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Numerical Methods
Jeffrey R. Chasnov
Adapted for:
Numerical Methods for EngineersClick to view a promotional video The Hong Kong University of Science and TechnologyDepartment of Mathematics
Clear Water Bay, Kowloon
Hong KongCopyright
c○2012 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.Preface
What follows were my lecture notes for Math 3311:Introduction to Numerical Meth- ods, taught at the Hong Kong University of Science and Technology. Math 3311, with two lecture hours per week, was primarily for non-mathematics majors and was required by several engineering departments. I also have some free online courses on Coursera. A lot of time and effort has gone into their production, and the video lectures for these courses are of high quality. You can click on the links below to explore these courses. If you want to learn differential equations, have a look atDifferential Equations for Engineers
If your interests are matrices and elementary linear algebra, tryMatrix Algebra for Engineers
If you want to learn vector calculus (also known as multivariable calculus, or calcu- lus three), you can sign up forVector Calculus for Engineers
And if your interest is numerical methods, have a go atNumerical Methods for Engineers
JeffreyR. Chasnov
Hong Kong
February 2021
iiiContents
1 IEEE Arithmetic
11.1 Definitions
11.2 Numbers with a decimal or binary point
11.3 Examples of binary numbers
11.4 Hex numbers
11.5 4-bit unsigned integers as hex numbers
11.6 IEEE single precision format:
21.7 Special numbers
21.8 Examples of computer numbers
31.9 Inexact numbers
31.9.1 Find smallest positive integer that is not exact in single precision
41.10 Machine epsilon
41.11 IEEE double precision format
51.12 Roundoff error example
52 Root Finding
72.1 Bisection Method
72.2 Newton"s Method
72.3 Secant Method
72.3.1 Estimatep2=1.41421356 using Newton"s Method. . . . . . . 8
2.3.2 Example of fractals using Newton"s Method
82.4 Order of convergence
92.4.1 Newton"s Method
92.4.2 Secant Method
103 Systems of equations
133.1 Gaussian Elimination
133.2LUdecomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Partial pivoting
163.4 Operation counts
183.5 System of nonlinear equations
204 Least-squares approximation
234.1 Fitting a straight line
234.2 Fitting to a linear combination of functions
245 Interpolation
275.1 Polynomial interpolation
275.1.1 Vandermonde polynomial
275.1.2 Lagrange polynomial
285.1.3 Newton polynomial
285.2 Piecewise linear interpolation
295.3 Cubic spline interpolation
305.4 Multidimensional interpolation
33v
CONTENTS
6 Integration
356.1 Elementary formulas
356.1.1 Midpoint rule
356.1.2 Trapezoidal rule
366.1.3 Simpson"s rule
366.2 Composite rules
366.2.1 Trapezoidal rule
376.2.2 Simpson"s rule
376.3 Local versus global error
386.4 Adaptive integration
397 Ordinary differential equations
417.1 Examples of analytical solutions
417.1.1 Initial value problem
417.1.2 Boundary value problems
427.1.3 Eigenvalue problem
437.2 Numerical methods: initial value problem
437.2.1 Euler method
447.2.2 Modified Euler method
447.2.3 Second-order Runge-Kutta methods
457.2.4 Higher-order Runge-Kutta methods
467.2.5 Adaptive Runge-Kutta Methods
477.2.6 System of differential equations
477.3 Numerical methods: boundary value problem
487.3.1 Finite difference method
487.3.2 Shooting method
507.4 Numerical methods: eigenvalue problem
517.4.1 Finite difference method
517.4.2 Shooting method
53vi CONTENTS
Chapter 1
IEEE Arithmetic
1.1Definitions
Bit = 0 or 1
Byte = 8 bits
Word = Reals: 4 bytes (single precision)
8 bytes (double precision)
= Integers: 1, 2, 4, or 8 byte signed1, 2, 4, or 8 byte unsigned
1.2Numbers with a decimal or binar ypoint
Decimal: 10
3102101100101102103104
Binary: 2
322212021222324
1.3Examples of binar ynumbers
Decimal Binary
1 1 2 10 3 11 4 1000.5 0.1
1.5 1.1
1.4Hex numbers
1.54-bit unsigned integers as hex numbers
Decimal Binary Hex
1 0001 1
2 0010 2
3 0011 3
10 1010 a
15 1111 f
11.6. IEEE SINGLE PRECISION FORMAT:
1.6IEEE single precision for mat:
s z}|{0e z}|{12345678f
z}|{9 31
# =(1)s2e1271.f where s = sign e = biased exponent p=e-127 = exponent1.f = significand (use binary point)
1.7Special numbers
Smallest exponent: e = 0000 0000, represents denormal numbers (1.f!0.f) Largest exponent: e = 1111 1111, represents¥, if f = 0 e = 1111 1111, represents NaN, if f6=0Number Range: e = 1111 1111 = 2
8- 1 = 255 reserved
e = 0000 0000 = 0 reserved so, p = e - 127 is1 - 127p254-127
-126p127Smallest positive normal number
= 1.0000 0000 00002126 '1.21038 bin: 0000 0000 1000 0000 0000 0000 0000 0000 hex: 00800000MATLAB: realmin("single")
Largest positive number
= 1.1111 1111 11112127 =(1+ (1223))2127 '2128'3.41038 bin: 0111 1111 0111 1111 1111 1111 1111 1111 hex: 7f7fffffMATLAB: realmax("single")
Zero bin: 0000 0000 0000 0000 0000 0000 0000 0000 hex: 00000000Subnormal numbers
Allow 1.f!0.f (in software)
Smallest positive number = 0.0000 0000 00012126 = 2232126'1.41045
2 CHAPTER 1. IEEE ARITHMETIC
1.8. EXAMPLES OF COMPUTER NUMBERS
1.8Examples of computer numbers
What is 1.0, 2.0 & 1/2 in hex ?
