Lesson 12 Differentiation and Integration 1
Lesson 12Differentiation and Integration
1
2
• We have seen two applications:
– signal smoothing
– root finding
• Today we look
– differentation
– integration
• These will form the basis for solving ODEs
3
Differentiation of Fourier series
4
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f(�) =��
k=��fk
k�
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f �(�) ���
k=��kfk
k�
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f �(�) ��
��, . . . , ����
���
� � ��
�
�� Ff
[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8
4
� *SV�JYRGXMSRW�SR�XLI�TIVMSHMG�MRXIVZEP� [I�LEZI�XLI�*SYVMIV�VITVIWIRXEXMSR
f(�) =��
k=��fk
k�
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f �(�) ���
k=��kfk
k�
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f �(�) ��
��, . . . , ����
���
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�� Ff
[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8
5
Pointwise convergence of derivative of at zero
500 1000 1500 2000
10-12
10-9
10-6
0.001
1
Numerical derivativeDirect interpolation
number of pointsnumber of points
f(�) = ecos(10��1)
0 500 1000 1500 2000
10-13
10-10
10-7
10-4
0.1
6
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f (N)(�) ���
k=��( k)N fk
k�
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f (N)(�) ��
��, . . . , ����
���
� � ��
�
��
N
Ff
[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8
7
Pointwise convergence of 10th derivative of at zero
Numerical 10th derivativeDirect interpolation
number of pointsnumber of points
f(�) = ecos(10��1)
500 1000 1500 200010-8
10-6
10-4
0.01
1
200 400 600 800 1000
10-14
10-11
10-8
10-5
0.01
10
8
Integration of Fourier series
9
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�f � =
��
k=��,k �=0
fk
kk� + f0� + C
� 8LMW�[MPP GSRZIVKI [LIRIZIV�XLI�*SYVMIV�WIVMIW�HSIW�
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� � �1�
01
� � �1�
�
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Ff +�e�0 Ff +C
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9
� % JYRGXMSR�HI½RI�HSR�XLI�TIVMSHMG�MRXIVZEP�LEW�XLI MRHI½RMXI�MRXIKVEP
�f � =
��
k=��,k �=0
fk
kk� + f0� + C
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01
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Ff +�e�0 Ff +C
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9
� % JYRGXMSR�HI½RI�HSR�XLI�TIVMSHMG�MRXIVZEP�LEW�XLI MRHI½RMXI�MRXIKVEP
�f � =
��
k=��,k �=0
fk
kk� + f0� + C
� 8LMW�[MPP GSRZIVKI [LIRIZIV�XLI�*SYVMIV�WIVMIW�HSIW�
� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�
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���, . . . , ��
�
�
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1�
� � �1�
01
� � �1�
�
�����������
Ff +�e�0 Ff +C
[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8
� 8LMW�MW WXEFPI FIGEYWI�XLI�IVVSV�MR�IEGL�GSQTYXIH fk MW�QYPXMTPMIH�F]�E FSYRHIHRYQFIV
10
Pointwise convergence of integral of at zero
number of points
f(�) = ecos(10��1)
200 400 600 800 1000
10-15
10-11
10-7
0.001
11
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Ff+f0
N !�N
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12
Differentiation of Taylor series
13
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f(z) =��
k=0
fkzk
� (IVMZEXMZI �
f �(z) =��
k=0
kfkzk�1
� 2YQIVMGEPP]� [I�SFXEMR�XLI (m � 1) � m QEXVM\�ETTVS\MQEXMSR�
f �(z) ��1 | · · · | zm�2
�
�
����
0 12
� � �m � 1
�
����T f
[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ
13
� *SV�JYRGXMSRW�MR�XLI�HMWO� [I�LEZI�XLI�8E]PSV�WIVMIW
f(z) =��
k=0
fkzk
� (IVMZEXMZI �
f �(z) =��
k=0
kfkzk�1
� 2YQIVMGEPP]� [I�SFXEMR�XLI (m � 1) � m QEXVM\�ETTVS\MQEXMSR�
f �(z) ��1 | · · · | zm�2
�
�
����
0 12
� � �m � 1
�
����T f
[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ
14
50 100 150 200
10-14
10-11
10-8
10-5
0.01
First derivative
Error approximating exp(z) for z = {.1,.5,1.}exp(.1i)
number of points
14
50 100 150 200
10-14
10-11
10-8
10-5
0.01
First derivative
50 100 150 200
10-8
10-5
0.01
10
104
107
10th derivative
Error approximating exp(z) for z = {.1,.5,1.}exp(.1i)
number of points
15
Integration of Taylor series
16
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��
k=0
fk
k + 1zk+1 + C
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�f(z) z �
�1 | · · · | zm
�
�
������
01
12
. . .1
m+1
�
������T f + C
[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ
� 2S[�XLMW�MW WXEFPI FSXL�SR�ERH�MRWMHI�XLI�YRMX�GMVGPI�
17
� -RXIKVEP � �f(z) z =
��
k=0
fk
k + 1zk+1 + C
� 2YQIVMGEPP]� [I�SFXEMR�XLI (m + 1) � m QEXVM\�ETTVS\MQEXMSR�
�f(z) z �
�1 | · · · | zm
�
�
������
01
12
. . .1
m+1
�
������T f + C
[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ
� 2S[�XLMW�MW WXEFPI FSXL�SR�ERH�MRWMHI�XLI�YRMX�GMVGPI�
18
First integral 10th integral
Error approximating exp(z) for z = {.1,.5,1.}exp(.1i)
number of points
50 100 150 200
10-14
10-11
10-8
10-5
0.01
50 100 150 200
10-16
10-14
10-12
10-10
10-8
19
Differentiation of Laurent series
20
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f(z) =��
k=��fkzk
� (IVMZEXMZI �
f �(z) =��
k=��kfkzk�1 =
��
k=��,k �=�1
(k + 1)fk+1zk
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f �(z) ��z��1 | · · · | z��1
��
���
. . .�
�
�� Ff
[LIVI f = (f(z1), . . . , f(zm))�
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21
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f(z) =��
k=��fkzk
� (IVMZEXMZI �
f �(z) =��
k=��kfkzk�1 =
��
k=��,k �=�1
(k + 1)fk+1zk
� 2YQIVMGEPP]� [I�SFXEMR�XLI��WUYEVI �ETTVS\MQEXMSR� [LIVI�[I GLERKIH�SYV�FEWMW�
f �(z) ��z��1 | · · · | z��1
��
���
. . .�
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�� Ff
[LIVI f = (f(z1), . . . , f(zm))�
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22
Integration of Taylor series
23
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f(z) =��
k=��fkzk
� -RXIKVEP��
f(z) z =��
k=��,k �=�1
1
k + 1fkzk+1 + f�1 z + C
� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�[LIVI�[I GLERKIH�FEWIW�
�f(z) z �
�z�+1 | · · · | z�+1
�
�
�����������
1�+1
. . .1
�10
1. . .
1�+1
�
�����������
Ff+e�1Ff z+C
[LIVI f = (f(z1), . . . , f(zm))�
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24
� *SV�JYRGXMSRW�SR�XLI�GMVGPI� [I�LEZI�XLI�0EYVIRX�WIVMIW
f(z) =��
k=��fkzk
� -RXIKVEP��
f(z) z =��
k=��,k �=�1
1
k + 1fkzk+1 + f�1 z + C
� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�[LIVI�[I GLERKIH�FEWIW�
�f(z) z �
�z�+1 | · · · | z�+1
�
�
�����������
1�+1
. . .1
�10
1. . .
