This document provides an overview of numerical methods. It defines numerical methods as algorithms that generate approximate solutions to mathematical problems through repetitive calculations. It notes that numerical methods are needed when analytical methods are not possible, too time-consuming or involve large data. The document also discusses sources of error in numerical solutions, different types of errors (absolute, relative, percentage), rounding numbers and significant digits.

1. Numerical
Methods
Lecture-1
[MA-200]
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2. Numerical Method
The process of obtaining a solution is to reduce the original problem to a
repetition of the same step or series of steps so that computations become
automatic such a process is called a numerical methods and a numerical method,
which can be used to solve a problem is called algorithm.
 Numerical methods are algorithms:
 Precise sets of rules to follow.
 Generate a Numerical approximation of the mathematical problem.
 Do not tell how good/poor approximation is.
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3. Need of Numerical Methods
 Analytical Methods may not exist at all to solve the problem.
 Analytical methods exist but are time consuming, very laborious to apply or
may be huge data is involved or may not be easily computed.
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4. Error
Error is the difference between the actual value and Approximated (Calculated)
value.
If we denote the actual value with 𝒙 and the calculated value with 𝒙∗
then error will be,
∈ = 𝐸 = 𝑥 − 𝑥∗
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5. Example
𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 = 𝟕. 𝟖𝟗𝟑 𝑎𝑛𝑑 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 = 𝟕. 𝟔𝟕𝟐
𝑆𝑜, 𝐸𝑟𝑟𝑜𝑟 = 𝟎. 𝟐𝟐𝟏
𝐸𝑟𝑟𝑜𝑟 𝑚𝑎𝑦 𝑏𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒, 𝑜𝑟 𝑒𝑟𝑟𝑜𝑟 𝑚𝑎𝑦 𝑏𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
𝐼𝑓 𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 = 𝟕. 𝟔𝟕𝟐 𝑎𝑛𝑑 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 = 𝟕. 𝟖𝟗𝟑
𝑆𝑜, 𝐸𝑟𝑟𝑜𝑟 = −𝟎. 𝟐𝟐𝟏
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6. Example
The derivative of a function 𝑓 𝑥 at a particular value of 𝑥 can be approximately
calculated by
𝑓′ 𝑥 ≡
𝑓 𝑥 + ℎ − 𝑓(𝑥)
ℎ
Find 𝑓′
2 for 𝑓 𝑥 = 7𝑒0.5𝑥
and ℎ = 0.3, Find
a) The approximate value of 𝑓′ 2
b) The actual value of 𝑓′ 2
c) The Error
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7. Cont…
a) 𝑓′ 𝑥 ≡
𝑓 𝑥+ℎ −𝑓(𝑥)
ℎ
,
𝑥 = 2 𝑎𝑛𝑑 ℎ = 0.3
𝑓′ 2 ≡
𝑓 2 + 0.3 − 𝑓(2)
0.3
𝑓′ 2 =
𝑓 2.3 − 𝑓(2)
0.3
=
22.107 − 19.028
0.3
= 10.265
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8. Cont…
b) 𝑓 𝑥 = 7𝑒0.5𝑥
𝑓′
𝑥 = 7 ∗ 0.5 ∗ 𝑒0.5𝑥
𝑓′ 2 = 7 ∗ 0.5 ∗ 𝑒0.5(2)
= 9.5140
c) Error = Actual Value – Approximated Value
𝐸𝑟𝑟𝑜𝑟 = 9.5140 − 10.265
= −0.75061
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9. Types of Error
We have three types of errors
 Absolute Error
 Relative Error
 Percentage Error
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10. Absolute Error
If 𝒙 is the actual value and 𝒙∗ is the calculated value, then
𝐴. 𝐸 = 𝐸𝑎 = |𝑥 − 𝑥∗|
For example,
𝐼𝑓 𝑝 = 0.3000 ∗ 101
𝑎𝑛𝑑 𝑝∗
= 0.3100 ∗ 101
𝐸𝑎 = −0.1 = 0.1
