na

5.3.Atableforfunctionsinxisgivenasfollows:x0.70.91.11.31.51.7xsin0.64420.78330.89120.96360.99750.9917Makeadivideddifferencetableaccordingly,andconstructNewtoninterpolationpolynomialsoftwo-point,three-pointandfour-pointtoapproximatesin(1.0).Solution:kx)(kxfk=1k=2k=3k=40.70.6442(x-0.7)0.90.78330.6955(x-0.7)(x-0.9)1.10.89120.5395-0.4150(x-0.7)(x-0.9)(x-1.1)1.30.96360.3620-0.4438-0.0896(x-0.7)(x-0.9)(x-1.1)(x-1.3)1.50.99750.1695-0.4813-0.0625(x-0.7)(x-0.9)(x-1.1)(x-1.3)(x-1.5)1.70.9917-0.0290-0.4963-0.0250(x-0.7)(x-0.9)(x-1.1)(x-1.3)(x-1.5)(x-1.7)Two-point)7.0(6955.06442.0)(2xxP85285.0)0.1sin(Three-point)9.0)(7.0(39.0)7.0(6955.06442.0)(3xxxxP84115.0)0.1sin(Four-point)1.1)(9.0)(7.0(0896.0)9.0)(7.0(39.0)7.0(6955.06442.0)(3xxxxxxxP841419.0)0.1sin(5.5Usingthesamefunctiontableinquestion4,(1)constructatableofupwinddifferenceandatableofdownwinddifferencerespectively;(2)constructtwo-point,three-pointandfour-pointNewtonupwindinterpolationformulatoapproximate)74.0sin(,andestimatetherelevanttruncationerrors;(3)constructtwo-point,three-pointandfour-pointNewtondownwindinterpolationformulatoapproximate)6.1sin(,andestimatetherelevanttruncationerrors;Solution:(1)kkkfff1and1kkkfffthetableofupwinddifferencekx)(kxfk=1k=2k=3k=4k=50.70.64420.90.78330.13911.10.89120.1079-0.03121.30.96360.0724-0.0355-0.00431.50.99750.0339-0.0385-0.00300.00131.70.9917-0.0058-0.0397-0.00120.00180.0005Thetableofdownwinddifferencekx)(kxfk=1k=2k=3k=4k=50.70.64420.1391-0.0312-0.00430.00130.00050.90.78330.1079-0.0355-0.00300.00181.10.89120.0724-0.0385-0.00121.30.96360.0339-0.03971.50.9975-0.00581.70.9917(2)usingNewtonupwindformula2.07.0,2.0,74.0hxthxTwo-point)74.0(1N672.000tff3221102.3)1(!2)74.0(httMRThree-point)74.0(2N675.0!2)1(020fttff4332109.2)2)(1(!3)74.0(htttMRFour-point)74.0(3N674.0!3)2)(1(!2)1(03020ftttfttff5443104.5)3)(2)(1(!4)74.0(httttMR(3)1.7+(-0.2)5.06.1ttTwo-point989.0)6.1(551ftfN015.0)1(!2)6.1(221httMRThree-point974.02)1()6.1(5552fttftfN0019.0)2)(1(!3)6.1(332htttMRFour-point9735.0!3)2)(1(2)1()6.1(55553ftttfttftfN0004.0)3)(2)(1(!4)6.1(443httttMR6.1.Findtheleastsquarespolynomialsofdegree1,2and3forthedatainthefollowingtable.ComputetheerrorEineachcase.xi0.000.150.310.500.600.75yi1.0001.0041.0311.1171.2231.422Solution:Degree1xaay105281.09295.082901.22911.131.2797.631.26101010aaaaaaxxP5281.09295.0)(10305.0))((6121iiixPyEDegree21495.13273.00115.16411.15182.07960.02911.182901.27960.02911.131.2797.62911.131.26210210210210aaaaaaaaaaaa221495.1)3273.0(0115.1)(xxxP461222104533.9))((iiixPyEDegree30711.10713.00166.09997.00378.12412.03493.05182.07960.06411.13493.05182.07960.02911.18290.25182.082901.27960.02911.131.2797.67960.02911.131.2632103210321032103210aaaaaaaaaaaaaaaaaaaa3230711.1)0713.0(0166.09997.0)(xxxxP461232101340.1))((iiixPyE6.2.Findtheleastsquarespolynomialapproximationsofdegree1and2ontheinterval[-1,1]forthefollowingfunctions,andcomputetheerrorEforeachcase.(1)f(x)=x2-2x+3(2)f(x)=x3(3)f(x)=1/(x+2)(4)f(x)=ex(5)xxxf2sin31cos21)(Solution:(1)degree1xaaxP101)(2310)32(,1)32(,010112111211110112011111100aadxxxxdxxadxxajdxxxxdxxadxxajxxP2310)(117778.0))()((11211dxxPxfEdegree222102)(xaxaaxP123)32(,2)32(,1)32(,0210112211421131112011211132112111101120112211111100aaadxxxxdxxadxxadxxajdxxxxdxxadxxadxxajdxxxxdxxadxxadxxaj2223)(xxxP0))()((11222dxxPxfE(2)degree1xaaxP101)(530,11,0101141121110113111110aadxxdxxaxdxajdxxxdxadxajxxP53)(10457.0))()((11211dxxPxfEdegree222102)(xaxaaxP06.00,2,11,0210115114211311120114113211211101131122111110aaadxxdxxadxxadxxajdxxdxxadxxaxdxajdxxdxxaxdxadxaj2223)(xxxP0457.0))()((11222dxxPxfE(3)degree1xaaxP101)(3ln333ln21)2(,1)2(1,01011112111101111111100aadxxxdxxadxxajdxxdxxadxxajxxP)3ln33(3ln21)(1211211104846.0))()((dxxPxfEdegree222102)(xaxaaxP1589.02958.04963.0)2(,2)2(,1)2(1,02101121142113111201111321121111011112211111100aaadxxxdxxadxxadxxajdxxxdxxadxxadxxajdxxdxxadxxadxxaj221589.02958.04963.0)(xxxP311222103584.0))()((dxxPxfE(4)degree1xaaxP101)(1037.11752.1,1,01011112111101111111100aadxxedxxadxxajdxedxxadxxajxxxxP1037.11752.0)(1111211105265.0))()((dxxPxfEdegree222102)(xaxaaxP5368.01037.19963.0,2,1,02101121142113111201111321121111011112211111100aaadxexdxxadxxadxxajdxxedxxadxxadxxajdxedxxadxxadxxajxxx225368.01037.19963.0)(xxxP211222101441.0))()((dxxPxfE(5)degree1xaaxP101)(43545.042075.0)2sin31cos21(,1)2sin31cos21(,01011112111101111111100aadxxxxdxxadxxajdxxxdxxadxxaj