Introduction to computing

1
Introduction to computing

• Dr. Asaf Varol
• avarol_at_mix.wvu.edu

2
Basic operation performed by a computer

• Arithmetic Operations Addition, subtraction,
  multiplication and division
• Logical operations the sign or the comparative
  magnitude of two numbers
• Data transfer Moving data from one location to
  another in the memory.
• Input-output operations Controlling the
  reading/writing of information into or out of the
  computer

3
On Digital Computers

• Digital computers store numbers in an entity (or
  unit) called a word which consists of a string of
  binary digits, or bits. Various number systems
  are used to represent mathematical numbers. Some
  commonly used number systems are hexadecimal
  (base 16), decimal (base 10), octal (base 8), and
  binary (base 2). For example, in a decimal
  system the number 8,410 is represented in powers
  of ten as
• 8?103 4?102 1?101 0?100 8000 400 10
  0 8,410

4
On Digital Computers (Contd)

• A method known as the doubling procedure is as
  follows. Given a decimal number N, it can be
  decomposed as
• N 2Q1 R1 (Q1 N/2 - remainder)
• Q1 2Q2 R2 (Q2 Q1/2 - remainder)
• . .
• . .
• . .
• Qk 0 Rk1 etc.
• The corresponding binary number is obtained by
  writing the remainders Rk1, Rk, ... , R1 in the
  reverse order as
• B Rk1RkRk-1 ... R1

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Example

• Convert the decimal number N 8,410, to a
  binary number.
• Solution
• Perform sequential division by 2 as follows
• 8,410 (2?4,205) 0 65 (2?32) 1
• 4,205 (2?2,102) 1 32 (2?16) 0
• 2,102 (2?1,051) 0 16 (2?8) 0
• 1,051 (2?525) 1 8 (2?4) 0
• 525 (2?262) 1 4 (2?2) 0
• 262 (2?131) 0 2 (2?1) 0
• 131 (2?65) 1 1 (2?0) 1
• The binary equivalent of 8,410 is then given by
  collecting the remainder digits from the last to
  the first
• 10000011011010 1?1213 0?212 0?211 0?210
  0?29 0?28 1?27 1?26 0?25 1?24 1?23
  0?22 1?21 0?20

6
Representations of Numbers

• Numbers are usually represented using the
  normal form notation. That is,
• x m.10E for 10-1 lt ?m? lt 1
• where for x ? 0, m is called the mantissa and E
  is the exponent. By convention, the number zero
  has the normal notation, 0.100.

7
Significant Digits

• If a number is written in standard decimal,
  floating-point form, or in normal form such that
• x 0.d1 d2 d3 ... dk?10n
• with d1 ? 0 and dk ? 0, we say that this number
  has k significant digits (or significant figures)
  which indicates those digits that can be used
  with a confidence relative to the true value of
  the number.

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Significant Digits (Contd)

• Note that the zeros which are used only to shift
  the decimal point are not counted as significant
  figures. The leading zeros may or may not be
  significant. For example,
• x 0.0002815 has 4 significant figures!
• x 1,200 may have 4, 3, or 2 significant
  figures!
• Some examples are
• 46.45072800 0.46450728?102 (with 8, 9, or
  10 significant digits)
• -335.12 -0.33512?103 (with 5
  significant digits)
• 0.00517 0.517?10-3 (with 3
  significant digits)
• 0.74 0.74?100 (with 2
  significant digits)

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Computer Representation of Numbers

• The decimal equivalent of the binary number
  represented in Figure 1.4.1 is given by
• -(0?26 0?25 0?24 1?23 0?22 1?21
  1?20)
• -(0 0 0 8 0 2 1)
• -11

• Figure 1.4.1 Binary representation of an integer
  using an
• 8 bit word (or Byte)

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Example

• Determine the largest integer that can be
  represented by an 8 bit machine.
• Solution
• Imax (1?26 1?25 1?24 1?23
  1?22 1?21 1?20)
• ( 64 32 16 8 4
  2 1 )
• (127)
• (27 - 1)
• In general
• Imax 2(n -1) - 1 Imin -2(n -1) -
  1
• For a binary computer utilizing 32 bit words,
• Imax 2,147,483,647

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Floating-Point Representation

• A floating-point number is written as
• x (sign)m.b(sign)E
• where m is the mantissa, b is the base (b 2
  for a binary system), and E is the exponent.

