4 À ìÀ O ƒEâ...

54
Œ˘%ˇ£˘ 18ø{ 4 ˘, O¯˘E˘ Email:[email protected]

Transcript of 4 À ìÀ O ƒEâ...

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êÆï�Ø%Ï£��Æ��

111888ùùùµµµ���������{{{

4�ÀìÀ�Æ, O�Å�Æ�EâÆ�

Email:[email protected]

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�ùÌ�SN

1 ����µÌ�0�.�KF��õ�ª��E�{!Ø��O¶©ã�5���{!�^���{�

2 ����� MATLAB ¢y3 �����A^

4 ��öSK

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3¢S¯K*ÿ¥, X�/�t&!/eYY iÿ!///mÿ±!�Æä�ã�×£�, ½u¤�Lp½*ÿÃã��, =U��Ü©lÑ�!$�´Ø5K�!DÕ�*ÿ: �*ÿ�. XÛWWWÖÖÖ"""���:::   ���êêêâââ½ö������,,,���***ÿÿÿ:::   ���¼¼¼êêê���´�ÆïÄ�ó§¢�¥²~���¯K.

ùa¯K�Ä�A:´: ¢S¯K´��1w�­�½­¡, �¼ê'XL�ª��, =�âÜ©*ÿ:�êâ, F"�Ñ,«�{��Ñ��*ÿ«�S?¿�*ÿ:�¼ê�.

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)ûù�¯K��Ð�{Ò´ÏÏϦ¦¦������BBBuuuOOO������¼¼¼êêêLLL���ªªª5CCCqqq%%%CCC¢S¯K.EEE,,,¼¼¼êêê���nnnØØØOOO���¯̄̄KKK:

nØO�¥²~����©E,½(J�¯K,�X¦�ê!È©�. XU�Ñ,«{{{üüü���CCCqqqOOO������{{{E,´�©­��k^�.d�|||^̂̂���¼¼¼êêê���)))���:::���¼¼¼êêê���, |^)ûcã¯K�g�, ÏÏϦ¦¦������CCCqqq���!!!BBBuuuOOO������{{{üüü¼¼¼êêê���OOO���555EEE,,,���!!!¦¦¦)))(((JJJ���¼¼¼êêê, �´êÆþ~^�Ãã��, ùa¼ê�CqL�½%C¯K, 3êêê������©©©!!!êêê���ÈÈÈ©©©!!!���©©©���§§§êêê���)))¥k2��A^.

êÆþ)ûda¯K�~^�{´���������[[[ÜÜÜ���{{{.

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~~~1 ��Ü��*ÿ,àã¥Y�, z� 1 Km � 1 �*ÿä¡, *ÿ(J�L1. Á�O x = 1.5, 2.6 Km ½Ù§?¿�*ÿ:?�p��íßÝ�.

L1µp��í*ÿ(J*ÿÕ:xi , Km 1 2 3 4 5 6*ÿßÝyi , mg/L 16 18 21 17 15 12

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�!����

1.��¯K�£ã

®�¼ê y = f (x ) 3 n + 1 �pÉ*ÿ:

x0, x1, x2, · · · , xn

þ�¼ê�(½*ÿ�)

yi = f (xi), i = 0, 1, · · · , n

�¦�^Cq­� φ(x ),¦ÙÏL¤kêâ:§=

φ(xi) = yi , i = 0, 1, · · · , n. (1)

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é?¿�*ÿ: x̂ (x̂ 6= xi , i = 0, 1, · · · n), �¦ f (x̂ ).

Q, φ(x ) ´ f (x ) �Cq¼ê, =φ(x ) ≈ f (x ), K φ(x̂ ) ���±�� f (x̂ ) �Cq�O�, = φ(x̂ ) ≈ f (x̂ ).

Ï~¡daï�¯K�������¯̄̄KKK, �ECq¼ê��{Ò¡����������{{{.

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2.�Vg

*ÿ: xi(i = 0, 1, . . . , n) ¡�������!!!:::, f (x ) ¡�������¼¼¼êêê½���¼¼¼êêê, φ(x ) �������¼¼¼êêê.

