chapter 3 interpolation and polynomial approximation

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Chapter 3 Interpolation and Polynomial Approximation. Numerical Analysis. ถ้ามีข้อมูลจากการวัด ซึ่งแทนความสัมพันธ์ของตัวแปรต้นและตัวแปรตาม แล้ว ต้องการทราบค่าตัวแปรตาม ณ จุดอื่นๆ ในช่วงของการวัด ต้องการทราบพฤติกรรมของฟังก์ชันที่แทนข้อมูล. Problem type. Theory of Weierstrass. - PowerPoint PPT Presentation

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Page 1: Chapter 3 Interpolation and Polynomial Approximation

Chapter 3Interpolation

andPolynomial Approximation

Numerical Analysis

Page 2: Chapter 3 Interpolation and Polynomial Approximation

Problem type

ถ�าม�ข�อม�ลจากการว�ด ซ��งแทนความส�มพ�นธ�ของต�วแปร ต�นและต�วแปรตาม แล�ว

◦ ต�องการทราบค าต�วแปรตาม ณ จ"ดอ#�นๆ ในช วงของการว�ด◦ต�องการทราบพฤต(กรรมของฟ*งก�ช�นท��แทนข�อม�ล

§¬¸ µ¦�¦³¤µ��°�� � � � � � � � � Weierstrass oµ� 𝑓 ¤ ·¥µ¤Â³ ªµ¤ n°ÁºÉ°� � � � � � � ሾ𝑎,𝑏ሿ³Įoε> 0Âoª ³¤¡®»µ¤� �𝑃 É·¥µ¤� � � � ሾ𝑎,𝑏ሿ ɤ » ¤ ·ªnµ� � � � � ȁ𝑓ሺ𝑥ሻ− 𝑃ሺ𝑥ሻȁ< 𝜀 宦´ »� � �𝑥∈ሾ𝑎,𝑏ሿ

Page 3: Chapter 3 Interpolation and Polynomial Approximation

Theory of Weierstrass

O x

y

nx 1x 0x

xf x

xPn x

2x

Page 4: Chapter 3 Interpolation and Polynomial Approximation

Interpolation: Overview

Ä®o𝑓ሺ𝑥ሻ= 𝑒𝑥³Įo 𝑃𝑖ሺ𝑥ሻÁ}¡®»µ¤Á¥rÁ°¦r� � � � 𝑖¡ r¦ 宦´ µ¦ ¦³¤µ� � � � � � �¦° »� � � 𝑥0 = 0

𝑃0ሺ𝑥ሻ= 1 𝑃1ሺ𝑥ሻ= 1+ 𝑥

𝑃2ሺ𝑥ሻ= 1+ 𝑥+ 𝑥22

𝑃3ሺ𝑥ሻ= 1+ 𝑥+ 𝑥22 + 𝑥36

𝑃4ሺ𝑥ሻ= 1+ 𝑥+ 𝑥22 + 𝑥36 + 𝑥424

𝑃5ሺ𝑥ሻ= 1+ 𝑥+ 𝑥22 + 𝑥36 + 𝑥424 + 𝑥5120

Page 5: Chapter 3 Interpolation and Polynomial Approximation

Interpolation: Overview𝑥 𝑓ሺ𝑥ሻ= 𝑒𝑥 𝑃1ሺ𝑥ሻ 𝑃2ሺ𝑥ሻ 𝑃3ሺ𝑥ሻ 𝑃4ሺ𝑥ሻ 𝑃5ሺ𝑥ሻ

-2.0 0.13533528 -1.00000000 1.00000000 -0.33333333 0.33333333 0.06666667

-1.5 0.22313016 -0.50000000 0.62500000 0.06250000 0.27343750 0.21015625

-1.0 0.36787944 0.00000000 0.50000000 0.33333333 0.37500000 0.36666667

-0.5 0.60653066 0.50000000 0.62500000 0.60416667 0.60677083 0.60651042

0.0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000

0.5 1.64872127 1.50000000 1.62500000 1.64583333 1.64843750 1.64869792

1.0 2.71828183 2.00000000 2.50000000 2.66666667 2.70833333 2.71666667

1.5 4.48168907 2.50000000 3.62500000 4.18750000 4.39843750 4.46171875

2.0 7.38905610 3.00000000 5.00000000 6.33333333 7.00000000 7.26666667

nµ¦³¤µ ³ ¹ÊÁ¤ºÉ°Äo¡®»µ¤ ¦ ¼� � � � � � � � � � � �

Page 6: Chapter 3 Interpolation and Polynomial Approximation

Interpolation: Overview° »¦¤Á¥rÁ°¦rɦ³ µ¥¦° »� � � � � � � � � 𝑥0 = 1 °� � 𝑓ሺ𝑥ሻ= 1𝑥Á¥ ÅoĦ¼¡®»µ¤ ¦¸� � � � � � � � 𝑛 Á}� � 𝑃𝑛ሺ𝑥ሻ= σ 𝑓ሺ𝑘ሻሺ1ሻ𝑘! ሺ𝑥− 1ሻ𝑘𝑛𝑖=0 = σ ሺ−1ሻ𝑘ሺ𝑥− 1ሻ𝑘𝑛𝑖=0

Ä µ¦ ¦³¤µ nµ� � � � � 𝑓ሺ3ሻà ¥¡®»µ¤� � 𝑃𝑛ሺ𝑥ሻ

Åonµ°� � � � 𝑃𝑛ሺ3ሻÁ}Å µ¤ µ¦µ� � � � � � 𝑛 0 1 2 3 4 5 6 7 𝑃𝑛ሺ3ሻ 1 -1 3 -5 11 -21 43 -85

Page 7: Chapter 3 Interpolation and Polynomial Approximation

Lagrange Polynomial

O

y

𝑥

Page 8: Chapter 3 Interpolation and Polynomial Approximation

Lagrange Polynomialoµ¦µ »� � � � � 2 »� � ൫𝑥0,𝑓ሺ𝑥0ሻ൯ ³൫𝑥1,𝑓ሺ𝑥1ሻ൯³ µ¤µ¦ ¦oµ¡®»µ¤ ¦®¹É� � � � � � � �

( ¤ µ¦Á o ¦� � � � ) Åo¹É°µÁ¥ Ħ¼¡®»µ¤Á}� � � � � � � � � � � 𝑃ሺ𝑥ሻ= 𝑥− 𝑥1𝑥0 − 𝑥1 𝑓ሺ𝑥0ሻ+ 𝑥− 𝑥0𝑥1 − 𝑥0 𝑓ሺ𝑥1ሻ

Á¤ºÉ° 𝑥= 𝑥0Á¦µÅ o� 𝑃1ሺ𝑥0ሻ= 𝑥0−𝑥1𝑥0−𝑥1 𝑓ሺ𝑥0ሻ+ 𝑥0−𝑥0𝑥1−𝑥0 𝑓ሺ𝑥1ሻ= 1∙𝑓ሺ𝑥0ሻ +0∙𝑓ሺ𝑥1ሻ= 𝑓ሺ𝑥0ሻ Á¤ºÉ°𝑥= 𝑥1Á¦µÅ o� 𝑃1ሺ𝑥1ሻ= 𝑥1−𝑥1𝑥0−𝑥1 𝑓ሺ𝑥0ሻ+ 𝑥1−𝑥0𝑥1−𝑥0 𝑓ሺ𝑥1ሻ= 0∙𝑓ሺ𝑥0ሻ +1∙𝑓ሺ𝑥1ሻ= 𝑓ሺ𝑥1ሻ ¹ÉÅo� � � 𝑃1ሺ𝑥ሻ ɤ » ¤ ·µ¤ o° µ¦� � � � � � � � �

Page 9: Chapter 3 Interpolation and Polynomial Approximation

Lagrange Polynomial

¡®»µ¤Á·Á o Énµ »� � � � � � � � � ൫𝑥0,𝑓ሺ𝑥0ሻ൯³ ൫𝑥1,𝑓ሺ𝑥1ሻ൯·¥µ¤Ä®o�

𝐿1,0ሺ𝑥ሻ= 𝑥−𝑥1𝑥0−𝑥1³ 𝐿1,1ሺ𝑥ሻ= 𝑥−𝑥0𝑥1−𝑥0 ʼnÁ}¢{r Á·Á o¹É�� � � � � � � � � � � � � �

Á¤ºÉ° 𝑥= 𝑥0, 𝐿1,0ሺ𝑥0ሻ= 1, 𝐿1,1ሺ𝑥0ሻ= 0 εĮo� 𝑃1ሺ𝑥0ሻ= 𝑓ሺ𝑥0ሻ Á¤ºÉ° 𝑥= 𝑥1, 𝐿1,0ሺ𝑥1ሻ= 0, 𝐿1,1ሺ𝑥1ሻ= 1 εĮo� 𝑃1ሺ𝑥1ሻ= 𝑓ሺ𝑥1ሻ ¦ ¸ÉªÅ� � � � 𝐿𝑛,𝑘ሺ𝑥ሻ¤ » ¤ ·ªnµ� � � �

𝐿𝑛,𝑘ሺ𝑥𝑖ሻ= ൜0, 𝑖 ≠ 𝑘1, 𝑖 = 𝑘 宦´ »� � � 𝑘 = 0,1,2,…,𝑛

Page 10: Chapter 3 Interpolation and Polynomial Approximation

Lagrange Polynomial·¥µ¤� 𝐿𝑛,𝑘ሺ𝑥ሻÃ¥�

𝐿𝑛,𝑘ሺ𝑥ሻ= ሺ𝑥− 𝑥0ሻሺ𝑥− 𝑥1ሻ…ሺ𝑥− 𝑥𝑘−1ሻሺ𝑥− 𝑥𝑘+1ሻ…ሺ𝑥− 𝑥𝑛ሻሺ𝑥𝑘 − 𝑥0ሻሺ𝑥𝑘 − 𝑥1ሻ…ሺ𝑥𝑘 − 𝑥𝑘−1ሻሺ𝑥𝑘 − 𝑥𝑘+1ሻ…ሺ𝑥𝑘 − 𝑥𝑛ሻ

¤ ³Á¥� � � � 𝐿𝑘ሺ𝑥ሻÂ� � 𝐿𝑛,𝑘ሺ𝑥ሻÁ¤ºÉ° ¦µ� � 𝑛 ´Á� � � � ·¥µ¤¡®»µ¤° ´� � � � � 𝑛 ° µ¦° rÄ µ¦ ¦³¤µ nµÄ nª� � � � � � � � � � � � � Ã¥�

𝑃𝑛ሺ𝑥ሻ= 𝐿0ሺ𝑥ሻ𝑓ሺ𝑥0ሻ+ ⋯+ 𝐿𝑛ሺ𝑥ሻ𝑓ሺ𝑥𝑛ሻ ®¦º° 𝑃𝑛ሺ𝑥ሻ= σ 𝐿𝑘ሺ𝑥ሻ𝑓ሺ𝑥𝑘ሻ𝑛𝑘=0

Page 11: Chapter 3 Interpolation and Polynomial Approximation

Lagrange Polynomial: Example 宦´ {®µµ¦ ¦³¤µ nµ� � � � � � � 𝑓ሺ𝑥ሻ= 1𝑥 ¤¤»·Ä®o� 𝑥0 = 2, 𝑥1 = 2.5³ 𝑥2 = 4 ³Åoo°¤¼� � �´ µ¦µ� � � � 𝑥 2 2.5 4 𝑓ሺ𝑥ሻ 0.5 0.4 0.25

µ o°¤¼oµ ooµÄ�o¡®»�µ¤ µ�¦°��r�¦³¤µ��nµ� � � � � � � � � � � � � � � � 𝑓ሺ3ሻ ³ εÅoÃ¥� � � � ®µ𝐿0,𝐿1,𝐿2Ã¥�

𝐿0ሺ𝑥ሻ= ሺ𝑥− 𝑥1ሻሺ𝑥− 𝑥2ሻሺ𝑥0 − 𝑥1ሻሺ𝑥0 − 𝑥2ሻ= ሺ𝑥− 2.5ሻሺ𝑥− 4ሻ

ሺ2− 2.5ሻሺ2− 4ሻ= 𝑥2 − 6.5𝑥+ 10 𝐿1ሺ𝑥ሻ= ሺ𝑥− 𝑥0ሻሺ𝑥− 𝑥2ሻ

ሺ𝑥1 − 𝑥0ሻሺ𝑥1 − 𝑥2ሻ= ሺ𝑥− 2ሻሺ𝑥− 4ሻሺ2.5− 2ሻሺ2.5− 4ሻ= −4𝑥2 + 24𝑥− 323

𝐿2ሺ𝑥ሻ= ሺ𝑥− 𝑥0ሻሺ𝑥− 𝑥1ሻሺ𝑥2 − 𝑥0ሻሺ𝑥2 − 𝑥1ሻ= ሺ𝑥− 2ሻሺ𝑥− 2.5ሻ

ሺ4− 2ሻሺ4− 2.5ሻ= 𝑥2 − 4.5𝑥+ 53

Page 12: Chapter 3 Interpolation and Polynomial Approximation

Lagrange Polynomial: Example

Åo� 𝑃2ሺ𝑥ሻ= σ 𝐿𝑛ሺ𝑥ሻ𝑓ሺ𝑥𝑘ሻ2𝑘=0

= 0.5ሺ𝑥2 − 6.5𝑥+ 10ሻ+ 0.4ቀ−4𝑥2+24𝑥−323 ቁ+ 0.25ቀ𝑥2−4.5𝑥+53 ቁ = 0.05𝑥2 − 0.425𝑥+ 1.15 Ä ³ Énµ¦·� � � � � � � 𝑓ሺ3ሻ= 13 = 0.333∙ nµ¦³¤µ ÉÅoµ� � � � � � � 𝑃2ሺ3ሻ= 0.325

Page 13: Chapter 3 Interpolation and Polynomial Approximation

Lagrange Polynomial: Example

° »¦¤Á¥rÁ°¦r¤¡ rÁ«¬Á®º°Ä¦¼� � � � � � � 𝑓ሺ𝑛+1ሻሺ𝜉ሻሺ𝑥−𝑥0ሻ𝑛+1ሺ𝑛+1ሻ! Á¤ºÉ° 𝜉°¥¼n¦³®ªnµ� 𝑥 ´� � 𝑥0

Ân¡®»µ¤ µ¦° r° ´� � � � � � � � 𝑛Äoo°¤¼µ » Énµ ´� � � � � � � � � � � 𝑥0,𝑥1,…,𝑥𝑛 ¼¦ nµ·¡ µ ° ¡®»µ¤ µ¦° r� � � � � � � � � � �

𝑓ሺ𝑥ሻ= 𝑃𝑛ሺ𝑥ሻ+ 𝑓ሺ𝑛+1ሻሺ𝜉ሻሺ𝑛+ 1ሻ! ሺ𝑥− 𝑥0ሻሺ𝑥− 𝑥1ሻ…ሺ𝑥− 𝑥𝑛ሻ

Á¤ºÉ°𝜉°¥¼n¦³®ªnµ� 𝑥 ´� � 𝑥0,𝑥1,…,𝑥𝑛

Page 14: Chapter 3 Interpolation and Polynomial Approximation

Divided Difference·¥µ¤ ¨ nµ ºÁºÉ°� � � � � � � ¨ nµ ºÁºÉ° É«¼¥r° ¢{r� � � � � � � � � � � � � � 𝑓Á¥ ´� � � � 𝑥𝑖 º°�

𝑓ሾ𝑥𝑖ሿ= 𝑓ሺ𝑥𝑖ሻ ¨ nµ ºÁºÉ° ÉÁ®º° ³ ¼·¥µ¤Ã¥ªµ¤ ¤¡´ rÁª¥ Á·� � � � � � � � � � � � � � � � � � � � (recursive) ¨ nµ� � �

ºÁºÉ° É®¹É°� � � � � � � � 𝑓Á¥ ´� � � � 𝑥𝑖³ 𝑥𝑖+1 º°�

𝑓ሾ𝑥𝑖,𝑥𝑖+1ሿ= 𝑓ሾ𝑥𝑖+1ሿ− 𝑓ሾ𝑥𝑖ሿ𝑥𝑖+1 − 𝑥𝑖

¨ nµ ºÁºÉ° É� � � � � � � 𝑘Á¥ ´� � � � 𝑥𝑖,𝑥𝑖+1,…,𝑥𝑖+𝑘 º°�

𝑓ሾ𝑥𝑖,𝑥𝑖+1,…𝑥𝑖+𝑘ሿ= 𝑓ሾ𝑥𝑖+1,…,𝑥𝑖+𝑘ሿ− 𝑓ሾ𝑥𝑖,…,𝑥𝑖+𝑘−1ሿ𝑥𝑖+𝑘 − 𝑥𝑖

Page 15: Chapter 3 Interpolation and Polynomial Approximation

Divided Difference¡®»µ¤� ¨ nµ ºÁºÉ° ° ·ª ´� � � � � � � � � � � ( oµ®oµ� � � ) ·¥µ¤Ã¥� � 𝑃𝑛ሺ𝑥ሻ= 𝑓ሾ𝑥0ሿ+ 𝑓ሾ𝑥0,𝑥1ሿሺ𝑥− 𝑥0ሻ+ 𝑓ሾ𝑥0,𝑥1,𝑥2ሿሺ𝑥− 𝑥0ሻሺ𝑥− 𝑥1ሻ+ ⋯+ 𝑓ሾ𝑥0,𝑥1,…,𝑥𝑛ሿሺ𝑥− 𝑥0ሻሺ𝑥− 𝑥1ሻ…ሺ𝑥− 𝑥𝑛−1ሻ

¡®»µ¤� ¨ nµ ºÁºÉ° ° ·ª ´� � � � � � � � � � � (¥o° ®´� � ) ·¥µ¤Ã¥� � 𝑃𝑛ሺ𝑥ሻ= 𝑓ሾ𝑥𝑛ሿ+ 𝑓ሾ𝑥𝑛−1,𝑥𝑛ሿሺ𝑥− 𝑥𝑛ሻ+ 𝑓ሾ𝑥𝑛−2,𝑥𝑛−1,𝑥𝑛ሿሺ𝑥− 𝑥𝑛ሻሺ𝑥− 𝑥𝑛−1ሻ+ ⋯+ 𝑓ሾ𝑥0,𝑥1,…,𝑥𝑛ሿሺ𝑥− 𝑥𝑛ሻሺ𝑥− 𝑥𝑛−1ሻ…ሺ𝑥− 𝑥1ሻ

Page 16: Chapter 3 Interpolation and Polynomial Approximation

Divided Difference𝒙 𝒇ሺ𝒙ሻ First divided difference Second divided difference 𝑥0 𝑓ሾ𝑥0ሿ

𝑓ሾ𝑥0,𝑥1ሿ= 𝑓ሾ𝑥1ሿ− 𝑓ሾ𝑥0ሿ𝑥1 − 𝑥0

𝑥1 𝑓ሾ𝑥1ሿ 𝑓ሾ𝑥0,𝑥1,𝑥2ሿ= 𝑓ሾ𝑥1,𝑥2ሿ− 𝑓ሾ𝑥0,𝑥1ሿ𝑥2 − 𝑥0

𝑓ሾ𝑥1,𝑥2ሿ= 𝑓ሾ𝑥2ሿ− 𝑓ሾ𝑥1ሿ𝑥2 − 𝑥1

𝑥2 𝑓ሾ𝑥2ሿ 𝑓ሾ𝑥1,𝑥2,𝑥3ሿ= 𝑓ሾ𝑥2,𝑥3ሿ− 𝑓ሾ𝑥1,𝑥2ሿ𝑥3 − 𝑥1

𝑓ሾ𝑥2,𝑥3ሿ= 𝑓ሾ𝑥3ሿ− 𝑓ሾ𝑥2ሿ𝑥3 − 𝑥2

𝑥3 𝑓ሾ𝑥3ሿ

Page 17: Chapter 3 Interpolation and Polynomial Approximation

Newton Divided Difference: Example

ε® nµ° ¢{rÉ®µ¥�»�� � � � � � � � � � � � �

𝑥 1.0 1.3 1.6 1.9 2.2 𝑓ሺ𝑥ሻ 0.7651977 0.6200860 0.4554022 0.2818186 0.1103623

Äoµ¦ ¦³¤µ nµÄ nªÃ¥¨ nµ ºÁºÉ° ° ·ª ´ ¦³¤µ nµ� � � � � � � � � � � � � � � � � � � � � � � � � 𝒇ሺ𝟏.𝟓ሻ

Page 18: Chapter 3 Interpolation and Polynomial Approximation

Newton Divided Difference: Example

xi

f(xi)=f[xi]

1st Divided

Diff

2nd Divided

Diff

3rd Divided

Diff

4th Divided

Diff

1.0 0.7651977

-0.4837057

1.3 0.6200860

-0.1087339

-0.5489460

0.0658784

1.6 0.4554022

-0.0494433

0.0018251

-0.5786120

0.0680685

1.9 0.2818186

0.0118183

-0.5715210

2.2 0.1103623

Page 19: Chapter 3 Interpolation and Polynomial Approximation

Newton Divided Difference: Example

¤ ¦³ · ·Í° ¨ nµ ºÁºÉ°� � � � � � � � � � � ( oµ®oµ� � � ) °� � Newton ÉÄ®o¡®»µ¤Ä µ¦ ¦³¤µ nµ� � � � � � �Ä nª³°¥¼nÄ�Â�ª�Â¥�¤»¤���°��µ¦µ�¡®»�µ¤�Ê�º°� � � � � � � � � � � � � � � � � 𝑃4ሺ𝑥ሻ= 0.7651977− 0.483705ሺ𝑥− 1ሻ− 0.1087339ሺ𝑥− 1ሻሺ𝑥− 1.3ሻ+ ⋯+ 0.0658784ሺ𝑥− 1ሻሺ𝑥− 1.3ሻሺ𝑥− 1.6ሻ+ 0.0018251ሺ𝑥− 1ሻሺ𝑥− 1.3ሻሺ𝑥− 1.6ሻሺ𝑥− 1.9ሻ ¹ÉÄ®o� � 𝑃4ሺ1.5ሻ= 0.511820

