Wai-Fah Chen.The Civil Engineering Handbook

Note that .

The disposition of the differences and sums relative to the function values is as shown (the arguments

are omitted in these cases in the interest of clarity).

In the forward difference scheme, the subscripts are seen to move forward into the difference table

and no fractional subscripts occur. In the backward difference scheme, the subscripts lie on diagonals

slanting backward into the table, while in the central difference scheme, the subscripts maintain their

positions and the odd-order subscripts are fractional.

All three, however, are merely alternative ways of labeling the same numerical quantities, as any

difference is the result of subtracting the number diagonally above it in the preceding column from that

diagonally below it in the preceding column, or, alternatively, it is the sum of the number diagonally

above it in the subsequent column with that immediately above it in its own column.

Calculus of Finite Differences

Forward difference scheme Backward difference scheme

Central difference scheme

∆

2

f

p

∆

p

2

=

∆∆

p

⋅=

∆ f

p 1+

f

p

–()=

∆=

p 1+

∆

p

–

f=

p 2+

f

p 1+

– f

p 1+

– f

p

+

f=

p 2+

2f

p 1+

– f

p

+

f

p

∆

p

0

∇

p

0

δ

p

0

≡≡≡

∆

1–

2–

f

2–

∆

2–

2

∆

1–

∆

2–

∆

3–

3

∆

0

2–

f

1–

∆

2–

2

∆

0

1–

∆

1–

∆

2–

3

∆

1

2–

f

0

∆

1–

2

∆

1

1–

∆

0

∆

1–

3

∆

2

2–

f

1

∆

0

2

∆

2

1–

∆

1

∆

0

3

∆

3

2–

f

2

∆

1

2

∇

3–

2–

f

2–

∇

1–

2

∇

2–

1–

∇

1–

∇

0

2

∇

2–

f

1–

∇

0

2

∇

1–

∇

0

∇

1

3

∇

1–

2–

f

0

∇

1

2

∇

0

1–

∇

1

∇

2

3

∇

0

2–

f

1

∇

2

∇

1–

∇

2

∇

3

∇

1

2–

f

2

∇

3

2

δ

2–

f

2–

δ

2–

2

δ

2–

4

δ

1

2

--

–

1–

δ

1

2

--

–

δ

1

2

--

–

3

δ

1–

2–

f

1–

δ

1–

2

δ

1–

4

δ

1

2

--

–

1–

δ

1

2

--

–

δ

1

2

--

–

3

δ

0

2–

f

0

δ

0

2

δ

0

4

δ

1

2

--

1–

δ

1

2

--

δ

1

2

--

3

δ

1

2–

f

1

δ

1

2

δ

1

4

δ

1

2

--

–

1–

δ

1

2

--

δ

1

2

--

3

δ

2

2–

f

2

δ

2

δ

2

4

In general,

If a polynomial of degree r is tabulated exactly, i.e., without any round-off errors, then the r th

differences are constant.

The following table enables the simpler operators to be expressed in terms of the others:

In addition to the above, there are other identities by means of which the above table can be extended,

such as

Note the emergence of Taylor’s series from

Interpolation

Finite difference interpolation entails taking a given set of points and ﬁtting a function to them. This

function is usually a polynomial. If the graph of

f (x) is approximated over one tabular interval by a chord

of the form

y = a + bx chosen to pass through the two points

the formula for the interpolated value is found to be

If the graph of f (x) is approximated over two successive tabular intervals by a parabola of the form

y = a + bx + cx

2

chosen to pass through the three points

∆

n

p

1

2

--

n–

δ

p

n

∇

p

1

2

--

n+

.

n

≡≡

E ∆

δµ

, ∇

E –– 1 ∆+ 1

µδ

1

2

---

δ

2

++ 1 ∇–()

1–

∆ E 1– ––

µδ

1

2

---

δ

2

+ ∇ 1 ∇–()

1–

δ

E

1

2

--

E

1

2

--

–

– ∆ 1 ∆+()

1

2

--

–

2

µ

2

1–()

1

2

--

∇ 1 ∇–()

1

2

--

–

∇ E

1–

– ∆ 1 ∆+()

1–

µδ

1

2

---

δ

2

– ––

µ

1

2

---

E

1

2

--

E

1

2

--

–

+





1

2

---

2 ∆+()1 ∆+()