1.0= (1)02(127127)1.0
bin: 0011 1111 1000 0000 0000 0000 0000 0000 hex: 3f80 00002.0= (1)02(128127)1.0
bin: 0100 00000 1000 0000 0000 0000 0000 0000 hex: 4000 00001/2= (1)02(126127)1.0
bin: 0011 1111 0000 0000 0000 0000 0000 0000 hex: 3f00 0000 1.9Inexact numbers
Example:
13 = (1)014 (1+13 so thatp=e127=2 ande=125=1283, or in binary,e=01111101. How is f=1/3 represented in binary? To compute binary number, multiply successively by 2 as follows:0.333... 0.
0.666... 0.0
1.333... 0.01
0.666... 0.010
1.333... 0.0101
etc. so that 1/3 exactly in binary is 0.010101.... With only 23 bits to representf, the number is inexact and we have f=01010101010101010101011, where we have rounded to the nearest binary number (here, rounded up). The machine number 1/3 is then represented as00111110101010101010101010101011
or in hex3eaaaaab.
CHAPTER 1. IEEE ARITHMETIC 3
1.10. MACHINE EPSILON
1.9.1 Find smallest positiv einteger that is not exact in single pre- cision LetNbe the smallest positive integer that is not exact. Now, I claim thatN2=2231.11...1,
andN1=2241.00...0.
The integerNwould then require a one-bit in the 224position, which is not avail- able. Therefore, the smallest positive integer that is not exact is 224+1=16777217.
In MATLAB, single(2
24) has the same value as single(224+1). Since single(224+1)
is exactly halfway between the two consecutive machine numbers 224and 224+2,
MATLAB rounds to the number with a final zero-bit in f, which is 2 24.1.10
Machine epsilon
Machine epsilon(emach)is the distance between 1 and the next largest number. If0d x+y=x(1+y/x), if 0y/xFindemach The number 1 in the IEEE format is written as
1=201.000...0,
with 23 0"s following the binary point. The number just larger than 1 has a 1 in the 23rd position after the decimal point. Therefore,
e mach=2231.192107. What is the distance between 1 and the number just smaller than 1? Here, the number just smaller than one can be written as 2 11.111...1=21(1+ (1223)) =1224
Therefore, this distance is 2
24=emach/2.
The spacing between numbers is uniform between powers of 2, with logarithmic spacing of the powers of 2. That is, the spacing of numbers between 1 and 2 is 2 23,
between 2 and 4 is 2 22, between 4 and 8 is 221, etc. This spacing changes for
denormal numbers, where the spacing is uniform all the way down to zero. Find the machine number just greater than5
A rough estimate would be 5(1+emach) =5+5emach, but this is not exact. The exact answer can be found by writing 5=22(1+14
so that the next largest number is 2 2(1+14
+223) =5+221=5+4emach.
4 CHAPTER 1. IEEE ARITHMETIC
1.11. IEEE DOUBLE PRECISION FORMAT
1.11 IEEE double precision for mat
Most computations take place in double precision, where round-off error is re- duced, and all of the above calculations in single precision can be repeated for double precision. The format is sz}|{0e z}|{ 1234567891011f
z}|{ 12 63
# =(1)s2e10231.f where s = sign e = biased exponent p=e-1023 = exponent 1.f = significand (use binary point)
1.12 Roundof ferror example
Consider solving the quadratic equation
x 2+2bx1=0,
wherebis a parameter. The quadratic formula yields the two solutions x =bpb 2+1. Consider the solution withb>0 andx>0 (thex+solution) given by x=b+pb 2+1. (1.1)
Asb!¥,
x=b+pb 2+1 =b+bp1+1/b2 =b(p1+1/b21) b 1+12b21
12b. Now in double precision, realmin2.210308and we would likexto be accurate to this value before it goes to 0 via denormal numbers. Therefore,xshould be computed accurately tob1/(2realmin)210307. What happens if we compute ( 1.1 ) directly? Thenx=0 whenb2+1=b2, or 1+1/b2=1. That is 1/b2=emach/2, orb=p2/
pe mach108. CHAPTER 1. IEEE ARITHMETIC 5
1.12. ROUNDOFF ERROR EXAMPLE
For a subroutine written to compute the solution of a quadratic for a general user, this is not good enough. The way for a software designer to solve this problem is to compute the solution forxas x=1b(1+p1+1/b2). In this form, if 1+1/b2=1, thenx=1/2bwhich is the correct asymptotic form. 6 CHAPTER 1. IEEE ARITHMETIC
Chapter 2
Root Finding
Solvef(x) =0 forx, when an explicit analytical solution is impossible. 2.1 Bisection Method
The bisection method is the easiest to numerically implement and almost always works. The main disadvantage is that convergence is slow. If the bisection method results in a computer program that runs too slow, then other faster methods may be chosen; otherwise it is a good choice of method. We want to construct a sequencex0,x1,x2,... that converges to the rootx=r that solvesf(x) =0. We choosex0andx1such thatx0
The number 1 in the IEEE format is written as
1=201.000...0,
with 23 0"s following the binary point. The number just larger than 1 has a 1 in the23rd position after the decimal point. Therefore,
e mach=2231.192107. What is the distance between 1 and the number just smaller than 1? Here, the number just smaller than one can be written as 211.111...1=21(1+ (1223)) =1224
Therefore, this distance is 2
24=emach/2.
The spacing between numbers is uniform between powers of 2, with logarithmic spacing of the powers of 2. That is, the spacing of numbers between 1 and 2 is 2 23,between 2 and 4 is 2