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Example

• Determine the smallest, positive, nonzero,
  floating point number that can be represented by
  an eight bit machine using binary system with one
  bit spared for the sign of the mantissa, one bit
  for the sign of the exponent, and two bits for
  the digits of the exponent

Solution m (0?23 0?22 0?21 1?20) m
( 0 0 0 1 ) 1 E
-(1?21) (1?20) -(21) - 3 Number 1?2-3
(which is equal to 0.1250 in decimal system)
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Errors
the approximation of numbers, accuracy, and
precision. Neither physical measurements nor
arithmetic calculations can be carried out
exactly. The engineer's motto should be There
is nothing which is absolutely correct or exact
in science Accuracy is the measure of how close
an estimated value or answer is to its true (or
exact) value. Since in many situations this
exact value is not known, the accuracy of an
answer is usually measured with respect to the
best-estimated value. Precision implies how
closely the repeatedly measured (or calculated)
values of a certain quantity agree with each
other. Thus it represents the number of
significant figures in representing that quantity
as a single average number with a spread
(variation) around its mean.

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Errors
True absolute error Et ?(true value) -
(approximate value)? Approximate absolute
error Ea ?(best estimate) - (approximate
value)? True relative error et ?(true
value) - (approximate value)?/?true value?
Approximate relative error ea ?(best
estimate) - (approximate value)?/?best estimate?

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Errors (Contd)

• For series, sequences, and iterations,
  approximate error can be defined as
• Approximate absolute error (for iterative
  calculations)
• Ea ?(current value) - (previous value)?
• Approximate relative error (for iterative
  calculation)
• ea (?(current value) - (previous
  value)?)/?current value?
• When the exact value of a quantity is not given
  (or known), it is not possible to calculate a
  true error. However, the approximate error can be
  used to calculate error bounds for an approximate
  answer. To this end, a theorem called Scarborough
  criteria can be very useful.
• Scarborough Criteria
• If the approximate relative error, ea lt
  0.5?10-m, then the result (or the answer) is
  correct to at least m significant digits.

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Example E1.5.1

• Calculate the number of terms that is necessary
  to estimate the value of ? to two significant
  digits from the Taylor series expansion for
  Arctan(x) about the base point x 0. The number
  ? 3.141592653589793 (to 16 significant digits)
  is related to this function by
• ? 4.0Arctan(1.0)
• Solution
• The infinite series (convergent in the range
  1.0 to 1.0) is given by

where n 1, 2, 3, ... Check your answer by
actually calculating ? from the above series.
Plot the true relative error and the approximate
relative error as a function of the number of
terms.
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Errors (Contd)

• Note that for such series, the definition of ea
  reduces to
• ea ?(current value) - (previous
  value)?/?current value?
• ea ?last term used?/?current sum?
• If we assume that the current sum is accurate to
  two significant digits (? 3.1), then
• ea ? 4? (-1)(n1) x2n-1/(2n-1)?/(3.1) at x 1
• Using Scarborough criteria,
• ea ? (4/3.1)/(2n-1) lt 0.5?10-2
• Solving this equation for n (by trial and error)
  gives n ? 130

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MatLab solution
19
Comparison of the true and approximate relative
error
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Computer errors Round off or Chop off errors

• Most computers chop off (or simply ignore) the
  digits beyond their capacity of representing
  them. That is, when the number of significant
  digits does not fit into the space allocated for
  the mantissa, some computers round the number.
• For example, a computer with a three digit
  mantissa will represent 68.501 as 0.068E03 when
  chopping off is used, or as 0.069E03 when
  rounding is used.