1

2

--

–

1

4

---

δ

2

+()

1

2

--

1

2

---

2 ∇–()1 ∇–()

1

2

--

–

Ee

hD

∆∇

1–

==

µ

E

1

2

----

–

1

2

---

δ

+ E

1

2

--

1

2

---

δ

–

1

2

---

hD()cosh= ==

δ

E

1

2

--

–

∆ E

1

2

--

∇∆∇()

1

2

--

== 2

1

2

---

hD()sinh= =

f

p

E

p

f

0

=

e

phD

f

0

=

f

0

phDf

0

1

2!

----

p

2

h

2

D

2

f

0

L++ +=

x

0

fx

0

(),() x

0

hfx

0

h+(),+( ),

f

x

0

ph+()fx

0

()pfx

0

h+()fx

0

()–[ ]+=

fx

0

()p∆f

0

+=

the formula for the interpolated value is found to be

Using polynomial curves of higher order to approximate the graph of f (x), a succession of interpolation

formulas involving higher differences of the tabulated function can be derived. These formulas provide,

in general, higher accuracy in the interpolated values.

Newton’s Forward Formula

Newton’s Backward Formula

Gauss’ Forward Formula

Gauss’ Backward Formula

Stirling’s Formula

Steffenson’s Formula

x

0

fx

0

(),() x

0

hfx

0

h+(),+( ), x

0

2hfx

0

2h+(),+( ),

f

x

0

ph+()fx

0

()pfx

0

h+()fx

0

()–[ ]+=

pp 1–()

2!

-------------------

fx

0

2h+()2fx

0

h+()– fx

0

()+[ ]+

f

0

pf

0

pp 1–()

2!

-------------------

∆

2

f

0

+∆+=

f

p

f

0

p∆

0

1

2!

----

pp 1–()∆

0

2

1

3!

----

pp 1–()p 2–()∆

0

3

L 0 p 1≤≤++ +=

f

p

f

0

p∇

0

1

2!

----

pp 1+()∇

0

2

1

3!

----

pp 1+()p 2+()∇

0

3

L 0 p 1≤≤++ +=

f

p

f

0

p

δ

1

2

--

G

2

δ

0

2

G

3

δ

1

2

--

3

G

4

δ

0

4

G

5

δ

1

2

--

5

L 0 p 1≤≤++ +++=

f

p

f

0

p

δ

1

2

--

–

G

∗

2

δ

0

2

G

3

δ

1

2

--

–

3

G

∗

4

δ

0

4

G

5

δ

1

2

--

–

5

L 0 p 1≤≤++ ++ +=

In the above, G

2n

pn1–+

2n





=

G

∗

2n

pn+

2n





=

G

2n 1+

pn+

2n 1+





=

f

p

f

0

1

2

---

p

δ

1

2

--

δ

1

2

--

–

+





1

2

---

p

2

δ

0

2

S

3

δ

1

2

--

3

δ

1

2

--

–

3

+





S

4

δ

0

4

L

1

2

---

p

1

2

-

--

≤≤–+ ++ ++=

f

p

f

0

1

2

---

pp 1+()

δ

1

2

--

1

2

---

p 1–()p

δ

1

2

--

–

– S

3

S

4

+()

δ

1

2

--

3

S

3

S

4

–()

δ

1

2

--

–

3

L

1

2

---

p

1

2

---

≤≤–+ + +=

Bessel’s Formula

Everett’s Formula

The coefﬁcients in the above two formulae are related to each other and to the coefﬁcients in the Gaussian

formulae by the identities

Also, for q  1 – p the following symmetrical relationships hold:

as can be seen from the tables of these coefﬁcients.