¤ ¦³ · ·Í° ¨ nµ ºÁºÉ°� � � � � � � � � � � (¥o° ®� � ) °� � Newton ÉÄ®o¡®»µ¤Ä µ¦ ¦³¤µ nµ� � � � � � �Ä nª³°¥¼nÄ�Â�ª�Â¥�¤»¤ nµ��°��µ¦µ�� � � � � � � � � � � � �

ค�าประมาณท� x=1.8 มค�าเท�าไร

Page 20: Chapter 3 Interpolation and Polynomial Approximation

Forward Divided DifferenceÄ®o𝑥0,𝑥1,…,𝑥𝑛®nµÁnµÇ´� � � � ℎ = 𝑥𝑖+1 − 𝑥𝑖 宦´ »� � � 𝑖 = 0,1,2,…,𝑛³𝑥= 𝑥0 + 𝑠ℎ ³Åo� � 𝑥− 𝑥𝑖 = ሺ𝑠− 𝑖ሻℎ ¼¦ ¨ nµ ºÁºÉ°� � � � � � � ( oµ®oµ� � � ) ³ µ¥Á}� � � � 𝑃𝑛ሺ𝑥ሻ= 𝑃𝑛ሺ𝑥0 + 𝑠ℎሻ = 𝑓ሾ𝑥0ሿ+ 𝑠ℎ𝑓ሾ𝑥0,𝑥1ሿ+ 𝑠ሺ𝑠− 1ሻℎ2𝑓ሾ𝑥0,𝑥1,𝑥2ሿ+ ⋯ +𝑠ሺ𝑠− 1ሻሺ𝑠− 2ሻ…ሺ𝑠− 𝑛+ 1ሻℎ𝑛𝑓ሾ𝑥0,𝑥1,…,𝑥𝑛ሿ

= 𝑠ሺ𝑠− 1ሻሺ𝑠− 2ሻ…ሺ𝑠− 𝑘+ 1ሻℎ𝑘𝑓ሾ𝑥0,𝑥1,…,𝑥𝑘ሿ𝑘=𝑛𝑘=0

®¦º°𝑃𝑛ሺ𝑥ሻ= 𝑃𝑛ሺ𝑥0 + 𝑠ℎሻ= σ ቀ𝑠𝑘ቁ𝑘!ℎ𝑘𝑓ሾ𝑥0,𝑥1,…,𝑥𝑘ሿ𝑘=𝑛𝑘=0

Page 21: Chapter 3 Interpolation and Polynomial Approximation

Forward Divided Difference·¥µ¤ ´ ¦ r° ¨ nµ ºÁºÉ° oµ®oµ° ε´� � � � � � � � � � � � � � � � � � � ሼ𝑃𝑛ሽ𝑛=0∞ Ä Ç� Ã¥�

∆𝑃𝑛 = 𝑃𝑛+1 − 𝑃𝑛 宦´ »� � � 𝑛 ≥ 0 ³ ¨ nµ ºÁºÉ° oµ®oµ° ¼Ç¤ ·¥µ¤Á}� � � � � � � � � � � � � � � � ∆𝑘𝑃𝑛 = ∆ሺ∆𝑘−1ሻ𝑃𝑛 宦´ »� � � 𝑘 ≥ 2 ³Åoªnµ� �

𝑓ሾ𝑥0,𝑥1ሿ= 𝑓ሺ𝑥1ሻ− 𝑓ሺ𝑥0ሻ𝑥1 − 𝑥0 = 1ℎ∆𝑓ሺ𝑥0ሻ 𝑓ሾ𝑥0,𝑥1,𝑥2ሿ= 12ℎቈ∆𝑓ሺ𝑥1ሻ− ∆𝑓ሺ𝑥0ሻℎ = 12ℎ2 ∆2𝑓ሺ𝑥0ሻ

³Åo¡ rɪÅ� � � � � 𝑓ሾ𝑥0,𝑥1,…,𝑥𝑘ሿ= 1𝑘!ℎ𝑘 ∆𝑘𝑓ሺ𝑥0ሻ

Page 22: Chapter 3 Interpolation and Polynomial Approximation

Forward Divided Difference

µ� � 𝑃𝑛ሺ𝑥0 + 𝑠ℎሻ= σ ቀ𝑠𝑘ቁ𝑘!ℎ𝑘𝑓ሾ𝑥0,𝑥1,…,𝑥𝑘ሿ𝑘=𝑛𝑘=0

Á¤ºÉ°Â� � 𝑓ሾ𝑥0,𝑥1,…,𝑥𝑘ሿ= 1𝑘!ℎ𝑘 ∆𝑘𝑓ሺ𝑥0ሻ

Åo ¼¦ ¨ nµ ºÁºÉ° oµ®oµ°� � � � � � � � � � � � � Newton Á¥ Åo°  ®¹Éº°� � � � � � � � �

𝑃𝑛ሺ𝑥ሻ= ቀ𝑠𝑘ቁ∆𝑘𝑓ሺ𝑥0ሻ

𝑘=𝑛𝑘=0

Page 23: Chapter 3 Interpolation and Polynomial Approximation

Backward Divided Difference 宦 ¼¦ ¨ nµ ºÁºÉ°� � � � � � � � (¥o° ®´� � ) 𝑃𝑛ሺ𝑥ሻ= 𝑓ሾ𝑥𝑛ሿ+ 𝑓ሾ𝑥𝑛−1,𝑥𝑛ሿሺ𝑥− 𝑥𝑛ሻ+ 𝑓ሾ𝑥𝑛−2,𝑥𝑛−1,𝑥𝑛ሿሺ𝑥− 𝑥𝑛ሻሺ𝑥− 𝑥𝑛−1ሻ+ ⋯+ 𝑓ሾ𝑥0,𝑥1,…,𝑥𝑛ሿሺ𝑥− 𝑥𝑛ሻሺ𝑥− 𝑥𝑛−1ሻ…ሺ𝑥− 𝑥1ሻ ®µÄo¦³¥³®nµÁnµ´Ã ¥� � � � � � � 𝑥= 𝑥𝑛 + 𝑠ℎ, 𝑥𝑖 = 𝑥𝑛 − (𝑛− 𝑖)ℎ εĮoÅo� � 𝑥− 𝑥𝑖 = ሺ𝑠+ 𝑛− 𝑖ሻℎ εĮoÅo� � 𝑃𝑛ሺ𝑥ሻ= 𝑃𝑛ሺ𝑥𝑛 + 𝑠ℎሻ= 𝑓ሾ𝑥𝑛ሿ+ 𝑠ℎ𝑓ሾ𝑥𝑛−1,𝑥𝑛ሿ+ 𝑠ሺ𝑠+ 1ሻℎ2𝑓ሾ𝑥𝑛−2,𝑥𝑛−1,𝑥𝑛ሿ+ ⋯+ 𝑠ሺ𝑠+ 1ሻሺ𝑠+ 2ሻ…ሺ𝑠+ 𝑛− 1ሻℎ𝑛𝑓ሾ𝑥0,𝑥1,…,𝑥𝑛ሿ

Page 24: Chapter 3 Interpolation and Polynomial Approximation

Backward Divided Difference¹ÉÅo� � � 𝑃𝑛ሺ𝑥0 + 𝑠ℎሻ= σ ቀ

−𝑠𝑘 ቁ𝑘!ℎ𝑘𝑓ሾ𝑥𝑛−𝑘,…,,𝑥𝑛−1,𝑥𝑛ሿ𝑘=𝑛𝑘=0

Ä®oሼ𝑃𝑛ሽ𝑛=0∞ Á}ε´ÄÇ ·¥µ¤ ´ ¦ r nµ ºÁºÉ° ¥o° ®Ã¥� � � � � � � � � � � � � � � � � �

∇𝑃𝑛 = 𝑃𝑛 − 𝑃𝑛−1 宦´ »� � � 𝑛 ≥ 1 ¨ nµ ºÁºÉ° ¥o° ® ° ¼Ç¤ ·¥µ¤Á}� � � � � � � � � � � � � � �

∇𝑘𝑃𝑛 = ∇ሺ∇𝑘−1ሻ𝑃𝑛 宦´ »� � � 𝑘 ≥ 2 ·ÉÊʪnµ� � �

𝑓ሾ𝑥𝑛−1,𝑥𝑛ሿ= 1ℎ ∇𝑓ሺ𝑥𝑛ሻ, 𝑓ሾ𝑥𝑛−2,𝑥𝑛−1,𝑥𝑛ሿ= 12ℎ2 ∇2𝑓ሺ𝑥𝑛ሻ ³Åo¡ rɪÅÁ}� � � � � � � 𝑓ሾ𝑥𝑛−𝑘,…,𝑥𝑛−1,𝑥𝑛ሿ= 1𝑘!ℎ𝑘 ∇k𝑓ሺ𝑥𝑛ሻ

Page 25: Chapter 3 Interpolation and Polynomial Approximation

Backward Divided Difference

µ� � 𝑃𝑛ሺ𝑥0 + 𝑠ℎሻ= σ ቀ−𝑠𝑘 ቁ𝑘!ℎ𝑘𝑓ሾ𝑥𝑛−𝑘,…,,𝑥𝑛−1,𝑥𝑛ሿ𝑘=𝑛𝑘=0

³ 𝑓ሾ𝑥𝑛−𝑘,…,𝑥𝑛−1,𝑥𝑛ሿ= 1𝑘!ℎ𝑘 ∇k𝑓ሺ𝑥𝑛ሻ Á¤ºÉ°൬– 𝑆𝑘 ൰= −−𝑠ሺ−𝑠−1ሻ…ሺ−𝑠−𝑘+1ሻ𝑘! = ሺ−1ሻ𝑘𝑠ሺ𝑠+1ሻ…ሺ𝑠+𝑘−1ሻ𝑘!

