Newtons Method Explanation

Explanation of Newton’s method

Dr. Ammar Hasan

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-4 -3 -2 -1 0 1 2 3 4

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10

Function (Blue line)

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-4 -3 -2 -1 0 1 2 3 4

0

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10

Function (Blue line) xk (Black cross)

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-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

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-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

Initial point x = -2,

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iteration 1

-4 -3 -2 -1 0 1 2 3 4

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10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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iteration 2

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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iteration 3

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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iteration 4

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

Initial point x = -0.5Initial point x = -0.5

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iteration 1

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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iteration 2

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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iteration 3

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

Initial point x = 0.6Initial point x = 0.6

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Initial point x = 0.6, iteration 1

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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30

Initial point x = 0.6, iteration 2

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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25

30

iteration 3

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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25

30

iteration 4

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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25

30

iteration 5

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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30

iteration 6

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

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25

30

iteration 7

-4 -3 -2 -1 0 1 2 3 4

0

5

10

Function (Blue line)

Quadratic approximation (Red line)

xk (Black cross)

sk (Green Asterisk)

Function of two variables

Initial point x = [0.7;0.7]Initial point x = [0.7;0.7]

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-3 -2 -1 0 1 2 3-3

-2

-1

-3 -2 -1 0 1 2 3-3

-2

-1

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3

0

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3

-3 -2 -1 0 1 2 3-3

-2

-1

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

-3 -2 -1 0 1 2 3-3

-2

-1