Bessel’s Formula (Unmodiﬁed)

In the above, S

2n 1+

1

2

--

pn+

2n 1+





=

S

2n 2+

p

2n 2+

---------------

pn+

2n 1+





=

S

2n 1+

S

2n 2+

+

pn1++

2n 2+





=

S

2n 1+

S

2n 2+

–

pn+

2n 2+





–=

f

p

f

0

p

δ

1

2

--

B

2

δ

0

2

δ

1

2

+





B

3

δ

1

2

--

3

B

4

δ

0

4

δ

1

4

+()B

5

δ

1

2

--

5

L 0 p 1≤≤++ ++ ++=

f

p

1 p–()f

0

pf

1

E

2

δ

0

2

F

2

δ

1

2

E

4

δ

0

4

F

4

δ

1

4

E

6

δ

0

6

F

6

δ

1

6

L 0 p 1≤≤++ ++++++=

B

2n

1

2

---

G

2n

1

2

---

E

2n

F

2n

+()≡≡

B

2n 1+

G

2n 1+

1

2

---

G

2n

1

2

---

F

2n

E

2n

–()≡–≡

E

2n

G

2n

G

2n 1+

B

2n

B

2n 1+

–≡–≡

F

2n

G

2n 1+

B

2n

B

2n 1+

+≡≡

B

2n

p() B

2n

q()≡

B

2n 1+

p() B–

2n 1+

q()≡

E

2n

p() F

2n

q()≡

F

2n

p() E

2n

q()≡

f

p

f

0

p

δ

1

2

--

B

2

δ

0

2

δ

1

2

+





B

3

δ

1

2

--

3

B

4

δ

0

4

δ

1

4

+

()B

5

δ

1

2

--

5

B

6

δ

0

6

δ

1

6

+





B

7

δ

1

2

--

7

L++ ++ ++ ++=

Lagrange’s Interpolation Formula

Newton’s Divided Difference Formula

where

The layout of a divided difference table is similar to that of an ordinary ﬁnite difference table.

where the ’s are deﬁned as follows:

and in general

and

fx()

xx

1

–()xx

2

–()… xx

n

–()

x

0

x

1

–()x

0

x

2

–()… x

0

x

n

–()

--------------------------------------------------------------------

fx

0

()=

xx

0

–()xx

2

–()… xx

n

–()

x

1

x

0

–()x

1

x

2

–()… x

1

x

n

–()

--------------------------------------------------------------------+ fx

1

()

L

xx

0

–()xx

1

–()… xx

n 1–

–()

x

n

x

0

–()x

n

x

1

–()… x

n

x

n 1–

–()

--------------------------------------------------------------------------

fx

n

()++

fx() f

0

xx

0

+()fx

0

x

1

,[] xx

0

–()xx

1

–()fx

0

x

1

x

2

,,[]+ +=

L xx

0

–()xx

1

–()… xx

n 1–

–()fx

0

x

1

, …, x

n

,[

]

++

fx

0

x

1

,[]

f

1

f

0

–

x

1

x

0

–

---------------=

fx

0

x

1

x

2

,,[]

fx

1

x

2

,[]fx

0

x

1

,[]–

x

2

x

0

–

------------------------------------------------=

fx

0

x

1

, …, x

k

,[ ]

fx

1

x

2

, …, x

k

,[ ] fx

0

x

1

, …, x

k 1–

,[ ]–

x

k

x

0

–

------------------------------------------------------------------------------------------=

x

1–

f

1–



1–

2



1–

4



1

2

--–



1

2

--

–

x

0

f

0



0

2



0

4



1

2

--



1

2

--

3

x

1

f

1



1

2



1

4



0

r

f

r



r

1

2

--

+

f

r 1+

f

r

–()x

r 1+

x

r

–()⁄≡,≡



r

2n



r

1

2

--

+

2n 1–



r

1

2

--

–

2n 1–

–

 

 

x

rn+

x

rn–

–( )⁄≡



2n 1+

r

1

2

--

+



r 1+

2n



r

2n

–( ) x

r 1 n++

x

rn–

–( )⁄≡

Iterative Linear Interpolation

Neville’s modiﬁcation of Aiken’s method of iterative linear interpolation is one of the most powerful

methods of interpolation when the arguments are unevenly spaced, as no prior knowledge of the order

of the approximating polynomial is necessary nor is a difference table required.

The values obtained are successive approximations to the required result and the process terminates

when there is no appreciable change. These values are, of course, useless if a new interpolation is required

when the procedure must be started afresh.

Deﬁning

the computation is laid out as follows:

As the iterates tend to their limit, the common leading ﬁgures can be omitted.