1�+1
�
�����������
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25
Integration of Chebyshev series
26
� ;I�[ERX�XS�GSQTYXI
�f(x) x =
��
k=0
fk
�Tk(x) x
� 8LI�½VWX�X[S QSQIRXW EVI�
T0(x) x = x+C = T1(x)+C ERH�
T1(x) x =x2
2+C =
T2(x) � T0(x)
4+C
� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�
� x
aTk(x) x =
� J�1� (x)
J�1� (a)
Tk(J(z))J �(z) z =1
4
� J�1� (x)
J�1� (a)
�zk + z�k
� �1 � 1
z2
�z
=1
4
� J�1� (x)
J�1� (a)
�zk + z�k � zk�2 � z�k�2
�z
=1
4
�zk+1
k + 1+
z1�k
1 � k� zk�1
k � 1� z�k�1
�k � 1
�+ C �2S[ z = J�1
� (x)�
=1
2(k + 1)
zk+1 + z�k�1
2� 1
2(k � 1)
zk�1 + z1�k
2+ C
=Tk+1(x)
2(k + 1)� Tk�1(x)
2(k � 1)+ C
26
� ;I�[ERX�XS�GSQTYXI
�f(x) x =
��
k=0
fk
�Tk(x) x
� 8LI�½VWX�X[S QSQIRXW EVI�
T0(x) x = x+C = T1(x)+C ERH�
T1(x) x =x2
2+C =
T2(x) � T0(x)
4+C
� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�
� x
aTk(x) x =
� J�1� (x)
J�1� (a)
Tk(J(z))J �(z) z =1
4
� J�1� (x)
J�1� (a)
�zk + z�k
� �1 � 1
z2
�z
=1
4
� J�1� (x)
J�1� (a)
�zk + z�k � zk�2 � z�k�2
�z
=1
4
�zk+1
k + 1+
z1�k
1 � k� zk�1
k � 1� z�k�1
�k � 1
�+ C �2S[ z = J�1
� (x)�
=1
2(k + 1)
zk+1 + z�k�1
2� 1
2(k � 1)
zk�1 + z1�k
2+ C
=Tk+1(x)
2(k + 1)� Tk�1(x)
2(k � 1)+ C
26
� ;I�[ERX�XS�GSQTYXI
�f(x) x =
��
k=0
fk
�Tk(x) x
� 8LI�½VWX�X[S QSQIRXW EVI�
T0(x) x = x+C = T1(x)+C ERH�
T1(x) x =x2
2+C =
T2(x) � T0(x)
4+C
� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�
� x
aTk(x) x =
� J�1� (x)
J�1� (a)
Tk(J(z))J �(z) z =1
4
� J�1� (x)
J�1� (a)
�zk + z�k
� �1 � 1
z2
�z
=1
4
� J�1� (x)
J�1� (a)
�zk + z�k � zk�2 � z�k�2
�z
=1
4
�zk+1
k + 1+
z1�k
1 � k� zk�1
k � 1� z�k�1
�k � 1
�+ C �2S[ z = J�1
� (x)�
=1
2(k + 1)
zk+1 + z�k�1
2� 1
2(k � 1)
zk�1 + z1�k
2+ C
=Tk+1(x)
2(k + 1)� Tk�1(x)
2(k � 1)+ C
26
� ;I�[ERX�XS�GSQTYXI
�f(x) x =
��
k=0
fk
�Tk(x) x
� 8LI�½VWX�X[S QSQIRXW EVI�
T0(x) x = x+C = T1(x)+C ERH�
T1(x) x =x2
2+C =
T2(x) � T0(x)
4+C
� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�
� x
aTk(x) x =
� J�1� (x)
J�1� (a)
Tk(J(z))J �(z) z =1
4
� J�1� (x)
J�1� (a)
�zk + z�k
� �1 � 1
z2
�z
=1
4
� J�1� (x)
J�1� (a)
�zk + z�k � zk�2 � z�k�2
�z
=1
4
�zk+1
k + 1+
z1�k
1 � k� zk�1
k � 1� z�k�1
�k � 1
�+ C �2S[ z = J�1
� (x)�
=1
2(k + 1)
zk+1 + z�k�1
2� 1
2(k � 1)
zk�1 + z1�k
2+ C
=Tk+1(x)
2(k + 1)� Tk�1(x)
2(k � 1)+ C
26
� ;I�[ERX�XS�GSQTYXI
�f(x) x =
��
k=0
fk
�Tk(x) x
� 8LI�½VWX�X[S QSQIRXW EVI�
T0(x) x = x+C = T1(x)+C ERH�
T1(x) x =x2
2+C =
T2(x) � T0(x)
4+C
� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�
� x
aTk(x) x =
� J�1� (x)
J�1� (a)
Tk(J(z))J �(z) z =1
4
� J�1� (x)
J�1� (a)
�zk + z�k
� �1 � 1
z2
�z
=1
4
� J�1� (x)
J�1� (a)
�zk + z�k � zk�2 � z�k�2
�z
=1
4
�zk+1
k + 1+
z1�k
1 � k� zk�1
k � 1� z�k�1
�k � 1
�+ C �2S[ z = J�1
� (x)�
=1
2(k + 1)
zk+1 + z�k�1
2� 1
2(k � 1)
zk�1 + z1�k
2+ C
=Tk+1(x)
2(k + 1)� Tk�1(x)
2(k � 1)+ C
26
� ;I�[ERX�XS�GSQTYXI
�f(x) x =
��
k=0
fk
�Tk(x) x