φ(xi) = yi , i = 0, 1, · · · , n ¡�������^̂̂���(OOOKKK).

Pa = min

0≤i≤n{xi}, b = max

0≤i≤n{xi}

K¹ xi(i = 0, 1, . . . , n) ���«m [a, b] ¡�������«««mmm,

x̂ (x̂ 6= xi , i = 0, 1, · · · , n ��*ÿ:) ¡�������:::,φ(x̂ ) ���¼ê f (x ) 3 x̂ ∈ [a, b] :?�������.

e x̂ ∈ [a, b] K¡�SSS���, ÄK¡�ííí.

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���555:

���{��^u��«mSÜ:�¼ê��O½ýÿ, |^T�{?1ª³íýÿ�, �?1áÏýÿ�O, é¥�Ïýÿ¿Ø·^

��¼ê�ÀJkNõ, Xkn¼ê!õ�ª�. Ù¥,õ�ª��´~�����{, ¡�õ�ª��. �A���¼ê¡���õ�ª.���333���¯̄̄KKKµµµ

1 XÛ�ECq¼ê£­�¤φ(x )§¦÷v¯K��¦º

2 ^Cq¼ê φ(x )O� f (x ),= f (x ) ≈ φ(x ), ¬�)õ��Ø�?

3 X���O?

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3. ����õ�ª��E�{µ3.1 — �½Xê�{

�g��

®�¼ê y = f (x ) 3ü�pÉ: x0 Ú x1 ?�¼ê�y0 Ú y1.

L: (x0, y0) Ú: (x1, y1) ���^��

���§��E�{�|^Ð�êÆ�:�ª½ü:ª:::���ªªªµ

y − y0 =y1 − y0x1 − x0

(x − x0)

üüü:::ªªª:y − y0x − x0

=y1 − y0x1 − x0

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�g��õ�ª

®�n�pØ�Ó�: x0, x1, x2, 9Ùn�:?�*ÿ� y0, y1, y2

¦�gõ�ª

P2(x ) = a0 + a1x + a2x2

¦ P2(xi) = yi , i = 0, 1, 2, Ù¥ a0, a1, a2 ��½Xê

d��OK, �¦ P2(xi) = yi , i = 0, 1, 2, �� a0 + a1x0 + a2x20 = y0

a0 + a1x1 + a2x21 = y1

a0 + a1x2 + a2x22 = y2

)dn��g�§|=�¦� a0, a1, a2.

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�� n g��õ�ª��E—�½Xê{

®�3 n + 1 �:pØ�Ó�: xi , i = 0, 1, · · · , n þ�¼ê� yi , i = 0, 1, · · · , n.�E n g��õ�ª

Pn(x ) = a0 + a1x + a2x2 + · · ·+ anx

n

¦ Pn(xi) = yi , i = 0, 1, · · · , n.d��OK Pn(xi) = yi , i = 0, 1, · · · , n k

a0 + a1x0 + a2x20 + · · ·+ anx

n0 = y0

a0 + a1x1 + a2x21 + · · ·+ anx

n1 = y1

· · · · · · · · ·a0 + a1xn + a2x

2n + · · ·+ anx

nn = yn

(2)

)d n + 1 ��5�§|���½Xê a0, a1, · · · , an .

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AAA:::555ººº

|^ n + 1 �*ÿ:�õ�±(½�� n gõ�ª.

�§(2) �XêÝ����Ý1 x0 x 2

0 · · · xn0

1 x1 x 21 · · · xn

1

1 x2 x 22 · · · xn

2...

......

. . ....

1 xn x 2n · · · xn

n

d�5�ê�£�: ��=�*ÿ: xi , i = 0, 1, · · · , npØ�Ó�, dÝ��÷�Ý, =�§|âk��).

� n v��, |^�½Xê�{(½��õ�ª¿Ø´��Ð��{.

½½½nnn 1 ÷v��^�(1)�gêØ�L n �õ�ª´�3 ����.

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3.2 .�KF(Lagrange) ���{

.�KF�5��

®�¼ê y = f (x ) 3ü�pÉ: x0 Ú x1 ?�¼ê�y0 Ú y1.