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Computer errors Round off or Chop off errors

• Subtractive cancellation
• This error occurs when subtracting two nearly
  equal numbers. Let us further explain this with
  an example.
• Example
• If x 40,000.01 and y 40,000, what is x minus
  y using a 3-digit mantissa?
• Solution
• 0.4000001x105
• -0.4000000x105
• _______________________
• 0.000 x105 0.0
• Smearing due to round-off errors
• Significant errors can occur when adding a large
  and a small number. For example, adding two
  temperatures, 0.4 K to 250 K, using a
  hypothetical decimal computer with a mantissa of
  3 digits, yields
• 0.250x103
• 0.0004x103
• _______________________
• 0.250 x103 250 K

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Example E1.5.5

• Consider the transient heat equation
• dT/dt -(Ta - T)/?h
• where t is the time, ?h is a characteristic
  thermal relaxation time, and Ta is the ambient
  temperature (the temperature of the
  surroundings).
• Use a finite difference method to find the
  variation of the temperature T of an object,
  initially at a temperature T0 950 K, after it
  is immersed in a fluid having an uniform
  temperature Ta 1000 K
• Tnew Told ?t(Ta - Told)/?h
• where ?t is the time increment between the old
  and new temperature and ?h 1000 sec. We can
  start at time t 0. Set Told T0 950 and
  march in time to calculate T at subsequent times.
  
• The results with ?t 0.1 and 0.001 sec are
  depicted in figure along with the exact solution
  
• T Ta (T0 - Ta)exp(-t/?h)

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Figure E.1.5.5 Illustration of smearing due to
round off
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Truncation error

• Truncation errors are those which usually arise
  from neglecting (cutting) a number of terms in a
  series formulation.
• The geometric series is given by
• in the interval -1 lt x lt 1 for n 0, 1, 2, 3
• when x 0.5, the exact value of ?(x) 2. If
  we represent this function by only three terms (n
  2), then
• ?(0.5) ? 1 x x2 1 0.5 (0.5)2 1.75
• Hence, the truncation error T.E. for n 2, is
  given by
• T.E. ?(0.5)exact - ?(0.5)approx 2 - 1.75
  0.25 Et true error
• or the true relative error,
• et Et/2.0 0.125 (12.5)
• This truncation error is the sum of all the
  terms left out from the infinite series.
• ?(x) 1/(1 - x) ? 1 x x2 (T.E.)n2

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Taylor series expansion approximation

• Taylor series is used in Analysis to derive
• Integration formulas, functional approximations,
  finite difference schemes and error analysis
• A convergent series is a series whose sequence of
  partial sums converges to a finite sum.
• A divergent series is a series that does not
  converge.
• The geometric series converges for -1lt x lt 1. As
  an example let us calculate f(x0.1)
• n0 S01 1.000
• n1 S11 x2 1.100
• n2 S21 x x2 1.110
• n8 S81.0/(1-0.1) 1.111111

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Determining of converges

• To determine whether a series is convergent or
  not, there are two most commonly applied tests
• The nth term test states that the series
• Sumq1q2q3qn
• Converges if qn approaches zero in the limit of
  the nth term must go to zero as n approaches 8.
• According to the ratio test, if a series with the
  sequence of the summation of its term q1, q2, q3,
  .qn has the property that.
• If Rlt1 or R0, the series converges absolutely.
• If Rgt1, the series diverges.
• If R1, the series may or may not converge.