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11. Relative Error
Relative error is the ratio of the absolute error to the actual value
𝑅. 𝐸 = 𝐸𝑅 =
𝐸𝑎
|𝑥|
=
|𝑥−𝑥∗|
|𝑥|
For example,
𝐼𝑓 𝑃 = 0.3000 ∗ 101
𝑎𝑛𝑑 𝑝∗
= 0.3100 ∗ 101
𝐸𝑅 =
𝐸𝑎
|𝑝|
=
0.1
|0.30000 ∗101|
= 0.03333 = 0.3333 ∗ 10−1
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12. Percentage Error
Relative Error expresses in terms of a percentage error
𝑃. 𝐸 =
𝐸𝑎
|𝑥|
∗ 100
For example,
𝐼𝑓 𝑃 = 0.3000 ∗ 101
𝑎𝑛𝑑 𝑝∗
= 0.3100 ∗ 101
𝐸𝑅 =
𝐸𝑎
|𝑝|
=
0.1
|0.30000 ∗101|
= 0.03333 = 0.3333 ∗ 10−1
𝑃. 𝐸 =
𝐸𝑎
|𝑝|
∗ 100 = 0.03333 ∗ 100 = 3.333%
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13. If we Don’t have Actual Value
 𝐴. 𝐸 = 𝐸𝑎 = |𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 − 𝑃𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒|
 𝑅. 𝐸 = 𝐸𝑅 =
𝐸𝑎
|𝑥|
=
|𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐸𝑠𝑖𝑡𝑖𝑚𝑎𝑡𝑒 −𝑃𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒|
|𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒|
𝑃. 𝐸 =
𝐸𝑎
|𝑥|
∗ 100
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14. Homework
 If 𝑃 = 0.3000 ∗ 10−3
𝑎𝑛𝑑 𝑝∗
= 0.3100 ∗ 10−3
, Find the absolute error and
relative error and percentage error.
 If 𝑃 = 0.3000 ∗ 104
𝑎𝑛𝑑 𝑝∗
= 0.3100 ∗ 104
, Find the absolute error and
relative error and percentage error
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15. Significant Digits
A significant digit in a number is a digit which gives a reliable information about
the size of a number
Rules:
 Every nonzero digit is significant E.g.,
 8.7456 has five significant digits
 495 has three significant digits
 Zero between nonzero digits are always significant. E.g.
 2047 has four significant digits,
 50.032 has five significant digits.
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16. Cont…
 Zeros before nonzero digits are not significant. E.g,
 0.0123 has three significant digits
 0.00000000123 has three significant digits
 Zeros after decimal point are significant. E.g,
 3.000 has four significant digits.
 Zeros behind significant numbers are sometimes significant E.g,
 2000 has one significant digit.
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17. Examples
 6.4320 ℎ𝑎𝑠 𝑓𝑖𝑣𝑒 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠
 0.06432 ℎ𝑎𝑠 𝑓𝑜𝑢𝑟 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠
 64 ℎ𝑎𝑠 𝑡𝑤𝑜 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠
 64.0 ℎ𝑎𝑠 𝑡ℎ𝑟𝑒𝑒 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠
 6.432 ∗ 104 ℎ𝑎𝑠 𝑓𝑜𝑢𝑟 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠
 6.43200 ℎ𝑎𝑠 𝑠𝑖𝑥 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠
 5000 ℎ𝑎𝑠 𝑜𝑛𝑒 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡
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18. Rounding the Numbers
Rule 1:
 If the last digit is less than 5 then it is simply dropped.
 e.g., 1.943 is rounded to 1.94
Rule 2:
 If the last digit is greater than 5 then the previous digit increased by one.
 e.g., 1.47 is rounded to 1.5
Rule 3:
 If the last digit is 5, then it is rounded to get nearest even
 e.g., 1.35 is rounded to 1.4 & 1.45 is rounded to 1.4
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