체비셰프 다항식

수학노트
http://bomber0.myid.net/ (토론)님의 2009년 11월 22일 (일) 13:49 판
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제1종 체비세프 다항식
  • \(n \geq 0 \), 다음과 같은 점화식에 의해여, \(T_n(x)\)을 정의
  • \(T_0(x) & = 1 \)
  • \(T_1(x) & = x\)
  • \(T_{n+1}(x) & = 2xT_n(x) - T_{n-1}(x). \)

 

삼각함수와의 관계

 

 

제1종 체비셰프 다항식 목록
  • 매쓰매티카 명령어 Do[Print["T_",n,"[x]=",ChebyshevT[n,x]],{n,0,20}]

T_0[x]=1
T_1[x]=x
T_2[x]=-1+2 x^2
T_3[x]=-3 x+4 x^3
T_4[x]=1-8 x^2+8 x^4
T_5[x]=5 x-20 x^3+16 x^5
T_6[x]=-1+18 x^2-48 x^4+32 x^6
T_7[x]=-7 x+56 x^3-112 x^5+64 x^7
T_8[x]=1-32 x^2+160 x^4-256 x^6+128 x^8
T_9[x]=9 x-120 x^3+432 x^5-576 x^7+256 x^9
T_10[x]=-1+50 x^2-400 x^4+1120 x^6-1280 x^8+512 x^10
T_11[x]=-11 x+220 x^3-1232 x^5+2816 x^7-2816 x^9+1024 x^11
T_12[x]=1-72 x^2+840 x^4-3584 x^6+6912 x^8-6144 x^10+2048 x^12
T_13[x]=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13
T_14[x]=-1+98 x^2-1568 x^4+9408 x^6-26880 x^8+39424 x^10-28672 x^12+8192 x^14
T_15[x]=-15 x+560 x^3-6048 x^5+28800 x^7-70400 x^9+92160 x^11-61440 x^13+16384 x^15
T_16[x]=1-128 x^2+2688 x^4-21504 x^6+84480 x^8-180224 x^10+212992 x^12-131072 x^14+32768 x^16
T_17[x]=17 x-816 x^3+11424 x^5-71808 x^7+239360 x^9-452608 x^11+487424 x^13-278528 x^15+65536 x^17
T_18[x]=-1+162 x^2-4320 x^4+44352 x^6-228096 x^8+658944 x^10-1118208 x^12+1105920 x^14-589824 x^16+131072 x^18
T_19[x]=-19 x+1140 x^3-20064 x^5+160512 x^7-695552 x^9+1770496 x^11-2723840 x^13+2490368 x^15-1245184 x^17+262144 x^19
T_20[x]=1-200 x^2+6600 x^4-84480 x^6+549120 x^8-2050048 x^10+4659200 x^12-6553600 x^14+5570560 x^16-2621440 x^18+524288 x^20

 

제2종 체비세프 다항식
  • \(n \geq 0 \), 다음과 같은 점화식에 의해여, \(U_n(x)\)을 정의
  • \(U_0(x) & = 1\)
  • \(U_1(x) & = 2x\)
  • \(U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)\)

 

 

삼각함수와의 관계
  •  \(U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}\)
  • 삼각함수

 

 

목록
  • 매쓰매티카 명령어 Do[Print["U_", n, "[x]=", ChebyshevU[n, x]], {n, 0, 20}]

