gauss-legendre求积公式作业

高等数值分析gauss-legendre求积公式作业第一题：1. 题目分析在高斯求积型公式中，取p(x)=1，区间为【-1，1】，相应的正交多项式为勒让德多项式。它的表达式为：-11fxdx=k=0nAkfxk+Rnf其中节点xkk=0n是勒让德求和公式的零点。2. 首先通过查表找出gauss-legendre求积公式的根xkk=0n和系数Ak。x3=0.7745966692 0.0000000000 -0.7745966692;c3=0.5555555556 0.8888888889 0.5555555556;x5=-0.9061798459386640 -0.5384693101056830 0.000000000000 0.5384693101056830 0.9061798459386640;c5=0.2369268850561890 0.4786286704993660 0.5688888888888880 0.4786286704993660 0.2369268850561890;x7=-0.9491079123427580 -0.7415311855993940 -0.4058451513773970 0.00 0.4058451513773970 0.7415311855993940 0.9491079123427580;c7=0.1294849661688690 0.2797053914892760 0.3818300505051180 0.4179591836734690 0.3818300505051180 0.2797053914892760 0.1294849661688690;3. 计算精确值，作为与高斯勒让德求积公式的对比。用通过高等数学的知识将原式化简为3620x2sin20xdx=cos(20x)36=-1.909041707762054此为精确值。4. 分别用三次、五次、七次计算单独和分段的积分值根据下列公式：abfxdx=-11fb-at+b+a2(b-a)2dt-11fxdxi=1ncif(xi)将原公式变化为：Fx=-111203x+92sin403x+9dx再使用求积公式可以计算出结果如下：次数计算值误差精确值1.9090417077620543次1.8981542620866110.0108874456754435次1.9087897316378750.0002519761241797次1.9090430926773800.0000013849153263次分段1.9090376249604140.0000040828016415次分段1.9090417078708700.0000000001088167次分段1.9090417077620490.000000000000005 由上表可以明显看出，次数越高计算越准确，分段比不分段计算要准确。附表：求积公式程序% 高斯法雷建德法参数设定x3=0.7745966692 0.0000000000 -0.7745966692;c3=0.5555555556 0.8888888889 0.5555555556;x5=-0.9061798459386640 -0.5384693101056830 0.00000 0.5384693101056830 0.9061798459386640;c5=0.2369268850561890 0.4786286704993660 0.5688888888888880 0.4786286704993660 0.2369268850561890;x7=-0.9491079123427580 -0.7415311855993940 -0.4058451513773970 0.00 0.4058451513773970 0.7415311855993940 0.9491079123427580;c7=0.1294849661688690 0.2797053914892760 0.3818300505051180 0.4179591836734690 0.3818300505051180 0.2797053914892760 0.1294849661688690;% 单区间y3=0;y5=0;y7=0;for i=1:3 y3=y3+f(x3(i)*c3(i);endfor i=1:5 y5=y5+f(x5(i)*c5(i);endfor i=1:7 y7=y7+f(x7(i)*c7(i);end% 6段 n=3z3=0;z5=0;z7=0;for i=1:3 z3=z3+f1(x3(i)*c3(i);endfor i=1:3 z3=z3+f2(x3(i)*c3(i);endfor i=1:3 z3=z3+f3(x3(i)*c3(i);endfor i=1:3 z3=z3+f4(x3(i)*c3(i);endfor i=1:3 z3=z3+f5(x3(i)*c3(i);endfor i=1:3 z3=z3+f6(x3(i)*c3(i);end% 6段n=5for i=1:5 z5=z5+f1(x5(i)*c5(i);endfor i=1:5 z5=z5+f2(x5(i)*c5(i);endfor i=1:5 z5=z5+f3(x5(i)*c5(i);endfor i=1:5 z5=z5+f4(x5(i)*c5(i);endfor i=1:5 z5=z5+f5(x5(i)*c5(i);endfor i=1:5 z5=z5+f6(x5(i)*c5(i);end% 6段 n=7for i=1:7 z7=z7+f1(x7(i)*c7(i);endfor i=1:7 z7=z7+f2(x7(i)*c7(i);endfor i=1:7 z7=z7+f3(x7(i)*c7(i);endfor i=1:7 z7=z7