Gauss’s Trigonometric Interpolation Formula

This is of greatest value when the function is periodic, i.e., a Fourier series expansion is possible.

where C

r

= N

r

(x)/N

r

(x

r

) and

This is similar to the Lagrangian formula.

Reciprocal Differences

These are used when the quotient of two polynomials will give a better representation of the interpolating

function than a simple polynomial expression.

f

rs,

x

s

x–()f

r

x

r

x–()f

s

–

x

s

x

r

–()

------------------------------------------------

≡

f

rst,,

x

t

x–()f

rs,

x

r

x–()f

st,

–

x

t

x

r

–()

------------------------------------------------------

≡

f

rstu,,,

x

u

x–()f

rst,,

x

t

x–()f

stu,,

–

x

u

x

r

–()

-------------------------------------------------------------

≡

x

1–

x

1–

x–()f

1–

f

1, 0–

x

0

x

0

x–()f

0

f

101,,–

f

01,

f

1012,,,–

x

1

x

1

x–()f

1

f

0, 1, 2

f

12,

x

2

x

2

x–()f

2

f

x() C

r

f

r

r 0=

n

∑

=

N

r

x()

xx

0

–()

2

------------------

sin

xx

1

–()

2

------------------

sin L

xx

r 1–

–()

2

-----------------------

sin

xx

r 1+

–()

2

-----------------------

sin L

xx

n

–()

2

------------------

sin=

A convenient layout is as shown below:

The interpolation formula is expressed in the form of a continued fraction expansion.

The expansion corresponding to Newton’s forward difference interpolation formula, in the sense of

the differences involved, is

while that corresponding to Gauss’ forward formula is

x

1–

f

1–

ρ

1

2

--

–

x

0

f

0

ρ

0

2

ρ

1

2

--

ρ

1

2

--

3

x

1

f

1

ρ

1

2

ρ

1

4

ρ

1

2

--

ρ

1

2

--

3

x

2

f

2

ρ

2

ρ

2

1

2

--

x

3

f

3

where

ρ

r

1

2

--

+

x

r 1+

x

r

–

f

r 1+

f

r

–

--------------------

≡

and

ρ

r

2

x

r 1+

x

r 1–

–

f

r

1

2

--

+

f

r

1

2

--

–

------------------------- f

r

+≡

In general,

ρ

r

1

2

--

+

2n 1+

x

rn1++

x

rn–

–

ρ

2n

r 1+

ρ

2n

r

–

-------------------------------

ρ

2n 1–

r

1

2

--

+

+≡

ρ

r

2n

x

rn+

x

rn–

–

ρ

2n 1–

r 1+

ρ

2n 1–

r

1

2

--

–

----------------------------------------------

ρ

r

2n 2–

+≡

fx() f

0

xx

0

–()

ρ

1

2

--

x

2

x

1

–()+

--------------------------------+=

ρ

1

f

0

– xx

2

–()+

ρ

1

2

--

3

ρ

1

2

--

x

4

x

3

–()+–

ρ

2

4

ρ

1

2

– xx

4

–()+

etc.

Probability

Deﬁnitions

A sample space S associated with an experiment is a set S of elements such that any outcome of the

experiment corresponds to one and only one element of the set. An event

E is a subset of a sample space

S. An element in a sample space is called a sample point or a simple event (unit subset of S).

Deﬁnition of Probability

If an experiment can occur in n mutually exclusive and equally likely ways, and if exactly m of these ways

correspond to an event

E, then the probability of E is given by

If E is a subset of S, and if to each unit subset of S a non-negative number, called its probability, is

assigned, and if

E is the union of two or more different simple events, then the probability of E, denoted

by

P(E ), is the sum of the probabilities of those simple events whose union is E.

Marginal and Conditional Probability

Suppose a sample space S is partitioned into rs disjoint subsets where the general subset is denoted by

E

i

∩ F

j

. Then the marginal probability of E

i

is deﬁned as

and the marginal probability of F

j

is deﬁned as

The conditional probability of E

i

, given that F

j

has occurred, is deﬁned as

and that of F

j

, given that E

i

has occurred, is deﬁned as

fx() f

0

xx

0

–()

ρ

1

2

--

x

2

x

1

–()+

--------------------------------+=

ρ

0

2

f

0

– x

3

x

1–

–()+

ρ

1

2

--

3

ρ

1

2

--

– x

4

x

2

–()+

ρ

0

4

ρ

0

2

– xx

2–

–()+

etc.