L: (x0, y0) Ú: (x1, y1) ���^��, ±:�ª�~

y − y0 =y1 − y0x1 − x0

(x − x0)

Ü¿äk�Ó y0, y1 ��,ò§�¤Xe/ª

P1(x ) = y0x − x1x0 − x1

+ y1x − x0x1 − x0

(3)

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eÚ?PÒ:

l0(x ) =x − x1x0 − x1

, l1(x ) =x − x0x1 − x0

K(3)ª�±�¤:

P1(x ) = y0l0(x ) + y1l1(x ) (4)

Ù¥ l0(x )÷v:

l0(x0) = 1, l0(x1) = 0;

l1(x ) ÷v:

l1(x0) = 0, l1(x1) = 1

¡ l0(x ), l1(x ) ��5��½�g.�KF���ÄÄļ¼¼êêê,¡(4)ª�...���KKKFFF���555������¼¼¼êêê½���ggg...���KKKFFF������úúúªªª

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�5��ļ꫿ã

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.�KF�Ô���úª

®� y = f (x ) 3pÉ�3 �: x0, x1 Ú x2 ?�¼ê�y0, y1 Ú y2

�¦��gêØ�L2�õ�ªy = P2(x ), ¦Ù÷v

P2(x0) = y0, P2(x1) = y1, P2(x2) = y2 (5)

�En�ļê: l0(x ), l1(x ), l2(x ), §�Ñ´�g¼ê,©O÷ve�^�:

l0(x0) = 1, l0(x1) = 0, l0(x2) = 0

l1(x0) = 0, l1(x1) = 1, l1(x2) = 0 (6)

l2(x0) = 0, l2(x1) = 0, l2(x2) = 1

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ļê l0(x ), l1(x ), l2(x ) L�ª�(½: ± l0(x ) �~,Äkl0(x ) ´�gõ�ª, � x1, x2 ´ l0(x ) �":, �l0(x ) �L�ª/ª�

l0(x ) = c(x − x1)(x − x2)

Ù¥ c ��½Xê, qd l0(x0) = 1, �\þª)Ñ

c =1

(x0 − x1)(x0 − x2),

l �� l0(x ) �L�ª�:

l0(x ) =(x − x1)(x − x2)

(x0 − x1)(x0 − x2)

Ón��:

l1(x ) =(x − x0)(x − x2)

(x1 − x0)(x1 − x2), l2(x ) =

(x − x1)(x − x0)

(x2 − x1)(x2 − x0).

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.�KF�Ô.��õ�ª

�â±þ?ا·���.�KF�Ô.��õ�ª�:

P2(x ) = y0l0(x ) + y1l1(x ) + y2l2(x ) (7)

