions to
mathematical problems when exact analytical solutions are unattainable. These methods
inherently involve approximation errors.
- Series are very accurate at a neighborhood of ๐ฅ = ๐.
- Taylor and Maclaurin series of ๐
= sin ๐ฅ up to 5
th
order,
a = 0
F = - x^3/6 + x
a = 2
F = sin(2) - (sin(2)(x - 2)^2)/2 + (sin(2)(x - 2)^4)/
- cos(2)(x - 2) - (cos(2)(x - 2)^3)/
a = 2*pi
F = x - 2pi - (x - 2pi)^3/
Maclaurin series of ๐
= sin ๐ฅ for different orders,
k = 2
F = x
k = 4
F = - x^3/6 + x
k = 10
F = x^9/362880 - x^7/5040 + x^5/120 - x^3/6 + x
Assignment 1
- Find the Maclaurin series of the following functions,
a) ๐(๐ฅ) = cos 2 ๐ฅ
b) ๐(๐ฅ) = ๐
๐ฅ
2
c) ๐
2
2
- Find the Taylor series of ๐
, around ๐ฅ = 3 , up to the 4
th
order. Evaluate
the percentage error and absolute error at ๐ฅ = 4. Plot ๐(๐ฅ) and its Taylor expansion.
- Use Maclaurin series to evaluate ๐(๐ฅ) = ๐
โ๐ฅ
sin ๐ฅ, at ๐ฅ = 1 with an absolute error less than
Determine the number of significant digits of accuracy of 2.7 2 with respect to ๐.
Show that ๐ฅ is an accurate approximation of ๐ to 3 significant digits when,
and suggest a value of ๐ that is accurate to 3 significant digits.