� 8LI�½VWX�X[S QSQIRXW EVI�
T0(x) x = x+C = T1(x)+C ERH�
T1(x) x =x2
2+C =
T2(x) � T0(x)
4+C
� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�
� x
aTk(x) x =
� J�1� (x)
J�1� (a)
Tk(J(z))J �(z) z =1
4
� J�1� (x)
J�1� (a)
�zk + z�k
� �1 � 1
z2
�z
=1
4
� J�1� (x)
J�1� (a)
�zk + z�k � zk�2 � z�k�2
�z
=1
4
�zk+1
k + 1+
z1�k
1 � k� zk�1
k � 1� z�k�1
�k � 1
�+ C �2S[ z = J�1
� (x)�
=1
2(k + 1)
zk+1 + z�k�1
2� 1
2(k � 1)
zk�1 + z1�k
2+ C
=Tk+1(x)
2(k + 1)� Tk�1(x)
2(k � 1)+ C
26
� ;I�[ERX�XS�GSQTYXI
�f(x) x =
��
k=0
fk
�Tk(x) x
� 8LI�½VWX�X[S QSQIRXW EVI�
T0(x) x = x+C = T1(x)+C ERH�
T1(x) x =x2
2+C =
T2(x) � T0(x)
4+C
� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�
� x
aTk(x) x =
� J�1� (x)
J�1� (a)
Tk(J(z))J �(z) z =1
4
� J�1� (x)
J�1� (a)
�zk + z�k
� �1 � 1
z2
�z
=1
4
� J�1� (x)
J�1� (a)
�zk + z�k � zk�2 � z�k�2
�z
=1
4
�zk+1
k + 1+
z1�k
1 � k� zk�1
k � 1� z�k�1
�k � 1
�+ C �2S[ z = J�1
� (x)�
=1
2(k + 1)
zk+1 + z�k�1
2� 1
2(k � 1)
zk�1 + z1�k
2+ C
=Tk+1(x)
2(k + 1)� Tk�1(x)
2(k � 1)+ C
27
� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE
�f(x) x =
��
k=0
fk
�Tk(x) x
= C + f0T1(x) + f1T2(x) � T0(x)
4+
1
2
��
k=2
fk
�Tk+1(x)
k + 1� Tk�1(x)
k � 1
�
= C � f1
4+
�f0 � f2
2
�T1(x) +
1
2
��
k=2
�fk�1 � fk+1
k
�Tk(x)
� 2YQIVMGEPP]� [I�ETTVS\MQEXI
f(x) ��1 | · · · | Tn�1(x)
�Cf
ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�
�f(x) x �
�1 | · · · | Tn(x)
�
�
����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�2) � 12(n�2)
12(n�1)
12n
�
����������
Cf
[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8
27
� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE
�f(x) x =
��
k=0
fk
�Tk(x) x
= C + f0T1(x) + f1T2(x) � T0(x)
4+
1
2
��
k=2
fk
�Tk+1(x)
k + 1� Tk�1(x)
k � 1
�
= C � f1
4+
�f0 � f2
2
�T1(x) +
1
2
��
k=2
�fk�1 � fk+1
k
�Tk(x)
� 2YQIVMGEPP]� [I�ETTVS\MQEXI
f(x) ��1 | · · · | Tn�1(x)
�Cf
ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�
�f(x) x �
�1 | · · · | Tn(x)
�
�
����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�2) � 12(n�2)
12(n�1)
12n
�
����������
Cf
[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8
27
� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE
�f(x) x =
��
k=0
fk
�Tk(x) x
= C + f0T1(x) + f1T2(x) � T0(x)
4+
1
2
��
k=2
fk
�Tk+1(x)
k + 1� Tk�1(x)
k � 1
�
= C � f1
4+
�f0 � f2
2
�T1(x) +
1
2
��
k=2
�fk�1 � fk+1
k
�Tk(x)
� 2YQIVMGEPP]� [I�ETTVS\MQEXI
f(x) ��1 | · · · | Tn�1(x)
�Cf
ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�
�f(x) x �
�1 | · · · | Tn(x)
�
�
����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�2) � 12(n�2)
12(n�1)
12n
�
����������
Cf
[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8
27
� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE
�f(x) x =
��
k=0
fk
�Tk(x) x
= C + f0T1(x) + f1T2(x) � T0(x)
4+
1
2
��
k=2
fk
�Tk+1(x)
k + 1� Tk�1(x)
k � 1
�
= C � f1
4+
�f0 � f2
2
�T1(x) +
1
2
��
k=2