¡ P2(x ) �...���KKKFFF���ÔÔÔ...������õõõ���ªªª½���ggg...���KKKFFF������õõõ���ªªª.

~~~2 �â~1 �êâ, ^�Ô�{O�√5 �Cq�.

))) ���C x = 5 �n: x0 = 1, x1 = 4, x2 = 9 ���!:, �â�Ô���úª(7), k

√5 ≈ P2(5) = 1× (5− 4)(5− 9)

(1− 4)(1− 9)

+2× (5− 1)(5− 9)

(4− 1)(4− 9)+ 3× (5− 1)(5− 4)

(9− 1)(9− 4)≈ 2.67

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í2µn g.�KF��õ�ª

n ggg...���KKKFFF������õõõ���ªªª�L«¤ n + 1 � n g��ļê l0(x ), l1(x ), · · · , ln(x ) ��5|Ü

Pn(x ) = y0l0(x ) + y1l1(x ) + . . .+ yn ln(x ) =n∑

k=0

yk lk(x )

(8)

ª¥ lk(x ) L«éAu������!!!::: xk � n ggg������ÄÄļ¼¼êêê½...���KKKFFFÄÄÄ���õõõ���ªªª, ÷v

lk(xi) =

{1, i = k

0, i 6= k, k = 0, 1, . . . , n (9)

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�c¡í�aq, �±d(9)ª�� lk(x ) �äNL�ª{lk(x ) =

(x−x0)...(x−xk−1)(x−xk+1)...(x−xn )(xk−x0)...(xk−xk−1)(xk−xk+1)...(xk−xn )

,

k = 0, 1, . . . , n;(10)

½{�¤

lk(x ) =n∏

j=0,j 6=k

(x − xj )

xk − xj, k = 0, 1, · · · , n.

w,, lk(x ), k = 0, 1, · · · , n �5Ã', �þ�gêØ�L n �õ�ª. Ïd, Ù�5|Ü Pn(x ) ½�gêØ�L n �õ�ª, �÷v��^�

Pn(xk) = yk , k = 0, 1, . . . , n.

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4. ��õ�ª�{�9Ø��O

PRn(x ) = f (x )− Pn(x ).

¡ Rn(x ) � n g��õ�ª����äääØØØ���, ½¡�������{{{���.

½½½nnn2 � x0, x1, · · · , xn ´«m [a, b] þ�pØ�Ó�:§¼ê f (x ) 3 [a, b] þk n �ëY�ê§�3(a, b) S�3 n + 1 �k.�ê, Pn(x ) ´Äuù n + 1�:þ� n g.�KF��õ�ª§Ké«m [a, b]þ�?Û�: x§7�3�: ξ ∈ (a, b) (ξ �6u x ),¦�

Rn(x ) = f (x )−Pn(x ) =f (n+1)(ξ)

(n + 1)!(x−x0)(x−x1) . . . (x−xn).

(11)

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��Ø�����ûu f (n+1)(ξ) Ú(x − x0)(x − x1) · · · (x − xn) ü�.

� f (x ) ��½L�ª��E,�, �O f (n+1)(ξ) �þ.´�©(J�.

���: x u��«mSÜ, cÙ´¥:NC�, Ø��é��, ���: u��«mà:NC½3��«m±(�) �, Ø��é��©

.�KF��æ����E�g�, �{�*!´n)!´?§. Ù��úª9ÙØ��O½n3ê�È©!ê��©�ê�O��{��{�O9ÙØ��O¥kX2� ­���A^.

bX3*ÿ:þØ=��¼ê�*ÿ�, Ó��®�T:�¼ê�ê�, KE�æ^.�KF��¥�Eļê�g�, �EÙ��õ�ª, ùҴͶ����A(Hermite) ��.

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pg��õ�ª�999���yyy���

��!:�ê8û½��õ�ª�gê���õ�ª�gê,p�, ¬�5ØÓ§Ý�ê���, =u)¤¢�999���(Runge)yyy���µµµ

«~µ

3«m [-5,5]þ, )¤ 11 ��å��!:,xi , i = 0, 1, · · · , 10.3�A��!:þO�¼ê

y(x ) = 1/(1 + x 2)

�¼ê���*ÿ�, y(xi), i = 0, 1, 2, · · · , 10.|^ù 11 �êâ:, �±)¤��10 g.�KF��õ�ª P10(x ).

aq/, )¤�� 5 g��õ�ª P5(x ) , �ã'���¼ê P10(x ),P5(x ) Ú�¼ê f (x ) , *­�[Ü�J.

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9���y�«¿ã

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5. ©ã��

pg��õ�ª�1w5p, ���5�r

e��!:�é'��8�, æ^$g��õ�ª�U��éÐ�Cq.©©©ããã���������ÄÄÄ���ggg���µµµ

1 r��«my©¤eZ��«m2 3z��«mþ^$gõ�ª?1��3 3����«mþÒ����©ã��¼ê

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�5

«m�y©�±´?¿�¶

��«mþ��õ�ª�gê�À���UäN¯KÀJ©

©ã��{Ï~k�Ð�Âñ5Ú­½5, �{{ü,�Ñ9�y�, ���¼êØX.�KFõ�ª1w~^�©ã���{k:

ÛÜ1w:©ã�5��!©ã�g��!©ãng���.�^��

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5.1 ©ã�5���{

P3��«m [a, b] þk n + 1 ���!:, Ø�P�:

a = x0 < x1 < . . . < xn = b (12)

��¼ê f (x ) 3þã!:�*ÿ�½¼ê��

f (x0) = y0, f (x1) = y1, . . . , f (xn) = yn (13)

�gë���ü�êâ:

(xi−1, yi−1), (xi , yi), i = 1, 2, · · · , n,

�� n ^òy�ã, §�|¤�^ò�,

P P(x ) �«m [a, b] þù^ò�L«�¼ê, ¡�©©©ããã������¼¼¼êêê.

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©ã�5��¼ê�Ä�5�

1 3z���f«mS P(x )Ñ´�5¼ê, 3«m[a, b] þ P(x ) ëY, �3��!:þ���êØëY.