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Example E.1.6.1
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Approximation of Functions Taylor Series

• Taylor's Theorem (Taylors Formula)
• Let ?(x) be a function defined and continuously
  differentiable in the closed interval from a to
  x, and it has continuous derivatives of all
  orders in the same interval. One can then
  express that function in terms of a power series
  as follows
• ?(x) ?(a) ?(1)(a)(x-a) ?(2)(a)(x-a)2/2!
  ?(3)(a)(x-a)3/3!
• where ?(n)(a) denotes the nth order derivative
  of ? with respect to the independent variable x,
  then, evaluated at the point x a. The point x
  a about which the expansion is built is known
  as the base point.
• If the Taylor series is truncated after a finite
  number of terms, then the sum of all the terms
  truncated is called the remainder Rn(x). That is
• ?(x) ?(a) ?(1)(a)(x-a) ?(2)(a)(x-a)2/2!
  ?(3)(a)(x-a)3/3!
• ... ? (n)(a)(x-a)n/n! Rn(x)

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Taylor Series

• The remainder Rn(x) can be written (Stein, 1967)
  in the integral form as
• where t is a dummy variable used for the purpose
  of integration.
• Using the first and second mean value theorems of
  calculus for integrals,
• Eq. 1.6.5 can be written as

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Taylor Series (Contd)

• Different forms of the Taylor series
• In Taylors Theorem, let a xi (meaning the ith
  point) and let x xi1 ah xih denote the
  next point. Then it takes the form
• ?(xi1) ?(xi) ?(1)(xi)h ?(2)(xi)h2/2!
  ?(3)(xi)h3/3! ?(n)(xi)hn/n!
• This form is particularly convenient for
  developing finite difference formulae. This
  equation can be further simplified with a change
  of notation
• ?i ?(xi) ?i1 ?(xi1) ?i(n) ?(n)(xi)
• Hence, we can write
• ?i1 ?i ?i(1)h ?i(2)h2/2!
  ?i(3)h3/3! ?i(n)hn/n! ...
• This form is convenient in deriving formulae for
  the integration of ordinary differential
  equations (see Chapter 6).
• Some simple formulae for computing the
  derivative of a function numerically are given
  here as an introduction to numerical
  differentiation.

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Numerical differentiation

• We truncate the Taylor series after the second
  term and solve for ?(1) ?? to obtain
• ?? (?i1-?i)/h
• or
• Forward Difference
• where h xi1-xi ?x. This is called a first
  order, forward difference approximation for the
  first derivative at the point x xi a. Note
  that the fundamental definition of the derivative
  of a function is
• Hence, Sf should become more and more accurate
  as we make the step size h smaller and smaller.
  However, one must be careful with round off
  errors
• Other formulae can also be derived with the help
  of Taylor series expansion. (see fig. below)
• Backward difference
• Central difference

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Graphical interpretation of finite difference
33
Application of Taylor Series

• Example
• Using Taylor series, expand the function ?(x)
  ln(1x) about the base point a 0. Also, using
  Taylor's formula, approximate this function as a
  first, second, and third degree polynomial.
  Compute ln(1.5) from the Taylor series with
  increasing an number of terms and make a table of
  the true and approximate errors as a function of
  the number of terms used in the summation.
  Further, estimate the truncation error.
• Solution
• ?(x) ln(1x)
• ?(1)(x) 1/(1x)
• ? (2)(x) -1/(1x)2
• ? (3)(x) (2)(1)/(1x)3
• ? (4)(x) -(3)(2)(1)/(1x)4
• . .
• ?(n)(x) (-1)n1(n-1)!/(1x)n
• a 0 ?(0) 0, ? (1)(0) 1, ? (2)(0) -1, ?
  (3)(0) 2, ? (4)(0) -6
• Substituting all of this into Eq. 1.6.4 yields
• ?(x) ln(1x) 0 x - x2/2 x3/3 - x4/4
  ... (-1)n-1 xn/n ..
• To determine for what values of x this series is
  convergent, apply the ratio test

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Application of Taylor Series (Contd)

• Polynomial approximation (truncate Taylor series
  with finite terms)
• One term ln(1x) ? x (straight line)
• Two terms ln(1x) ? x - x2/2 (parabola)
• Three terms ln(1x) ? x - x2/2 x3/3 (cubic
  polynomial)
• These approximate functions are plotted in the
  figure below in comparison with the original
  function, ?(x) ln(1x). It is seen that as we
  take more and more terms, the approximate
  function (polynomial) represents the original
  function better and better in the neighborhood of
  the base point x a 0.