U_0[x]=1
U_1[x]=2 x
U_2[x]=-1+4 x^2
U_3[x]=-4 x+8 x^3
U_4[x]=1-12 x^2+16 x^4
U_5[x]=6 x-32 x^3+32 x^5
U_6[x]=-1+24 x^2-80 x^4+64 x^6
U_7[x]=-8 x+80 x^3-192 x^5+128 x^7
U_8[x]=1-40 x^2+240 x^4-448 x^6+256 x^8
U_9[x]=10 x-160 x^3+672 x^5-1024 x^7+512 x^9
U_10[x]=-1+60 x^2-560 x^4+1792 x^6-2304 x^8+1024 x^10
U_11[x]=-12 x+280 x^3-1792 x^5+4608 x^7-5120 x^9+2048 x^11
U_12[x]=1-84 x^2+1120 x^4-5376 x^6+11520 x^8-11264 x^10+4096 x^12
U_13[x]=14 x-448 x^3+4032 x^5-15360 x^7+28160 x^9-24576 x^11+8192 x^13
U_14[x]=-1+112 x^2-2016 x^4+13440 x^6-42240 x^8+67584 x^10-53248 x^12+16384 x^14
U_15[x]=-16 x+672 x^3-8064 x^5+42240 x^7-112640 x^9+159744 x^11-114688 x^13+32768 x^15
U_16[x]=1-144 x^2+3360 x^4-29568 x^6+126720 x^8-292864 x^10+372736 x^12-245760 x^14+65536 x^16
U_17[x]=18 x-960 x^3+14784 x^5-101376 x^7+366080 x^9-745472 x^11+860160 x^13-524288 x^15+131072 x^17
U_18[x]=-1+180 x^2-5280 x^4+59136 x^6-329472 x^8+1025024 x^10-1863680 x^12+1966080 x^14-1114112 x^16+262144 x^18
U_19[x]=-20 x+1320 x^3-25344 x^5+219648 x^7-1025024 x^9+2795520 x^11-4587520 x^13+4456448 x^15-2359296 x^17+524288 x^19
U_20[x]=1-220 x^2+7920 x^4-109824 x^6+768768 x^8-3075072 x^10+7454720 x^12-11141120 x^14+10027008 x^16-4980736 x^18+1048576 x^20

 

 

sin 함수의 배각공식

\(x=\sin\theta\), \(\sqrt{1-x^2}=\cos\theta\)로 이해

 

xU_0[Sqrt[1-x]]=x
xU_1[Sqrt[1-x]]=2 x Sqrt[1-x^2]
xU_2[Sqrt[1-x]]=3 x-4 x^3
xU_3[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-1+2 x^2)
xU_4[Sqrt[1-x]]=x (5-20 x^2+16 x^4)
xU_5[Sqrt[1-x]]=2 x Sqrt[1-x^2] (3-16 x^2+16 x^4)
xU_6[Sqrt[1-x]]=7 x-56 x^3+112 x^5-64 x^7
xU_7[Sqrt[1-x]]=-8 x Sqrt[1-x^2] (-1+10 x^2-24 x^4+16 x^6)
xU_8[Sqrt[1-x]]=x (9-120 x^2+432 x^4-576 x^6+256 x^8)
xU_9[Sqrt[1-x]]=2 x Sqrt[1-x^2] (5-80 x^2+336 x^4-512 x^6+256 x^8)
xU_10[Sqrt[1-x]]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11
xU_11[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-3+70 x^2-448 x^4+1152 x^6-1280 x^8+512 x^10)
xU_12[Sqrt[1-x]]=x (13-364 x^2+2912 x^4-9984 x^6+16640 x^8-13312 x^10+4096 x^12)
xU_13[Sqrt[1-x]]=2 x Sqrt[1-x^2] (7-224 x^2+2016 x^4-7680 x^6+14080 x^8-12288 x^10+4096 x^12)
xU_14[Sqrt[1-x]]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15
xU_15[Sqrt[1-x]]=-16 x Sqrt[1-x^2] (-1+42 x^2-504 x^4+2640 x^6-7040 x^8+9984 x^10-7168 x^12+2048 x^14)
xU_16[Sqrt[1-x]]=x (17-816 x^2+11424 x^4-71808 x^6+239360 x^8-452608 x^10+487424 x^12-278528 x^14+65536 x^16)
xU_17[Sqrt[1-x]]=2 x Sqrt[1-x^2] (9-480 x^2+7392 x^4-50688 x^6+183040 x^8-372736 x^10+430080 x^12-262144 x^14+65536 x^16)
xU_18[Sqrt[1-x]]=19 x-1140 x^3+20064 x^5-160512 x^7+695552 x^9-1770496 x^11+2723840 x^13-2490368 x^15+1245184 x^17-262144 x^19
xU_19[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-5+330 x^2-6336 x^4+54912 x^6-256256 x^8+698880 x^10-1146880 x^12+1114112 x^14-589824 x^16+131072 x^18)
xU_20[Sqrt[1-x]]=x (21-1540 x^2+33264 x^4-329472 x^6+1793792 x^8-5870592 x^10+12042240 x^12-15597568 x^14+12386304 x^16-5505024 x^18+1048576 x^20)

 

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