2 31 i �f«m [xi−1, xi ] þ P(x )�L�ª�

P(x ) = yi−1x − xi

xi−1 − xi+ yi

x − xi−1

xi − xi−1, xi−1 ≤ x ≤ xi (14)

3 éAuz���!:���ļê�

l0(x ) =

{ x−x1x0−x1

, x ∈ [x0, x1]

0, Ù¦.

li(x ) =

x−xi−1

xi−xi−1, x ∈ [xi−1, xi ]

x−xi+1

xi−xi+1, x ∈ [xi , xi+1], i = 1, · · · ,n − 1

0, Ù¦

ln(x ) =

{ x−xn−1

xn−xn−1, x ∈ [xn−1, xn ]

0, Ù¦.

4 3��«mþ P(x ) �±L«�: P(x ) =n∑

i=0li(x ) · yi , x ∈ [a, b].

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©ã�5��¼ê�Ø��O

½n

��½��!: a = x0 < x1 < . . . < xn = b, 9!:þ�¼ê� f (xi) = yi , i = 0, 1, · · · , n, f ′′(x ) 3 [a, b] þ�3, Ké?¿� x ∈ [a, b], k

|R(x )| = |f (x )− P(x )| ≤ h2M

8,

Ù¥

h = max1≤i≤n

{|xi − xi−1|}, M = maxx∈[a,b]

{|f ′′(x )|}.

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í2

aq��{, Ó���±�E©ã�g(�Ô.) ��Ú©ãng��õ�ª.

±©ã�g��õ�ª�~, Äkr��«m [a, b ¿©¤Ûê���!:, X:

a = x0 < x1 < x2 < · · · < x2n−1 < x2n = b.

±���n� x2i , x2i+1, x2i+2 ���f«m[x2i , x2i+2], i = 0, 1, · · · , n − 1.

e®�3¤k��: xi , i = 0, 1, · · · , 2n þ�¼ê�,K3z�f«mþÑk3 �êâ:, �±�E�g.�KF��õ�ª.

�daí...

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'u©ã��¼ê�A:5º

1 ©ã��¼ê3��f«m�à:?(q�:) 1w§ÝØp, X©ã�5��3��!:?���êØ�3.

2 eØü��!:?�¼ê�,Ó���±��Ù�ê�,|^Ó��g���±�EÙ��õ�ª, X Taylor ÐmªÒ´��A~.

3 |^©ã��¼êP(x ) ≈ f (x ) �A5, Kw,k

P ′(x ) ≈ f ′(x ),b∫a

P(x )dx ≈b∫a

f (x )dx , ù�Cqg�2

�^uê��©!ê�È©!�©�§ê�)�.4 k�ÿ,¢S¯KØ��¦3à:?��¼ê����êëY, ��¦���ê$��p��êëY,w,^©ã����ª®Ø÷v¯K��¦.

5 XÛd�½��X�êâ:�Ñ�^�N(�ÛÜz�)'�1w(X���êëY)�Cq­�º

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5.2 �^���{

/�^0�c u@Ïó§±ã¤^��«±ãóä, §´��[�!Lk�5½��­�7�½��^, �ã�Äkrêâ:£±3²¡þ, 2r��Lk�5�[�^(¡����^̂̂)�­, �âêâ:�©Ù, N�Ù�­ �Ú§Ý, ,�^Øc�½Ù�/G, ¦Ù�>ÏLùêâ:, ù�÷�^>=�±Ñ�^1w�­�, XdxÑ�­�¡��^­�.

Xêâ:�õ�, �^A��^, ©ã�¤þãó�, ù�A�4���^̂̂ëëë���:::������±±±111www. é±ã^�^xÑ�­�, ?1êÆÄ�ÚVz, ? �Ñ�^¼ê�VgÚêÆ�..

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�^¼ê�½Â

®�¼ê f (x ) 3«m [a, b] þ� n + 1 �:

a = x0 < x1 < · · · < xn = bn

?�¼ê�(½*ÿ�): f (xi) = yi , i = 0, 1, · · · , n. S (x ) ´��©ã½Â���¼ê, e S (x ) ÷vXe^�, K¡S (x )� f (x ) �nnnggg���^̂̂������¼¼¼êêê,{¡nnnggg���^̂̂.

1 S (x ) 3z�f«m [xi , xi+1] þ´��ngõ�ª, P� Si(x ), i = 0, 1, · · · , n − 1;

2 S (xi) = yi = f (xi), i = 0, 1, · · · , n;3 S (x ) 3«m (a, b) þkëY����ê.

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Ó�, S (x )3à:?÷vXe>.^���:

a. gggddd>>>...½½½ggg,,,>>>...^̂̂���(free boundarycondition)µ

S ′′(x0) = f ′′(a), S ′′(xn) = f ′′(b).

AÏ/ S ′′(x0) = S ′′(xn) = 0(g,�^).

b. ���½½½>>>...^̂̂���(clamped boundary condition):

S ′(x0) = f ′(x0), S′(xn) = f ′(xn).

c. ±±±ÏÏÏ>>>���^̂̂���(periodic boundary condition):

S ′(x0) = S ′(xn), S′′(x0) = S ′′(xn)

(�,ÙcJ^�´ S (x0) = S (xn) ).

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'ung�^¼ê½Â�A:5º:

��!:� n + 1 �, ©n �f«m.

3z�f«m [xi , xi+1] þ, Ù��õ�ª��� 3 gõ�ª, z� 3 gõ�ªk 4 ��½ëê, �(½ S (x )�I� 4n �®�^�.

d^�(2), �� S (xi), i = 0, 1, 2, · · · , n, � n + 1 �^�.

d^�(3), =3¤k��S:?, ÷v

Si(xi+1) = Si+1(xi+1), i = 0, 1, · · · , n − 2,

S ′i (xi+1) = S ′i+1(xi+1), i = 0, 1, · · · , n − 2,

S ′′i (xi+1) = S ′′i+1(xi+1), i = 0, 1, · · · , n − 2.

� 3(n − 1) = 3n − 3 �^�.

d^�(2)!(3)�� 4n − 2 ���^�,ÿI 2 ü�^�â���(½ S (x ), ù�:�d>.^�5�y.

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�!����� MATLAB ¢y

1. ~^ MATLAB ����¼ê