�fk�1 � fk+1
k
�Tk(x)
� 2YQIVMGEPP]� [I�ETTVS\MQEXI
f(x) ��1 | · · · | Tn�1(x)
�Cf
ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�
�f(x) x �
�1 | · · · | Tn(x)
�
�
����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�2) � 12(n�2)
12(n�1)
12n
�
����������
Cf
[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8
28
First integral 10th integral
Error approximating exp(x) for x = .1
number of points
10 20 30 40 50
10-13
10-10
10-7
10-4
0.1
10 20 30 40 50
10-14
10-12
10-10
10-8
29
Differentiation of Chebyshev series
30
� 0IX�W�XV]�HIGSQTSWMRK�XLI�XIVQW�
f �(x) =��
k=0
fkT �k(x)
� ;I�[ERX�XS�VI[VMXI T �k(x) MR�XIVQW�SJ T0(x), . . . , Tk�1(x)
� 0IX�W�XV]�ER�I\TIVMQIRX�
9WI Tk(x) = k x� WS T �k(x) = k k x�
1�x2
8LIVIJSVI� [I�GER RYQIVMGEPP] IZEPYEXI
CTk(x) = C
�
��T �
k(x1)���
T �k(xn)
�
��
XS�KIX�XLI�GSIJ½GMIRXW
30
� 0IX�W�XV]�HIGSQTSWMRK�XLI�XIVQW�
f �(x) =��
k=0
fkT �k(x)
� ;I�[ERX�XS�VI[VMXI T �k(x) MR�XIVQW�SJ T0(x), . . . , Tk�1(x)
� 0IX�W�XV]�ER�I\TIVMQIRX�
9WI Tk(x) = k x� WS T �k(x) = k k x�
1�x2
8LIVIJSVI� [I�GER RYQIVMGEPP] IZEPYEXI
CTk(x) = C
�
��T �
k(x1)���
T �k(xn)
�
��
XS�KIX�XLI�GSIJ½GMIRXW
30
� 0IX�W�XV]�HIGSQTSWMRK�XLI�XIVQW�
f �(x) =��
k=0
fkT �k(x)
� ;I�[ERX�XS�VI[VMXI T �k(x) MR�XIVQW�SJ T0(x), . . . , Tk�1(x)
� 0IX�W�XV]�ER�I\TIVMQIRX�
9WI Tk(x) = k x� WS T �k(x) = k k x�
1�x2
8LIVIJSVI� [I�GER RYQIVMGEPP] IZEPYEXI
CTk(x) = C
�
��T �
k(x1)���
T �k(xn)
�
��
XS�KIX�XLI�GSIJ½GMIRXW
31
�CT �
0(x) | · · · | CT �10(x)
�=
31
0 1. 0 3. 0 5. 0 7. 0 9. 00 0 4. 0 8. 0 12. 0 16. 0 20.0 0 0 6. 0 10. 0 14. 0 18. 00 0 0 0 8. 0 12. 0 16. 0 20.0 0 0 0 0 10. 0 14. 0 18. 00 0 0 0 0 0 12. 0 16. 0 20.0 0 0 0 0 0 0 14. 0 18. 00 0 0 0 0 0 0 0 16. 0 20.0 0 0 0 0 0 0 0 0 18. 00 0 0 0 0 0 0 0 0 0 20.0 0 0 0 0 0 0 0 0 0 0
�CT �
0(x) | · · · | CT �10(x)
�=
31
0 1. 0 3. 0 5. 0 7. 0 9. 00 0 4. 0 8. 0 12. 0 16. 0 20.0 0 0 6. 0 10. 0 14. 0 18. 00 0 0 0 8. 0 12. 0 16. 0 20.0 0 0 0 0 10. 0 14. 0 18. 00 0 0 0 0 0 12. 0 16. 0 20.0 0 0 0 0 0 0 14. 0 18. 00 0 0 0 0 0 0 0 16. 0 20.0 0 0 0 0 0 0 0 0 18. 00 0 0 0 0 0 0 0 0 0 20.0 0 0 0 0 0 0 0 0 0 0
Problem: the operation is dense!
�CT �
0(x) | · · · | CT �10(x)
�=
32� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR
� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX
�
�����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�3) � 12(n�3)
12(n�2)
12(n�1)
�
�����������
u = Cf
� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�
un�2 = 2(n � 1)fn�1
un�3 = 2(n � 2)fn�2
un�4 = 2(n � 3)fn�3 + un�2
���
u0 = f1 +u2
2
� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#
8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]
32
� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR
� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX
�
�����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�3) � 12(n�3)
12(n�2)
12(n�1)
�
�����������
u = Cf
� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�
un�2 = 2(n � 1)fn�1
un�3 = 2(n � 2)fn�2
un�4 = 2(n � 3)fn�3 + un�2
���
u0 = f1 +u2
2
� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#
8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]
32
� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR
� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX
�
�����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�3) � 12(n�3)
12(n�2)
12(n�1)
�
�����������
u = Cf
� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�
un�2 = 2(n � 1)fn�1
un�3 = 2(n � 2)fn�2
un�4 = 2(n � 3)fn�3 + un�2
���
u0 = f1 +u2
2
� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#
8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]
32
� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR
� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX
�
�����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�3) � 12(n�3)
12(n�2)
12(n�1)
�
�����������
u = Cf
� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�
un�2 = 2(n � 1)fn�1
un�3 = 2(n � 2)fn�2
un�4 = 2(n � 3)fn�3 + un�2
���
u0 = f1 +u2
2
� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#
8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]
32
� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR
� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX
�
�����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�3) � 12(n�3)
12(n�2)
12(n�1)
�
�����������
u = Cf
� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�
un�2 = 2(n � 1)fn�1
un�3 = 2(n � 2)fn�2
un�4 = 2(n � 3)fn�3 + un�2
���
u0 = f1 +u2
2
� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#
8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]
32
� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR
� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX
�
�����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�3) � 12(n�3)
12(n�2)
12(n�1)
�
�����������
u = Cf
� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�
un�2 = 2(n � 1)fn�1
un�3 = 2(n � 2)fn�2
un�4 = 2(n � 3)fn�3 + un�2
���
u0 = f1 +u2
2
� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#
8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]
32
� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR
� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX
�
�����������
� 14
1 � 12
14 � 1
4� � �
� � �1
2(n�3) � 12(n�3)
12(n�2)
12(n�1)
�
�����������
u = Cf
� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�
un�2 = 2(n � 1)fn�1
un�3 = 2(n � 2)fn�2
un�4 = 2(n � 3)fn�3 + un�2
���
u0 = f1 +u2
2
� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#
8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]
33
First derivative 10th derivative
Error approximating exp(x) for x = .1
number of points
10 20 30 40 50
10-13
10-10
10-7
10-4
0.1
10 20 30 40 50
10-5
10-4
0.001
0.01
0.1
1
10