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Application of Taylor Series (Contd)
36
Case Study

• Case Study C1.7.1 Numerical Evaluation of
  Derivatives
• Using the approximate forward difference formula
  for the derivative of a function, calculate
  numerically the derivatives of the following
  functions at the specified points. Make a table
  for each case showing the variation of the exact
  derivative, the numerical derivative, and the
  absolute true error with the step size h. Let h
  vary between 1.0 and 1.E-20 decreasing each time
  by a factor of 10 times.
• (a.) ?(x) Cos(-10x2) at x 0
• (b.) ?(x) e-ln(1/x) at x 1.0
• (c.) ?(x) x/(53x-5) at x 10
• Note ??(x0) ? ?(x0 h) -?(x0)/h.

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Case Study (Contd)

• Let us first determine analytical derivatives
• (a) ? '(x) 20xSin(-10x) ? '(0) 0.
• (b) ? '(x) e-ln(1/x)/x 1 ? '(1) 1.0
• (c) ? '(x) (5 18x-5)/(5 3x-5)2 ? '(10)
  0.2000048
• Numerical results for parts (a.), (b.), and (c.)
  are tabulated in Tables C1.7.1a-c, and shown in
  Figures C1.7.1a-c. For case (b.), note that ?(x)
  eln(x) x. Hence, df/dx 1.0. We obtain the
  exact derivative with any value of h that is not
  very small so that the round off error becomes
  significant. When h becomes very small, we are
  adding a small number h to a large number that
  cannot be handled by the computer. For example,
  for small h
• ?(1 h) - ?(1)/h ? ?(1) - ?(1)/h
  0.0 (wrong !)

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Case Study (Contd)

• Table Variation of the True Error with Step Size
  for Calculating the Derivative of ?(x)
  Cos(-10x2)
• Exact Derivative 0.000000E00
• Numerical Derivative at x 0.000000E00
• h df/dx True Error
• 1.000000E00 -1.839072E00 1.839072E00
• 1.000000E-01 -4.995835E-02 4.995835E-02
• 1.000000E-02 -4.999999E-05 4.999999E-05
• 9.999999E-04 -4.999999E-08 4.999999E-08
• 9.999999E-05 -5.000016E-11 5.000016E-11
• 9.999999E-06 -4.878910E-14 4.878910E-14
• 9.999999E-07 0.000000E00 0.000000E00
• 9.999999E-08 0.000000E00 0.000000E00
• 9.999999E-09 0.000000E00 0.000000E00
• 9.999999E-10 0.000000E00 0.000000E00
• 9.999999E-11 0.000000E00 0.000000E00
• 9.999999E-12 0.000000E00 0.000000E00
• 9.999999E-13 0.000000E00 0.000000E00
• 9.999999E-14 0.000000E00 0.000000E00
• 9.999999E-15 0.000000E00 0.000000E00

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Case Study (Contd)
40
End of Chapter 1

41
References

• Celik, Ismail, B., Introductory Numerical
  Methods for Engineering Applications, Ararat
  Books Publishing, LCC., Morgantown, 2001
• Fausett, Laurene, V. Numerical Methods,
  Algorithms and Applications, Prentice Hall, 2003
  by Pearson Education, Inc., Upper Saddle River,
  NJ 07458
• Rao, Singiresu, S., Applied Numerical Methods
  for Engineers and Scientists, 2002 Prentice Hall,
  Upper Saddle River, NJ 07458
• Mathews, John, H. Fink, Kurtis, D., Numerical
  Methods Using MATLAB Fourth Edition, 2004
  Prentice Hall, Upper Saddle River, NJ 07458
• Varol, A., Sayisal Analiz (Numerical Analysis),
  in Turkish, Course notes, Firat University, 2001