~~~^̂̂������¼¼¼êêê: interp1

NNN^̂̂���ªªª: interp1(x,y,cx,’methods’)

���ªªª`̀̀²²²:

x, y ©OL«�êâ:�î!p�I�þ, x7LüN;cx �I�O�����I:(½ê|�þ), 5¿cxØU�Ñx �«m��;’methods’��À���{, �âI��l±eo��{¥?À��:

1 ’nearest’ — ��C:��.2 ’linear’ — ©ã�5��("��).3 ’spline’ — ©ãng�^��.4 ’cubic’ — �/©ãng��.

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~~~3 Eæ^~1 �êâ, ©O|^þão��{À�?1��

�S

x0=1:6; y0=[16 18 21 17 15 12];cx=1:0.1:6;y1=interp1(x0,y0,cx,’nearest’);y2=interp1(x0,y0,cx,’linear’);y3=interp1(x0,y0,cx,’spline’);y4=interp1(x0,y0,cx,’cubic’);subplot(2,2,1), plot(x0,y0,’o’,cx,y1,’-r’),axis([0,7,10,24]),title(’Nearest Interpolant’)subplot(2,2,2), plot(x0,y0,’o’,cx,y2,’-k’)axis([0,7,10,24]),title(’Linear Interpolant’)subplot(2,2,3), plot(x0,y0,’o’,cx,y3,’-b’)axis([0,7,10,24]),title(’Spline Interpolant’)subplot(2,2,4), plot(x0,y0,’o’,cx,y4,’-k’)axis([0,7,10,24]),title(’Cubic Interpolant’)

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$1(J

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~~~4 312hS, z�1hÿþ�g§Ý, §Ý�g�: 5, 8,9, 15, 25, 29, 31, 30, 22, 25, 27, 24. Á�O33.2, 6.5,7.1, 11.7h��§Ý�.