³� Åo� “ ¼¦ ¨ nµ ºÁºÉ° ¥o° ®´ °� � � � � � � � � � � Newton” Á}� � 𝑃𝑛ሺ𝑥ሻ= σ ሺ−1ሻ𝑘ቀ𝑠𝑘ቁ∇𝑘𝑓ሺ𝑥𝑛ሻ𝑘=𝑛𝑘=0

Page 26: Chapter 3 Interpolation and Polynomial Approximation

Newton Divided Difference: Example

ε® nµ° ¢{rÉ®µ¥�»�� � � � � � � � � � � � �

𝑥 1.0 1.3 1.6 1.9 2.2 𝑓ሺ𝑥ሻ 0.7651977 0.6200860 0.4554022 0.2818186 0.1103623

Äoµ¦ ¦³¤µ nµÄ nªÃ¥¨ nµ ºÁºÉ° ° ·ª ´ ¦³¤µ nµ� � � � � � � � � � � � � � � � � � � � � � � � � 𝑓ሺ1.1ሻ ³ 𝑓ሺ2.0ሻ

Ä µ¦ ¦³¤µ nµ� � � � � 𝑓ሺ1.1ሻÄ®oÁº° Äoo°¤¼ÉÄ o� � � � � 𝑥= 1.1 ¤µ É »É��º°� � � � � � 𝑥0 = 1.0 ¹ÉÅo� � �ℎ = 0.3,𝑠= 13 ®¦º°𝑃4ሺ1.1ሻ= 𝑃4൬1.0+ 13ሺ0.3ሻ൰ Ä µ¦ ¦³¤µ nµ� � � � � 𝑓ሺ2.0ሻÄ®oÁº° Äoo°¤¼ÉÄ o� � � � � 𝑥= 2.0 ¤µ É »É��º°� � � � � � 𝑥4 = 2.2 ¹ÉÅo� � �ℎ = 0.3,𝑠= −23 ®¦º°𝑃4ሺ2.0ሻ= 𝑃4൬2.2+ቀ−23ቁሺ0.3ሻ൰

Page 27: Chapter 3 Interpolation and Polynomial Approximation

Example (Newton Divided Difference)

xi

f(xi)=f[xi]

1st Divided

Diff

2nd Divided

Diff

3rd Divided

Diff

4th Divided

Diff

1.0 0.7651977

-0.4837057

1.3 0.6200860

-0.1087339

-0.5489460

0.0658784

1.6 0.4554022

-0.0494433

0.0018251

-0.5786120

0.0680685

1.9 0.2818186

0.0118183

-0.5715210

2.2 0.1103623

Page 28: Chapter 3 Interpolation and Polynomial Approximation

Newton Divided Difference: ExampleεĮoÅo� �

𝑃4ሺ1.1ሻ= 𝑃4ቆ1.0+ 13ሺ0.3ሻቇ= 0.7651977+ 13ሺ0.3ሻሺ−0.483705ሻ+ 13൬−23൰ሺ0.3ሻ2ሺ−0.1087339ሻ+ 13൬−23൰൬−53൰ሺ0.3ሻ3ሺ0.0658784ሻ+ 13൬−23൰൬−53൰൬−83൰ሺ0.3ሻ4ሺ0.0018251ሻ= 0.7196480

Page 29: Chapter 3 Interpolation and Polynomial Approximation

Newton Divided Difference: Example

xi

f(xi)=f[xi]

1st Divided

Diff

2nd Divided

Diff

3rd Divided

Diff

4th Divided

Diff

1.0 0.7651977

-0.4837057

1.3 0.6200860

-0.1087339

-0.5489460

0.0658784

1.6 0.4554022

-0.0494433

0.0018251

-0.5786120

0.0680685

1.9 0.2818186

0.0118183

-0.5715210

2.2 0.1103623

Page 30: Chapter 3 Interpolation and Polynomial Approximation

Newton Divided Difference: Example

εĮoÅo� �

𝑃4ሺ2.0ሻ= 𝑃4ቆ2.2+൬−23൰ሺ0.3ሻቇ= 0.1103623− 23ሺ0.3ሻሺ−0.5715210ሻ− 23൬13൰ሺ0.3ሻ2ሺ0.0118183ሻ− 23൬13൰൬43൰ሺ0.3ሻ3ሺ0.0680685ሻ− 23൬13൰൬43൰൬73൰ሺ0.3ሻ4ሺ0.0018251ሻ= 0.2238754

Page 31: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation

การท��พห"นามม�ด�กร�ส�งข�,นจะท-าให�ค าประมาณด�ข�,น การประมาณค าในช วงของ Hermite นอกจากจะใช�ค า

ฟ*งก�ช�น ณ จ"ดท��ก-าหนดแล�ว ย�งใช�ค าอน"พ�นธ�อ�นด�บหน��ง ณ จ"ดท��ก-าหนดน�,นด�วย ส-าหร�บข�อม�ลจ-านวน n+1 ต�ว

พห"นาม Hermite จะม�ด�กร� 2n+1

Page 32: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation¡®»µ¤� Hermite

oµ� 𝑓∈𝐶1ሾ𝑎,𝑏ሿ³ 𝑥0,𝑥1,…,𝑥𝑛 ∈ሾ𝑎,𝑏ሿÁ} » nµÇ´Âoª¡®»µ¤®¹ÉÁ¥ª ɤ� � � � � � � � � � � � �¦ Éε »Â³¤ nµÁ®¤º°� � � � � � 𝑓 ´° »¡´ rÁ®¤º°� � � � � � 𝑓′ � 𝑥0,𝑥1,…,𝑥𝑛 º°¡®»µ¤ ¦¸o°¥� � � � �

ªnµ®¦º°Ánµ´� � � � 2𝑛+ 1 ¹É¤¦¼Á}� � � � � 𝐻2𝑛+1ሺ𝑥ሻ= σ 𝑓൫𝑥𝑗൯𝐻𝑛,𝑗ሺ𝑥ሻ𝑛𝑗=0 + σ 𝑓′൫𝑥𝑗൯𝐻 𝑛,𝑗ሺ𝑥ሻ𝑛𝑗=0

Á¤ºÉ° 𝐻𝑛,𝑗ሺ𝑥ሻ= ቀ1− 2൫𝑥− 𝑥𝑗൯𝐿𝑛,𝑗′ ൫𝑥𝑗൯𝐿𝑛,𝑗2 ሺ𝑥ሻቁ 𝐻 𝑛,𝑗ሺ𝑥ሻ= ൫𝑥− 𝑥𝑗൯𝐿𝑛,𝑗2 ሺ𝑥ሻ Ä ÉÊ� � � 𝐿𝑛,𝑗ሺ𝑥ሻ º°¡®»µ¤ ¤ ¦³ · ·Í É� � � � � � 𝑗 °� � Langrange ɤ ¦¸� � � 𝑛

Page 33: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation ¼¦ nµ·¡ µ ° ¡®»µ¤� � � � � � � � Hermite

oµ� 𝑓∈𝐶ሺ2𝑛+2ሻሾ𝑎,𝑏ሿÂoª

𝑓ሺ𝑥ሻ= 𝐻2𝑛+1ሺ𝑥ሻ+ 𝑓ሺ2𝑛+2ሻ൫𝜉ሺ𝑥ሻ൯ሺ2𝑛+2ሻ! ሺ𝑥− 𝑥0ሻ2 …ሺ𝑥− 𝑥𝑛ሻ2

Á¤ºÉ° 𝜉ሺ𝑥ሻ∈ሺ𝑎,𝑏ሻ

ªµ¤ ¤¡´ r¦³®ªnµ ¨ nµ ºÁºÉ° ´° »¡´ r� � � � � � � � � � � � � � � oµ� 𝑓∈𝐶𝑛ሾ𝑎,𝑏ሿ³ 𝑥0,…,𝑥𝑛 ∈ሾ𝑎,𝑏ሿÂoª¤ 𝜉ሺ𝑥ሻ∈ሺ𝑎,𝑏ሻ ɪnµ�

𝑓ሾ𝑥0,𝑥1,…,𝑥𝑛ሿ= 𝑓𝑛ሺ𝜉ሻ𝑛!

Page 34: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolationµ¦ ¦oµ¡®»µ¤� � � Hermite ³Á¦·É¤ µ µ¦� � � � ·¥µ¤ ε´� � � ሼ𝑧𝑘ሽ𝑘=02𝑛+1Ã¥�

𝑧2𝑖 = 𝑧2𝑖+1 = 𝑥𝑖 ሺ𝑖 = 0,1,…,𝑛ሻ µ Ê ¦oµ µ¦µ ¨ nµ ºÁºÉ° °� � � � � � � � � � � � � � � 𝑧0,𝑧1,…,𝑧2𝑛+1 Á¡¦µ³ªnµ

𝑧2𝑖 = 𝑧2𝑖+1 = 𝑥𝑖 宦´ »nµ� � � � 𝑖 ´ Ê� � � � 𝑓ሾ𝑧2𝑖,𝑧2𝑖+1ሿ= 𝑓ሾ𝑧2𝑖+1ሿ−𝑓ሾ𝑧2𝑖ሿ𝑧2𝑖+1−𝑧2𝑖 ¹®µÅ¤nÅo� � �

Ânµ ªµ¤ ¤¡´ r� � � � � � 𝑓ሾ𝑥0,𝑥1,…,𝑥𝑛ሿ= 𝑓𝑛ሺ𝜉ሻ𝑛! ³Åoªnµ� � 𝑓ሾ𝑥0,𝑥1ሿ= 𝑓′ሺ𝜉ሻ 宦´ µ� � � 𝜉∈ሺ𝑥0,𝑥1ሻ oª¥Á®»Ê� � � lim𝑥1→𝑥0 𝑓ሾ𝑥0,𝑥1ሿ= 𝑓′ሺ𝑥0ሻ ε° Á¥ª ´� � � � � � 𝑓ሾ𝑧2𝑖,𝑧2𝑖+1ሿ= 𝑓′ሺ𝑥𝑖ሻ

Page 35: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation¦¼Â ¨ nµ ºÁºÉ° ° ¡®»µ¤� � � � � � � � � � � � Hermite

oµ� 𝑓∈𝐶1ሾ𝑎,𝑏ሿ³ 𝑥0,𝑥1,…,𝑥𝑛 ∈ሾ𝑎,𝑏ሿÁ} » nµÇ´Âoª� � � � � � � �

𝐻2𝑛+1ሺ𝑥ሻ= 𝑓ሾ𝑥0ሿ+ 𝑓ሾ𝑧0,𝑧1,…,𝑧𝑘ሿሺ𝑥− 𝑧0ሻሺ𝑥− 𝑧1ሻ…ሺ𝑥− 𝑧𝑘−1ሻ2𝑛+1𝑘=1

Á¤ºÉ°𝑧2𝑘 = 𝑧2𝑘+1 = 𝑥𝑘³ 𝑓ሾ𝑧2𝑖,𝑧2𝑖+1ሿ= 𝑓′ሺ𝑥𝑘ሻ 宦´ »� � � 𝑖 =0,1,…,𝑛

Page 36: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation𝑧 𝑓ሾ𝑧ሿ First divided difference Second divided difference 𝑧0 = 𝑥0 𝑓ሾ𝑧0ሿ= 𝑓ሾ𝑥0ሿ