PE()

m

n

----=

PE

i

() = P(E

i

F

j

)∩

j 1=

s

∑

PF

j

() = PE

i

F

j

∩()

i 1=

r

∑

PE

i

F

j

⁄()

PE(

i

F

j

)∩

PF

j

)(

-------------------------

, PF

j

()0≠=

PF

j

E

i

⁄()

PE(

i

F

j

)∩

PE

i

)(

-------------------------

, PE

i

()0≠=

Probability Theorems

1. If φ is the null set, P(φ ) = 0.

2. If S is the sample space, P(S) = 1.

3. If E and F are two events,

4. If E and F are mutually exclusive events,

5. If E and E

′

are complementary events,

6. The conditional probability of an event E, given an event F, is denoted by P(E/F) and is deﬁned as

where P(F ) ≠ 0.

7. Two events E and F are said to be independent if and only if

E is said to be statistically independent of F if P(E/F ) = P(E ) and P(F/E ) = P(F ).

8. The events E

1

, E

2

, K, E

n

are called mutually independent for all combinations if and only if every

combination of these events taken any number at a time is independent.

9. Bayes Theorem.

If E

1

, E

2

, K, E

n

are n mutually exclusive events whose union is the sample space S, and E is any

arbitrary event of

S such that P(E ) ≠ 0, then

Random Variable

A function whose domain is a sample space S and whose range is some set of real numbers is called a

random variable, denoted by

X. The function X transforms sample points of S into points on the x-axis.

X will be called a discrete random variable if it is a random variable that assumes only a ﬁnite or

denumerable number of values on the

x-axis. X will be called a continuous random variable if it assumes

a continuum of values on the

x-axis.

Probability Function (Discrete Case)

The random variable X will be called a discrete random variable if there exists a function f such that

f (x

i

) ≥ 0 and for i = 1, 2, 3, K and such that for any event E,

where means sum f (x) over those values x

i

that are in E and where f (x) = P[X = x].

PE F∪()PE() PF() PE F∩()–+=

PE F∪()PE() PF()+=

PE() 1 PE′()–=

PE F⁄()

PE( F )∩

PF)(

-----------------------=

PE F∩()PE()PF()⋅=

PE

k

E⁄()

PE

k

()PE E

k

⁄()⋅

PE

j

()PE E

j

⁄()⋅[ ]

j 1=

n

∑

--------------------------------------------------=

fx

i

()

i

∑

1=

PE() P X is in E[]fx()

E

∑

= =

Σ

E

The probability that the value of X is some real number x is given by f (x) = P [X = x], where f is called

the probability function of the random variable

X.

Cumulative Distribution Function (Discrete Case)

The probability that the value of a random variable X is less than or equal to some real number x is

deﬁned as

where the summation extends over those values of i such that x

i

≤ x.

Probability Density (Continuous Case)

The random variable X will be called a continuous random variable if there exists a function f such that

f (x) ≥ 0 and for all x in interval −∞ < x < ∞ and such that, for any event E,

f (x) is called the probability density of the random variable X. The probability that X assumes any given

value of

x is equal to zero, and the probability that it assumes a value on the interval from a to b, including

or excluding either endpoint, is equal to

Cumulative Distribution Function (Continuous Case)

The probability that the value of a random variable X is less than or equal to some real number x is

deﬁned as

From the cumulative distribution, the density, if it exists, can be found from

From the cumulative distribution

Mathematical Expectation

Expected Value

Let X be a random variable with density f (x). Then the expected value of X, E (X), is deﬁned to be

Fx() P X x≤()=

Σfx

i

(), ∞– x ∞<<=

fx()xd

∞–

∞

∫

1=

PE() P X is in E()fx()xd

E

∫

= =

fx()xd

a

b

∫

Fx() P X x≤(), ∞– x ∞<<=

fx() x.d

∞–

∞

∫

=

fx()

dF x()

dx

--------------=

Pa X b≤≤()P X b≤()P X a≤()–=

Fb() Fa()–=

E X() xf x()

x

∑

=

Wai-Fah Chen.The Civil Engineering Handbook

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