�S

hours=1:12;temps=[5 8 9 15 25 29 31 30 22 25 27 24];tval=[3.2 6.5 7.1 11.7];t=interp1(hours, temps, tval)%�5��T=interp1(hours, temps, tval,’spline’)%ng�^��

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O�(J

�5���{:t = 10.2000 30.0000 30.9000 24.9000

ng�^���{:T = 9.6734 30.0427 31.1755 25.3820

(((JJJµµµdddµ�{ØÓ, (J¬k�½�É. Ï���´���O½ßÿ�L§, A^ØÓ��O�{ò��ØÓ�(J.

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2. ng�^��3 MATLAB ¥�¢y

MATLAB �âØÓ�é�, JøNõ�^u¦�ng�^���¼ê, ~^�kXe�¼êµ

(1) interp1 ¼ê

NNN^̂̂���ªªª: y=interp1(x0,y0,cx,’spline’)

õõõUUU`̀̀²²²µµµ x0,y0 ´�½êâ:�î�IÚp�I�þ, cx ´��:. -spline.À�%@/�Û(0à:(not-a-knot end conditions) ^�, �£��: cx ?�¼ê�y.

���555:/�Û(0´vkäN�½��«mà:�>.^�, é>.^��?næ�Cq�{, =r�1 1�Ú1 2 �ngõ�ª�n��ê��, é�� 1 �Ú�ê1 2 �ngõ�ª��Ó�/?n.

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(2) spline ¼ê

NNN^̂̂���ªªªIµµµ y=spline(x0,y0,cx)

õõõUUU`̀̀²²²µµµ � y=interp1(x0,y0,cx,’spline’) �Ó.

NNN^̂̂���ªªªIIµµµ pp=spline(x0,y0)

õõõUUU`̀̀²²²µµµ %@/�Û(0à:^�, �£�^���©ãõ�ª�/ª(�.

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�~

Xx0=1:6;y0=[16 18 21 17 15 12];pp=spline(x0,y0)w«(JXeµpp =form: ’pp’breaks: [1 2 3 4 5 6]coefs: [5x4 double]pieces: 5order: 4dim: 1

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(((JJJ`̀̀²²²: (�/ª�’pp’, =©ãõ�ª(piecewisepolynomial), �k 6 �*ÿ:(breaks), ©ãõ�ª�Xê� 5× 4 �V°ÝÝ, 5�f«m(pieces), z�f«mk4��£Oëê(order)§����(dim).

XJF"?�Ú�ÑäNz�ã�ng�^õ�ª�äNL�ª, K�UYÑ\Xe·-:pp.coefs�£(J�:ans =-2.2222 7.1667 -2.9444 16.0000-2.2222 0.5000 4.7222 18.00003.1111 -6.1667 -0.9444 21.0000-1.2222 3.1667 -3.9444 17.0000-1.2222 -0.5000 -1.2778 15.0000

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(J`²:

z1XêéA�ã�ng�^��õ�ª�Xê, =�ãng�^��õ�ª�L�/ª�

S (x ) ==−2.2222(x − 1)3 + 7.1667(x − 1)2 − 2.9444(x − 1) + 16, 1 ≤ x ≤ 2;−2.2222(x − 2)3 + 0.5(x − 2)2 + 4.7222(x − 2) + 18, 2 ≤ x ≤ 3;3.1111(x − 3)3 − 6.1667(x − 3)2 − 0.9444(x − 3) + 21, 3 ≤ x ≤ 4;−1.2222(x − 4)3 + 3.1667(x − 4)2 − 3.9444(x − 4) + 17, 4 ≤ x ≤ 5;−1.2222(x − 5)3 − 0.5(x − 5)2 − 1.2778(x − 5) + 15, 5 ≤ x ≤ 6;