𝑓ሾ𝑧0,𝑧1ሿ= 𝑓′ሺ𝑥0ሻ 𝑧1 = 𝑥0 𝑓ሾ𝑧1ሿ= 𝑓ሾ𝑥0ሿ 𝑓ሾ𝑧0,𝑧1,𝑧2ሿ= 𝑓ሾ𝑧1,𝑧2ሿ− 𝑓ሾ𝑧0,𝑧1ሿ𝑧2 − 𝑧0

𝑓ሾ𝑧1,𝑧2ሿ= 𝑓ሾ𝑧2ሿ− 𝑓ሾ𝑧1ሿ𝑧2 − 𝑧1

𝑧2 = 𝑥1 𝑓ሾ𝑧2ሿ= 𝑓ሾ𝑥1ሿ 𝑓ሾ𝑧1,𝑧2,𝑧3ሿ= 𝑓ሾ𝑧2,𝑧3ሿ− 𝑓ሾ𝑧1,𝑧2ሿ𝑧3 − 𝑧1

𝑓ሾ𝑧2,𝑧3ሿ= 𝑓′ሺ𝑥1ሻ 𝑧3 = 𝑥1 𝑓ሾ𝑧3ሿ= 𝑓ሾ𝑥1ሿ 𝑓ሾ𝑧2,𝑧3,𝑧4ሿ= 𝑓ሾ𝑧3,𝑧4ሿ− 𝑓ሾ𝑧2,𝑧3ሿ𝑧4 − 𝑧2

𝑓ሾ𝑧3,𝑧4ሿ= 𝑓ሾ𝑧4ሿ− 𝑓ሾ𝑧3ሿ𝑧4 − 𝑧3

𝑧4 = 𝑥2 𝑓ሾ𝑧4ሿ= 𝑓ሾ𝑥2ሿ 𝑓ሾ𝑧3,𝑧4,𝑧5ሿ= 𝑓ሾ𝑧4,𝑧5ሿ− 𝑓ሾ𝑧3,𝑧4ሿ𝑧5 − 𝑧3

𝑓ሾ𝑧4,𝑧5ሿ= 𝑓′ሺ𝑥2ሻ 𝑧5 = 𝑥2 𝑓ሾ𝑧5ሿ= 𝑓ሾ𝑥2ሿ

Page 37: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation: Example

µ o°¤¼Ä µ¦µ¡¦o°¤ oª¥nµ° »¡´ r ¦oµ µ¦µ ¨ nµ ºÁºÉ° Á¡ºÉ°Äo¡®»µ¤� � � � � � � � � � � � � � � � � � � � � � � � Hermite ¦³¤µ nµ°� � � � � 𝑓ሺ1.5ሻ 𝑥 𝑓ሺ𝑥ሻ 𝑓′ሺ𝑥ሻ

1.3 0.6200860 -0.5220232 1.6 0.4554022 -0.5698959 1.9 0.2818186 -0.5811571

Page 38: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation: Example𝑧 𝑓ሾ𝑧ሿ 1st D Diff. 2nd D Diff. 3rd D Diff. 4th D Diff. 5th D Diff. 1.3 0.6200860 -0.5220232

1.3 0.6200860 -0.08977427 -0.5489460 0.0663657

1.6 0.4554022 -0.0698330 0.0026663 -0.5698959 0.0679655 -0.0027738

1.6 0.4554022 -0.0290537 0.0010020 -0.5786120 0.0685667

1.9 0.2818186 -0.0084837 -0.5811571

1.9 0.2818186

Page 39: Chapter 3 Interpolation and Polynomial Approximation

Hermite Interpolation: Example

µ� � 𝐻2𝑛+1ሺ𝑥ሻ= 𝑓ሾ𝑥0ሿ+ σ 𝑓ሾ𝑧0,𝑧1,…,𝑧𝑘ሿሺ𝑥− 𝑧0ሻሺ𝑥− 𝑧1ሻ…ሺ𝑥− 𝑧𝑘−1ሻ2𝑛+1𝑘=1 Á¦µ³Åo� � 𝐻5ሺ1.5ሻ= 0.6200860+ሺ1.5− 1.3ሻሺ−0.5220232ሻ+ሺ1.5− 1.3ሻ2ሺ−0.0897427ሻ+ሺ1.5− 1.3ሻ2ሺ1.5− 1.6ሻሺ0.0663657ሻ+ሺ1.5− 1.3ሻ2ሺ1.5− 1.6ሻ2ሺ0.0026663ሻ+ሺ1.5− 1.3ሻ2ሺ1.5− 1.6ሻ2ሺ1.5− 1.9ሻሺ−0.0027738ሻ= 0.5118277

Page 40: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline Interpolation ความแม นย-าในการประมาณอาจส�งข�,น เม#�อใช�พห"นามท��

ม�ด�กร�ส�ง แต เม#�อด�กร�ท��ส�งข�,นมากอาจจะม�การกว�ดแกว ง ของเส�นโค�งส�งข�,นด�วย ซ��งจะส งผลให�ค าประมาณม�

ความคลาดเคล#�อนมากข�,นก2ได� ว(ธ�หน��งท��ใช�แก�ป*ญหาค#อ แบ งช วงท�,งหมดออกเป5นช วงย อยๆ แล�วสร�างพห"นาม

ประจ-าแต ละช วงย อย เร�ยกว า “การประมาณโดยพห"นามเป5นช วงๆ””

ถ�าให�ท"กสองค� ของจ"ดแทนช วงหน��งช วง การเช#�อมจ"ด ของข�อม�ลด�วยเส�นตรงก2ค#อว(ธ�ท��ง ายท��ส"ด แต ก2จะท-าให�

เส�นโค�งไม เร�ยบ แนวทางอ#�นค#อ การใช�พห"นาม Hermite แต ก2ต�องม�

ข�อม�ลของอน"พ�นธ�อ�นด�บหน��งของท"กจ"ด

Page 41: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline Interpolation

การประมาณโดยพห"นามเป5นส วนๆ ท��พบบ อยท��ส"ดค#อ การใช�พห"นามก-าล�งสามระหว างค� ของจ"ด ท��เร�ยกว า

Cubic Spline

พห"นามก-าล�งสาม ม�ค าคงต�ว 4 ค า โดยท��วไปแล�ว อน"พ�นธ�ของ Cubic Spline ไม จ-าเป5นต�องเท าก�บอน"พ�นธ�

ของฟ*งก�ช�นจร(ง แม�ท��จ"ดน(ยาม

Page 42: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline InterpolationÄ®o¢{r� � � � 𝑓 ·¥µ¤� � � ሾ𝑎,𝑏ሿ³¤Á ° »� � � � � � 𝑎 = 𝑥0 < 𝑥1 < ⋯ < 𝑥𝑛 = 𝑏 ª�¦³¤µ ε µ¤� � � � 𝑆 °� � 𝑓 º°¢{r É ° o° µ¤ÁºÉ° Å n°Å Ê� � � � � � � � � � � � � � � � 1. 𝑆Á}¡®»µ¤ ε µ¤Á¥Â oª¥� � � � � � � � � � 𝑆𝑗 宦´ nª¥n°¥� � � �𝑥𝑗,𝑥𝑗+1൧, 𝑗=0,1,…,𝑛− 1 2. 𝑆൫𝑥𝑗൯= 𝑓൫𝑥𝑗൯ ( 𝑗= 0,1,…,𝑛 )

3. 𝑆𝑗+1൫𝑥𝑗+1൯= 𝑆𝑗൫𝑥𝑗+1൯ ( 𝑗= 0,1,…,𝑛− 2 ) 4. 𝑆𝑗+1′ ൫𝑥𝑗+1൯= 𝑆𝑗′൫𝑥𝑗+1൯ ( 𝑗= 0,1,…,𝑛− 2 ) 5. 𝑆𝑗+1′′ ൫𝑥𝑗+1൯= 𝑆𝑗′′൫𝑥𝑗+1൯ ( 𝑗= 0,1,…,𝑛− 2 ) 6. Á ° ÁºÉ° Å ° Á o°Ä o°®¹Én°Å ÊÁ}¦·� � � � � � � � � � � � � � � � � � � � � � �

a. 𝑆′′ሺ𝑥0ሻ= 𝑆′′ሺ𝑥𝑛ሻ= 0 ( ° ¦¦¤ µ·®¦º° ° °· ¦³� � � � � � � )

b. 𝑆′ሺ𝑥0ሻ= 𝑓′ሺ𝑥0ሻ³ 𝑆′ሺ𝑥𝑛ሻ= 𝑓′ሺ𝑥𝑛ሻ ( ° ¥¹� � � )

Page 43: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline Interpolation𝑆൫𝑥𝑗൯= 𝑓൫𝑥𝑗൯ 𝑗= 0,1,…,𝑛 𝑆𝑗+1൫𝑥𝑗+1൯= 𝑆𝑗൫𝑥𝑗+1൯ 𝑗= 0,1,…,𝑛− 2 𝑆𝑗+1′ ൫𝑥𝑗+1൯= 𝑆𝑗′൫𝑥𝑗+1൯ 𝑗= 0,1,…,𝑛− 2 𝑆𝑗+1′′ ൫𝑥𝑗+1൯= 𝑆𝑗′′൫𝑥𝑗+1൯ 𝑗= 0,1,…,𝑛− 2

𝑥0 𝑥1 𝑥𝑗 𝑥𝑗+2 𝑥𝑗+1 𝑥𝑛

𝑆𝑗 𝑆𝑗+1

Page 44: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline Interpolation

·¥µ¤¡®»µ¤ ε µ¤Ä¦¼� � � � � � 𝑆𝑗ሺ𝑥ሻ= 𝑎𝑗 + 𝑏𝑗൫𝑥− 𝑥𝑗൯+ 𝑐𝑗൫𝑥− 𝑥𝑗൯2 + 𝑑𝑗൫𝑥− 𝑥𝑗൯3 𝑗= 0,1,…,𝑛− 1 µÁºÉ° Å o°� � � � � � 2 Åoªnµ� 𝑆𝑗൫𝑥𝑗൯= 𝑎𝑗 = 𝑓൫𝑥𝑗൯ µÁºÉ° ÅÄ o°� � � � � � � 3 Åo� 𝑎𝑗+1 = 𝑆𝑗+1൫𝑥𝑗+1൯= 𝑆𝑗൫𝑥𝑗+1൯

= 𝑎𝑗 + 𝑏𝑗൫𝑥𝑗+1 − 𝑥𝑗൯+ 𝑐𝑗൫𝑥𝑗+1 − 𝑥𝑗൯2 + 𝑑𝑗൫𝑥𝑗+1 − 𝑥𝑗൯3 𝑗= 0,1,…,𝑛− 2 Ä®oℎ𝑗 =൫𝑥𝑗+1 − 𝑥𝑗൯ 宦 � 𝑗= 0,1,…,𝑛− 1 oµ·¥µ¤� � 𝑎𝑛 = 𝑓ሺ𝑥𝑛ሻÂoª ³Åoªnµ� �

¤ µ¦ É� � 1 𝑎𝑗+1 = 𝑎𝑗 + 𝑏𝑗ℎ𝑗 + 𝑐𝑗ℎ𝑗2 + 𝑑𝑗ℎ𝑗3

Page 45: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline Interpolation 𝑆𝑗′ሺ𝑥ሻ= 𝑏𝑗 + 2𝑐𝑗൫𝑥 − 𝑥𝑗൯+ 3𝑑𝑗൫𝑥 − 𝑥𝑗൯2 ¹É� � 𝑆𝑗′൫𝑥𝑗൯= 𝑏𝑗 𝑗= 0,1,…,𝑛− 1 oµ·¥µ¤� � 𝑏𝑛 = 𝑆′ሺ𝑥𝑛ሻà ¥� ÁºÉ° ÅÄ o°� � � � � 4. 𝑆𝑗+1′ ൫𝑥𝑗+1൯= 𝑆𝑗′൫𝑥𝑗+1൯ nµ� � � 𝑥= 𝑥𝑗+1

Ä ¤ µ¦ n° ®oµ ³Åo� � � � � � � 𝑆𝑗+1′ ൫𝑥𝑗+1൯= 𝑆𝑗′൫𝑥𝑗+1൯= 𝑏𝑗 + 2𝑐𝑗൫𝑥𝑗+1 − 𝑥𝑗൯+ 3𝑑𝑗൫𝑥𝑗+1 − 𝑥𝑗൯2 ¤ µ¦ É� � 2 𝑏𝑗+1 = 𝑏𝑗 + 2𝑐𝑗ℎ𝑗 + 3𝑑𝑗ℎ𝑗2

Page 46: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline Interpolation

𝑆𝑗′′ሺ𝑥ሻ= 2𝑐𝑗 + 6𝑑𝑗൫𝑥 − 𝑥𝑗൯ Á¤ºÉ°Â nµ� � � 𝑥= 𝑥𝑗 ³Åo� � 𝑆𝑗′′൫𝑥𝑗൯= 2𝑐𝑗 É º°� � � 𝑐𝑛 = 12𝑆′′ሺ𝑥𝑛ሻÂ³Ä oÁºÉ° ÅÄ o°� � � � � � 5. ³Åo� �

¤ µ¦ É� � 3 𝑐𝑗+1 = 𝑐𝑗 + 3𝑑𝑗ℎ𝑗 𝑗= 0,1,…,𝑛− 1 µ ¤ µ¦ É� � � � 3 µ¤µ¦ ®µnµ� � 𝑑𝑗Å o�

𝑑𝑗 = ൫𝑐𝑗+1−𝑐𝑗൯3ℎ𝑗

Page 47: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline InterpolationÂ� � 𝑑𝑗 nµ¨ Ä ¤ µ¦ É� � � � � � 1³ ¤ µ¦ É� � 2 ³Åo� � ¤ µ¦ É� � 4 𝑎𝑗+1 = 𝑎𝑗 + 𝑏𝑗ℎ𝑗 + 13ℎ𝑗2൫2𝑐𝑗 + 𝑐𝑗+1൯ ³

¤ µ¦ É� � 5 𝑏𝑗+1 = 𝑏𝑗 + ℎ𝑗൫𝑐𝑗 + 𝑐𝑗+1൯ 𝑗= 0,1,…,𝑛− 1 Ã¥µ¦Âo ¤ µ¦ É� � � � � 4®µ𝑏𝑗 ³Åo� � ¤ µ¦ É� � 6 𝑏𝑗 = 1ℎ𝑗൫𝑎𝑗+1 − 𝑎𝑗൯− ℎ𝑗3 ൫2𝑐𝑗 + 𝑐𝑗+1൯ Á¨É¥ ¦¦ Ä®o ¨ ®¹É³Å�o� � � � � � � � � � �

𝑏𝑗−1 = 1ℎ𝑗−1൫𝑎𝑗 − 𝑎𝑗−1൯− ℎ𝑗−13 ൫2𝑐𝑗−1 + 𝑐𝑗൯

Page 48: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline Interpolation

 nµ� � � 𝑏𝑗³ 𝑏𝑗−1 ¨ Ä� � � ¤ µ¦ É� � 5 ɦ´ ¦¦ ¸ ¤µ®¹É³Å�o¦³�� ¤�µ¦� � � � � � � � � � � � � �

¤ µ¦ É� � 7 ℎ𝑗−1𝑐𝑗−1 + 2൫ℎ𝑗−1 + ℎ𝑗൯𝑐𝑗 + ℎ𝑗𝑐𝑗+1 = 3ℎ𝑗൫𝑎𝑗+1 − 𝑎𝑗൯−3ℎ𝑗−1൫𝑎𝑗 − 𝑎𝑗−1൯ 𝑗= 1,…,𝑛− 1 ¦³ ʤ ªÂ¦Å¤n¦µ nµº°� � � � � � � � � ൛𝑐𝑗ൟ𝑗=0𝑛 nª� ൛ℎ𝑗ൟ𝑗=0𝑛−1

³൛𝑎𝑗ൟ𝑗=0𝑛 Á}nµÉ ª ɦµ nµ� � � � � � � � � � �

Á¤ºÉ° ¦µ nµ� � � ൛𝑐𝑗ൟ𝑗=0𝑛 Âoª ³®µnµ ª ÉÁ®º°� � � � � � ൛𝑏𝑗ൟ𝑗=0𝑛−1Åoµ� � � ¤ µ¦ É� � 6³ ൛𝑑𝑗ൟ𝑗=0𝑛−1 µ� � ¤ µ¦ É� � 3Âoª 夵 ¦oµ� � ൛𝑆𝑗ൟ𝑗=0𝑛−1

Page 49: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Exampleµ o°¤¼� � � 𝑥 1 2 3 4 5 𝑓ሺ𝑥ሻ 0 1 0 1 0

®µ� � Spline ε µ¤Ä nª� � � � � ሾ1,2ሿ, ሾ2,3ሿ, ሾ3,4ሿ³ ሾ4,5ሿ É ° o° ´ÁºÉ° Å ° °· ¦³� � � � � � � � � � �

Ä®o 𝑆𝑗ሺ𝑥ሻ= 𝑎𝑗 + 𝑏𝑗൫𝑥− 𝑥𝑗൯+ 𝑐𝑗൫𝑥− 𝑥𝑗൯2 + 𝑑𝑗൫𝑥− 𝑥𝑗൯3 (𝑗= 0,1,2,3)

Ã¥� 𝑆= 𝑆0 ∪𝑆1 ∪𝑆2 ∪𝑆3 ¹É� � 𝑆0ሺ𝑥ሻ= 𝑎0 + 𝑏0ሺ𝑥− 1ሻ+ 𝑐0ሺ𝑥− 1ሻ2 + 𝑑0ሺ𝑥− 1ሻ3 宦´ nª� � � ሾ1,2ሿ

𝑆1ሺ𝑥ሻ= 𝑎1 + 𝑏1ሺ𝑥− 2ሻ+ 𝑐1ሺ𝑥− 2ሻ2 + 𝑑1ሺ𝑥− 2ሻ3 宦´ nª� � � ሾ2,3ሿ 𝑆2ሺ𝑥ሻ= 𝑎2 + 𝑏2ሺ𝑥− 3ሻ+ 𝑐2ሺ𝑥− 3ሻ2 + 𝑑2ሺ𝑥− 3ሻ3 宦´ nª� � � ሾ3,4ሿ 𝑆3ሺ𝑥ሻ= 𝑎3 + 𝑏3ሺ𝑥− 4ሻ+ 𝑐3ሺ𝑥− 4ሻ2 + 𝑑3ሺ𝑥− 4ሻ3 宦´ nª� � � ሾ4,5ሿ

Page 50: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example

µÁºÉ° ÅÄ o°� � � � � � � 2. 𝑆𝑗൫𝑥𝑗൯= 𝑎𝑗 = 𝑓൫𝑥𝑗൯ 宦 � 𝑗= 0,1,2,3,4Á¦µÅ o� 𝑎0 = 0, 𝑎1 = 1, 𝑎2 = 0, 𝑎3 = 1, 𝑎4 = 0 Ä®oℎ𝑗 = ൫𝑥𝑗+1 − 𝑥𝑗൯ 宦 � 𝑗= 0,1,2,3Á¦µÅ o� ℎ0 = ℎ1 = ℎ2 = ℎ3 = 1 µÁºÉ° Å ° °· ¦³ Éà ¥rε® Á¦µÅ o� � � � � � � � � � � � � � , 𝑐0 = 12𝑆′′ሺ𝑥0ሻ= 0, 𝑐𝑛 = 12𝑆′′ሺ𝑥𝑛ሻ= 0

Page 51: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Exampleµ ¤ µ¦ É� � � � 7Á¤ºÉ°Â� � 𝑖 = 1,2,3Å o�

ℎ0𝑐0 + 2ሺℎ0 + ℎ1ሻ𝑐1 + ℎ1𝑐2 = 3ℎ1ሺ𝑎2 − 𝑎1ሻ− 3ℎ0ሺ𝑎1 − 𝑎0ሻ ℎ1𝑐1 + 2ሺℎ1 + ℎ2ሻ𝑐2 + ℎ2𝑐3 = 3ℎ2ሺ𝑎3 − 𝑎2ሻ− 3ℎ1ሺ𝑎2 − 𝑎1ሻ ℎ2𝑐2 + 2ሺℎ2 + ℎ3ሻ𝑐3 + ℎ3𝑐4 = 3ℎ3ሺ𝑎4 − 𝑎3ሻ− 3ℎ2ሺ𝑎3 − 𝑎2ሻ Á¤ºÉ°Â nµ� � � 𝑎𝑗,ℎ𝑗 ³Åo� � 4𝑐1 + 𝑐2 = 3ሺ0− 1ሻ− 3ሺ1− 0ሻ= −6 𝑐1 + 4𝑐2 + 𝑐3 = 3ሺ1− 0ሻ− 3ሺ0− 1ሻ= 6 𝑐2 + 4𝑐3 = 3ሺ0− 1ሻ− 3ሺ1− 0ሻ= −6 ¹ÉÅoªnµ� � � 𝑐1 = −157 , 𝑐2 = 187 , 𝑐3 = −157

Page 52: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example

µ ¤ µ¦ É� � � � 6 𝑏𝑗 = 1ℎ𝑗൫𝑎𝑗+1 − 𝑎𝑗൯− ℎ𝑗3 ൫2𝑐𝑗 + 𝑐𝑗+1൯ ³Åo� � 𝑏0 = 1ℎ0ሺ𝑎1 − 𝑎0ሻ− ℎ03 ሺ2𝑐0 + 𝑐1ሻ= ሺ1− 0ሻ− 13ቀ0− 157ቁ= 127

𝑏1 = 1ℎ1ሺ𝑎2 − 𝑎1ሻ− ℎ13 ሺ2𝑐1 + 𝑐2ሻ= ሺ0− 1ሻ− 13ቀ−307 + 187ቁ= −37 𝑏2 = 1ℎ2ሺ𝑎3 − 𝑎2ሻ− ℎ23 ሺ2𝑐2 + 𝑐3ሻ= ሺ1− 0ሻ− 13ቀ367 − 157ቁ= 0 𝑏3 = 1ℎ3ሺ𝑎4 − 𝑎3ሻ− ℎ33 ሺ2𝑐3 + 𝑐4ሻ= ሺ0− 1ሻ− 13ቀ−307 + 0ቁ= 37

Page 53: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example

³ µ� � ¤ µ¦ É� � 3 𝑑𝑗 = ൫𝑐𝑗+1−𝑐𝑗൯3ℎ𝑗 ³Åo� � 𝑑0 = 13ℎ0ሺ𝑐1 − 𝑐0ሻ= 13ቀ−157 − 0ቁ= −57 𝑑1 = 13ℎ1ሺ𝑐2 − 𝑐1ሻ= 13ቀ187 + 157ቁ= 117

𝑑2 = 13ℎ2ሺ𝑐3 − 𝑐2ሻ= 13ቀ−157 − 187ቁ= −117

𝑑3 = 13ℎ3ሺ𝑐4 − 𝑐3ሻ= 13ቀ0+ 157ቁ= 57

Page 54: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example

ª ¦³¤µ� � � Spline ε µ¤ ¦³ ° oµ¥� � � � � �

𝑆0ሺ𝑥ሻ= 0+ 127 ሺ𝑥− 1ሻ+ 0− 57ሺ𝑥− 1ሻ3 𝑆1ሺ𝑥ሻ= 1− 37ሺ𝑥− 2ሻ− 157 ሺ𝑥− 2ሻ2 + 117 ሺ𝑥− 2ሻ3 𝑆2ሺ𝑥ሻ= 0+ 0+ 187 ሺ𝑥− 3ሻ2 − 117 ሺ𝑥− 3ሻ3 𝑆3ሺ𝑥ሻ= 1+ 37ሺ𝑥− 4ሻ− 157 ሺ𝑥− 4ሻ2 + 57ሺ𝑥− 4ሻ3

Page 55: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example®µ� � Spline ε µ¤ ° ¥¹ ÉÄo¦³¤µ¢{rÉ� � � � � � � � � � � � � 𝑓ሺ𝑥ሻ= 3𝑥𝑒𝑥 − 𝑒2𝑥 � . »� � 𝑥= 1.03 µ o°¤¼Ä µ¦µ� � � � � � 𝑥 1.0 1.02 1.04 1.06 𝑓ሺ𝑥ሻ 0.765789386 0.795366779 0.822688170 0.847522258 𝑓′ሺ𝑥ሻ 1.5315787 1.1754977

Ä ÉÊ� � � 𝑛 = 3Ä®o 𝑆𝑗ሺ𝑥ሻ= 𝑎𝑗 + 𝑏𝑗൫𝑥− 𝑥𝑗൯+ 𝑐𝑗൫𝑥− 𝑥𝑗൯2 + 𝑑𝑗൫𝑥− 𝑥𝑗൯3 Ã¥É� � 𝑥0 = 1.0, 𝑥1 = 1.02, 𝑥2 = 1.04, 𝑥3 = 1.06

ℎ =൫𝑥𝑗+1 − 𝑥𝑗൯= 0.02 ( 𝑗= 0,1,2) ³𝑆= 𝑆0 ∪𝑆1 ∪𝑆2

Page 56: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example

µÁºÉ° ÅÄ o°� � � � � � � 2. ɪnµ� 𝑆𝑗൫𝑥𝑗൯= 𝑎𝑗 = 𝑓൫𝑥𝑗൯ 宦 � 𝑗= 0,1,2,3Ä®o 𝑎0 = 𝑓ሺ𝑥0ሻ= 0.7657894 𝑎1 = 𝑓ሺ𝑥1ሻ= 0.7953668 𝑎2 = 𝑓ሺ𝑥2ሻ= 0.8226882 𝑎3 = 𝑓ሺ𝑥3ሻ= 0.8475223 µÁºÉ° Å ° ¥¹Éªnµ� � � � � � � � � 𝑆′ሺ𝑥0ሻ= 𝑓′ሺ𝑥0ሻ³ 𝑆′ሺ𝑥𝑛ሻ= 𝑓′ሺ𝑥𝑛ሻÁ¦µÄ®o

𝑏0 = 1.5315787 𝑏3 = 1.1754977

Page 57: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Exampleµ¦³ ¤ µ¦Ä� � � � � � ¤ µ¦ É� � 6Á¤ºÉ° 𝑏𝑗 = 1ℎ𝑗൫𝑎𝑗+1 − 𝑎𝑗൯− ℎ𝑗3 ൫2𝑐𝑗 + 𝑐𝑗+1൯ 宦 �𝑗= 0,1,2³ nµ� � � 𝑎0,𝑎1,𝑎2,𝑎3³ 𝑏0,𝑏3Åo¦³ ¤ µ¦� � � � (1) 1.5315787 = 𝑏0 = 1.47887− 0.01333𝑐0 − 0.006667𝑐1

𝑏1 = 1.36607− 0.01333𝑐1 − 0.006667𝑐2 𝑏2 = 1.241722− 0.01333𝑐2 − 0.006667𝑐3 µ¦³ ¤ µ¦Ä� � � � � � ¤ µ¦ É� � 5 𝑏𝑗+1 = 𝑏𝑗 + ℎ𝑗൫𝑐𝑗 + 𝑐𝑗+1൯ 宦� 𝑗= 0,1,2³

 nµ� � � 𝑏0,𝑏3Åo¦³ ¤ µ¦� � � � (2) 𝑏1 = 1.5315787+ 0.02𝑐0 + 0.02𝑐1 𝑏2 = 𝑏1 + 0.02𝑐1 + 0.02𝑐2 1.1754977 = 𝑏3 = 𝑏2 + 0.02𝑐2 + 0.02𝑐3

Page 58: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: ExampleÂo¦³ ¤ µ¦� � � � (1) ³ (2) Ã¥µ¦¦ª¤ ° ¤ µ¦Å o¦³ ¤ µ¦� � � � � � � � 3 Á}� � 𝑐0 + 0.5𝑐1 = −4.2687857 𝑐0 + 0.33333𝑐1 − 0.33333𝑐2 = 8.275435 𝑐2 − 𝑐3 = 9.9305985 𝑐0 − 2𝑐2 − 𝑐3 = 17.80405 Âo¦³ ¤ µ¦ É� � � � � (3) Åo� 𝑐0 = 46.109654, 𝑐1 = 100.75688 𝑐2 = 12.745401, 𝑐3 = 2.814803

Page 59: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example

 nµ¨ Ħ³ ¤ µ¦� � � � � � � � � (2) ¦ª¤ ´ÁºÉ° Å ° ¥¹Á¦µÅ o� � � � � � � � � 𝑏0 = 1.5315787, 𝑏1 = 2.6245232 𝑏2 = 0.8642936, 𝑏3 = 1.1754977  nµÄ� � � � ¤ µ¦ É� � 3Á¤ºÉ° 𝑗= 0,1,2Å o� 𝑑0 = −2447.7756 , 𝑑1 = 1891.7047, 𝑑2 = −165.50997

Page 60: Chapter 3 Interpolation and Polynomial Approximation

Cubic Spline: Example´ Ê� � � � 𝑆= 𝑆0 ∪𝑆1 ∪𝑆2à ¥ É� � 𝑆0ሺ𝑥ሻ= 0.7657894+ 1.5315787ሺ𝑥− 1ሻ+ 46.109654ሺ𝑥− 1ሻ2− 2447.7756ሺ𝑥− 1ሻ3 𝑆1ሺ𝑥ሻ= 0.7953668+ 2.6245232ሺ𝑥− 1.02ሻ+ 100.75688ሺ𝑥− 1.02ሻ2 + 1891.7047ሺ𝑥− 1.02ሻ3 𝑆2ሺ𝑥ሻ= 0.8226882+ 0.8642936ሺ𝑥− 1.04ሻ+ 12.745401ሺ𝑥− 1.04ሻ2 − 165.50997ሺ𝑥− 1.04ሻ3 ´ Ê nµ¦³¤µ °� � � � � � � � � 𝑓ሺ1.03ሻ º°� 𝑆1ሺ1.03ሻ= 0.7953668+ 2.6245232ሺ1.03− 1.02ሻ+ 100.75688ሺ1.03− 1.02ሻ2 + 1891.7047ሺ1.03− 1.02ሻ3

Page 61: Chapter 3 Interpolation and Polynomial Approximation

Least Square Methodในกรณท�ค�าข�อม�ลท�ว�ดมาน��น อาจมความคลาดเคล��อน การประมาณค�าฟ�งก ช�นภายใต�เง��อนไขท�ว�า ค�าท�ได�จากการว�ดจะต�องเท�าก�บค�าประมาณของฟ�งก ช�น ณ จ&ดต�างๆ ก(อาจจะท)าให�ค�าประมาณย+�งคลาดเคล��อนไปจากค�าท�ควรจะเป,นนอกจากน� ถ�ามข�อม�ลจ)านวนหน.�ง เช�น n+1 และใช�การประมาณพห&นามในช�วงเช�น พห&นามลากรองจ พห&นามผลต�างส�บเน��องของน+วต�น ก(จะได�พห&นามดกร n ซ.�งอาจมการกว�ดแกว�งของเส�นโค�งแทนท�จะเป,นโค�งของพห&นามท�มดกรต)�ากว�าภายใต�สมม&ต+ฐานเหล�าน� เราอาจส�งเกตล�กษณะของการเรยงต�วของข�อม�ลแล�วจ.งสร�างพห&นามต�วประมาณท�มดกรท�เหมาะสม ท�ประมาณได�ดท�ส&ด (ในบางล�กษณะ) โดยไม�จ)าเป,นจะต�องผ�านจ&ดข�อม�ลท&กจ&ด ซ.�งจะต�องหาส�มประส+ทธ+8ของพห&นามน��น

Page 62: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method

Page 63: Chapter 3 Interpolation and Polynomial Approximation

Least Square Methodoµo°¤¼Á¦¥ ª ´ oµ¥´Á o ¦ Á¦µ³ ¦³¤µ oª¥¡®»µ¤ ¦¸� � � � � � � � � � � � � � � � � � � 1 ®¦º°¢{ r Á·� � � � � �

Á o�

Ä®o𝑎𝑥𝑖 + 𝑏Á}nµÉ� � � � 𝑖 ° Á o ¦ ÉÄo¦³¤µÂ³� � � � � � � � � 𝑦𝑖Á}nµ¦· ° ¢{r ɼ� � � � � � � � � � � � � �¦³¤µÁ¤ºÉ° o° µ¦ ¦oµ¢{r ¦³¤µ ÉÄ®onµÄ onµ¦· É »³¡·�µ¦�µ¢{��r���°�� � � � � � � � � � � � � � � � � � � � � � � � � � � � �ªµ¤ µÁ ºÉ°� � � � � 𝐸ሺ𝑎,𝑏ሻ³®µnµ� 𝑎,𝑏 ÉÁ®¤µ³ ¤�

Á¦µ°µ ·¥µ¤ ° ªµ¤ µÁ ºÉ°� � � � � � � � � 𝐸ሺ𝑎,𝑏ሻ Åo®µ¥Â Án� � � � � ¢{r ° ªµ¤ µÁ ºÉ°  ¤··Â¤ r� � � � � � � � � � � � � � � � (Minimax Error Function) 𝐸∞ሺ𝑎,𝑏ሻ= mȁx𝑖=1,2,…,10ሼȁ𝑦𝑖 −ሺ𝑎𝑥𝑖 + 𝑏ሻȁሽ ¢{r ° ªµ¤ µÁ ºÉ°  ÁÉ¥Á ¤ ¼¦ r� � � � � � � � � � � � � � � � � � � (Absolute derivation Error Function) 𝐸𝑙ሺ𝑎,𝑏ሻ= σ ȁ𝑦𝑖 −ሺ𝑎𝑥𝑖 + 𝑏ሻȁ10𝑖=1

Page 64: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method¢{r´ ° ªµ¤ µÁ ºÉ° ε ° o°¥ »¦ª¤� � � � � � � � � � � � � � � � (Total Square Error)

𝐸2ሺ𝑎,𝑏ሻ= σ ൫𝑦𝑖 −ሺ𝑎𝑥𝑖 + 𝑏ሻ൯210𝑖=1

µ¦®µnµÉε »ÎµÅ�oÃ�¥Â�o ¤�µ¦� � � � � � � � �

0 = 𝜕𝜕𝑎 σ ൫𝑦𝑖 −ሺ𝑎𝑥𝑖 + 𝑏ሻ൯210𝑖=1 = 2σ ሺ𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏ሻሺ−𝑥𝑖ሻ10𝑖=1

³ 0 = 𝜕𝜕𝑏 σ ൫𝑦𝑖 −ሺ𝑎𝑥𝑖 + 𝑏ሻ൯210𝑖=1 = 2σ ሺ𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏ሻሺ−1ሻ10𝑖=1

Á¤ºÉ° ° ¦¼³Å�o� � � � � � 𝑎σ 𝑥𝑖2𝑚𝑖=1 + 𝑏σ 𝑥𝑖𝑚𝑖=1 = σ 𝑥𝑖𝑦𝑖𝑚𝑖=1

³ 𝑎σ 𝑥𝑖𝑚𝑖=1 + 𝑏𝑚= σ 𝑦𝑖𝑚𝑖=1

Page 65: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method

Á¤ºÉ°Âo¦³ ¤ µ¦Á¡ºÉ°®µnµ� � � � � 𝑎 ³𝑏 ³Åo� � 𝑎 = 𝑚σ 𝑥𝑖𝑦𝑖𝑚𝑖=1 −σ 𝑥𝑖𝑚𝑖=1 σ 𝑦𝑖𝑚𝑖=1𝑚σ 𝑥𝑖2𝑚𝑖=1 −൫σ 𝑥𝑖𝑚𝑖=1 ൯

2

𝑏= σ 𝑥𝑖2𝑚𝑖=1 σ 𝑦𝑖𝑚𝑖=1 −σ 𝑥𝑖𝑦𝑖𝑚𝑖=1 σ 𝑥𝑖𝑚𝑖=1𝑚σ 𝑥𝑖2𝑚𝑖=1 −൫σ 𝑥𝑖𝑚𝑖=1 ൯2

¹É³ εÅ nµÄ¢{r´Á·Á o� � � � � � � � � � � � � � � � 𝑃1ሺ𝑥ሻ= 𝑦= 𝑎𝑥+ 𝑏

Page 66: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method: Example®µ ¤ µ¦Á o ¦ Ã¥ª· ε ° o°¥ » ɦ³¤µ µ o°¤¼Éε® Ä®o� � � � � � � � � � � � � � � � � � � � � � � 𝑥𝑖 1 2 3 4 5 6 7 8 9 10 𝑦𝑖 1.3 3.5 4.2 5.0 7.0 8.8 10.1 12.5 13.0 15.6

µ� � 𝑎 = 𝑚σ 𝑥𝑖𝑦𝑖𝑚𝑖=1 −σ 𝑥𝑖𝑚𝑖=1 σ 𝑦𝑖𝑚𝑖=1𝑚σ 𝑥𝑖2𝑚𝑖=1 −൫σ 𝑥𝑖𝑚𝑖=1 ൯2

𝑏= σ 𝑥𝑖2𝑚𝑖=1 σ 𝑦𝑖𝑚𝑖=1 −σ 𝑥𝑖𝑦𝑖𝑚𝑖=1 σ 𝑥𝑖𝑚𝑖=1𝑚σ 𝑥𝑖2𝑚𝑖=1 −൫σ 𝑥𝑖𝑚𝑖=1 ൯2

ɳ o° ®µ� � � � � σ 𝑥𝑖𝑚𝑖=1 , σ 𝑦𝑖𝑚𝑖=1 , σ 𝑥𝑖2𝑚𝑖=1 , σ 𝑥𝑖𝑦𝑖𝑚𝑖=1

Page 67: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method: Example𝑥𝑖 𝑦𝑖 𝑥𝑖2 𝑥𝑖𝑦𝑖 1 1.3 1 1.3 2 3.5 4 7.0 3 4.2 9 12.6 4 5.0 16 20.0 5 7.0 25 35.5 6 8.8 36 52.8 7 10.1 49 70.7 8 12.5 64 100.0 9 13.0 81 117.0 10 15.6 100 156.0 55 81.0 385 572.4

µ� � 𝑎 = 𝑚σ 𝑥𝑖𝑦𝑖𝑚𝑖=1 −σ 𝑥𝑖𝑚𝑖=1 σ 𝑦𝑖𝑚𝑖=1𝑚σ 𝑥𝑖2𝑚𝑖=1 −൫σ 𝑥𝑖𝑚𝑖=1 ൯2

𝑏= σ 𝑥𝑖2𝑚𝑖=1 σ 𝑦𝑖𝑚𝑖=1 −σ 𝑥𝑖𝑦𝑖𝑚𝑖=1 σ 𝑥𝑖𝑚𝑖=1𝑚σ 𝑥𝑖2𝑚𝑖=1 −൫σ 𝑥𝑖𝑚𝑖=1 ൯2

³Åo� � 𝑎 = 10ሺ572.4ሻ−55ሺ81ሻ10ሺ385ሻ−ሺ55ሻ2 = 1.538 𝑏= 385ሺ81ሻ−55ሺ572.4ሻ10ሺ385ሻ−ሺ55ሻ2 = −0.360

𝑃ሺ𝑥𝑖ሻ= 1.538𝑥𝑖 − 0.360 1.18 2.72 4.25 5.79 7.33 8.87 10.41 11.94 13.48 15.02

𝐸= ൫𝑦𝑖 − 𝑃ሺ𝑥𝑖ሻ൯210𝑖=1 ≈ 2.34

Page 68: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method: Example

0 1 2 3 4 5 6 7 8 9 10-2

0

2

4

6

8

10

12

14

16

Page 69: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method ¡®»µ¤� 𝑃𝑛ሺ𝑥ሻ= σ 𝑎𝑘𝑥𝑘𝑛𝑘=0 ɤ ¦¸� � � 𝑛 < 𝑚− 1 È µ¤µ¦ ¦oµÅooª¥¦³Á¥ª· ε °� � � � � � � � � � �o°¥ »ª· µ¦ º°®µ� � � � � 𝑎0,…,𝑎𝑛Á¡ºÉ° εĮonµ·¡ µ ε ° o°¥ »¦ª¤� � � � � � � � � �

𝐸= σ ൫𝑦𝑖 − 𝑃𝑛ሺ𝑥𝑖ሻ൯2𝑛𝑖=1

¤ nµÉε »¹É��o°�Å�oªnµ� � � � � � � � 𝜕𝐸𝜕𝑎𝑗 = 0 宦 � 𝑖 = 0,1,2,…,𝑛 ¹ÉÅo¦³ ¤ µ¦� � � � � � 𝑛+ 1 ¤ µ¦Ä� �𝑎𝑗 𝑎0 𝑥𝑖0

𝑚𝑖=1 + 𝑎1 𝑥𝑖1

𝑚𝑖=1 + 𝑎2 𝑥𝑖2

𝑚𝑖=1 + ⋯+ 𝑎𝑛 𝑥𝑖𝑛

𝑚𝑖=1 = 𝑦𝑖𝑥𝑖0

𝑚𝑖=1

𝑎0 𝑥𝑖1𝑚

𝑖=1 + 𝑎1 𝑥𝑖2𝑚

𝑖=1 + 𝑎2 𝑥𝑖3𝑚

𝑖=1 + ⋯+ 𝑎𝑛 𝑥𝑖𝑛+1𝑚𝑖=1 = 𝑦𝑖𝑥𝑖1

𝑚𝑖=1

⋮ 𝑎0 𝑥𝑖𝑛

𝑚𝑖=1 + 𝑎1 𝑥𝑖𝑛+1𝑚

𝑖=1 + 𝑎2 𝑥𝑖𝑛+2𝑚𝑖=1 + ⋯+ 𝑎𝑛 𝑥𝑖2𝑛𝑚

𝑖=1 = 𝑦𝑖𝑥𝑖𝑛𝑚

𝑖=1

Page 70: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method: Example

µ o°¤¼Ä µ¦µ Éε® Ä®o ¦³¤µ nµ¢{r´ oª¥¡®»µ¤ ¦ ° oª¥¦³Á�¥�ª·�� � � � � � � � � � � � � � � � � � � � � � � � � � � �ε ° o°¥ »ÁȤ®nª¥� � � � � � � 𝑥𝑖 0 0.25 0.50 0.75 1.00 𝑦𝑖 1.0000 1.2840 1.6487 2.1170 2.7183

Ä ÉÊ� � � 𝑛 = 2,𝑚 = 5 ¤ µ¦ · Ê µ¤ ¤ µ¦ º°� � � � � � � � 5𝑎0 + 2.5𝑎1 + 1.875𝑎2 = 8.7680 2.5𝑎0 + 1.875𝑎1 + 1.5625𝑎2 = 5.4514 1.875𝑎0 + 1.5625𝑎1 + 1.3828𝑎2 = 4.4015

Page 71: Chapter 3 Interpolation and Polynomial Approximation

Least Square Method: Example

Á¤ºÉ°Âo¦³ ¤ µ¦ ³Åo� � � � � � 𝑎0 = 1.0052, 𝑎1 = 0.8641, 𝑎2 = 0.8437 ¡®»µ¤ ÉÅoº°� � � � 𝑃2ሺ𝑥ሻ= 1.0052+ 0.8641𝑥+ 0.84370.8437𝑥2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5