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dng�^��õ�ª pp, O�3��: cx ?�¼ê�µ�N^ ppval() ¼ê

N^�ª�µ y=ppval(pp,cx)

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(3) csape ¼ê

NNN^̂̂���ªªª: pp=csape(x0,y0,conds, valconds)

���ªªª`̀̀²²²:csape ¼ê��£�´©¡ 3 gõ�ª� pp /ª.conds ´�(½�^���>.^�, Ù���

’complete’, "�^�, >.����ê;’not-a-knot’, �Û(^�;’periodic’, ±Ï5>.^�;’second’ , �½>.����ê�;’variational’, g,�^, �½>.����ê��".�AÏ>.�^�?n, �±ÏL conds �����^�1�þ5L«§^��þ������ 1 ½ 2.NNN^̂̂¼¼¼êêê���ªªª�: pp=csape(x0,y0 ext,conds). Ù¥y0 ext=[left, y0, right], ùp left L«�>.���, right L«m>.���. conds �1����L«�>.�^�, 1����L«m>.�^�, X conds=[2,1]L«�>.´���ê, m>.´���ê, éA��d left Ú right �Ñ.

valconds �>.^��.

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(4) csaps ¼ê

NNN^̂̂���ªªª: pp=csaps(x0,y0,p)

���ªªª`̀̀²²²:Ù¥, p ��Ïf( 0 < p < 1), p ��,�êâ��C. AO/,e p=0, K�©ã�5[Ü, e p=1, K�g,�^. �£�´ pp /ª, ¢y1w[Ü.

(5) csapi ¼ê

NNN^̂̂���ªªª: pp=csapi(x0,y0)

õõõUUU`̀̀²²²: �£�´ pp /ª, ¢yng�^¼ê�1w[Ü.

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(6) Ù§9ϼê: fnplt ¼ê

NNN^̂̂���ªªª: fnplt(pp)

õõõUUU`̀̀²²²: pp ´©¡�^��õ�ª¼ê, /Ïufnplt ¼ê±���¼ê�Jã

?�Ú/: eIO�¤��^¼ê3���þ cx ?�¼ê�, K�ÏLN^¼ê

y=fnval(pp,cx)5¢y.

(7) splinetool óä�

ù´��;�¦)�^���<Å�pª MATLAB óä�. Öö�±ÃI)äN����{Úúª�EL§, �±���\êâ, ÀJ�«�{§ÏL'�[Ü�J§(½���{Ú��¼ê.

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~~~5 |^ y = sin(x ) )¤«m [0, 10] þ� 11 �êâ:,,�|^ spline ¼ê¦\�����:�¼ê�¿±�ã/.

ë� MATLAB §SXe

x = 0:10;y = sin(x);xx = 0:.25:10;yy = spline(x,y,xx);plot(x,y,’o’,xx,yy,’-b’)legend(’sin(x)’,’spline interpolation’)

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��(Jã«

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n!�����{�A^

ê��©: f ′(x ) ≈ P ′(x )

ê�È©:∫ b

af (x )dx ≈

∫ b

aP(x )dx

�©�§ê�): k��©�{

�ÓÆ�|^©ã�5��í�þãCqO�úª.

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������öööSSS

1. ®�√x 3 x =100,121,169 ?�¼ê�, ©O^�5

Ú�Ô.���{¦√110 �Cq�¿©ÛØ�.

2. ^ x = 1 Ú x = 4 �¼ê f (x ) = x 1/2 ����5��¼ê, ¿¦ x = 2 Ú x = 3 �¼ê f (x ) �Cq�.3. N�À�Ú� h, âU¦©ã�5��¼êÚ sin x �Ø��u 1/2× 10−6.4. ¦÷ve�^��ng�^��¼ê s(x ):

s(1) = s(2) = 1, s(3) = 2, s ′(1) = 0, s ′(3) = 3.