如何在R中生成正交多项式？
我们可以说正交是vertical的同义词。如果两个多项式的内积（内积是点积的泛化）为零，则称它们为正交多项式。在R中，我们可以通过使用poly函数来找到正交积，如以下示例所示。
例1
> x<-rnorm(10)> x
输出结果
[1] 1.53798786 -0.85463326 2.39444451 0.82559418 -2.22197322 -1.04243823[7] -0.04693054 -0.68691236 -1.63040923 -1.42408865
示例
> orthogonal_x<-poly(x,degree=2)> orthogonal_x
输出结果
1 2[1,] 0.41743651 -0.01687537
[2,] -0.12158589 -0.21414848
[3,] 0.61038362 0.54027924
[4,] 0.25694468 -0.28489882
[5,] -0.42962751 0.57201520
[6,] -0.16389559 -0.14490046
[7,] 0.06037767 -0.37137903
[8,] -0.08380084 -0.26556522
[9,] -0.29635683 0.15165057
[10,] -0.24987583 0.03382235
attr(,"coefs")
attr(,"coefs")$alpha
[1] -0.3149359 0.5267254
attr(,"coefs")$norm2
[1] 1.00000 10.00000 19.70309 32.71111
attr(,"degree")
[1] 1 2
attr(,"class")
[1] "poly" "matrix"
例2
> orthogonal_x_degree3<-poly(x,degree=3)> orthogonal_x_degree3
输出结果
1 2 3[1,] 0.41743651 -0.01687537 -0.41882840
[2,] -0.12158589 -0.21414848 0.23310682
[3,] 0.61038362 0.54027924 0.42176833
[4,] 0.25694468 -0.28489882 -0.44362203
[5,] -0.42962751 0.57201520 -0.48208095
[6,] -0.16389559 -0.14490046 0.25573226
[7,] 0.06037767 -0.37137903 -0.07651475
[8,] -0.08380084 -0.26556522 0.19221655
[9,] -0.29635683 0.15165057 0.11263775
[10,] -0.24987583 0.03382235 0.20558443
attr(,"coefs")
attr(,"coefs")$alpha
[1] -0.3149359 0.5267254 -0.1157637
attr(,"coefs")$norm2
[1] 1.00000 10.00000 19.70309 32.71111 59.69013
attr(,"degree")
[1] 1 2 3
attr(,"class")
[1] "poly" "matrix"
例子3
> orthogonal_x_degree5<-poly(x,degree=5)> orthogonal_x_degree5
输出结果
1 2 3 4 5[1,] 0.41743651 -0.01687537 -0.41882840 -0.53824181 -0.4545569
[2,] -0.12158589 -0.21414848 0.23310682 0.05839481 -0.2859929
[3,] 0.61038362 0.54027924 0.42176833 0.21609548 0.0994605
[4,] 0.25694468 -0.28489882 -0.44362203 0.06801947 0.5307426
[5,] -0.42962751 0.57201520 -0.48208095 0.33629533 -0.1868526
[6,] -0.16389559 -0.14490046 0.25573226 -0.09590512 -0.1828276
[7,] 0.06037767 -0.37137903 -0.07651475 0.48513908 0.1252141
[8,] -0.08380084 -0.26556522 0.19221655 0.19189132 -0.3055896
[9,] -0.29635683 0.15165057 0.11263775 -0.37914934 0.4409641
[10,] -0.24987583 0.03382235 0.20558443 -0.34253923 0.2194384
attr(,"coefs")
attr(,"coefs")$alpha
[1] -0.3149359 0.5267254 -0.1157637 0.1206771 -0.1404103
attr(,"coefs")$norm2
[1] 1.00000 10.00000 19.70309 32.71111 59.69013 67.13315 68.51457
attr(,"degree")
[1] 1 2 3 4 5
attr(,"class")
[1] "poly" "matrix"
例子4
> orthogonal_y_degree3<-poly(y,degree=3)> orthogonal_y_degree3
输出结果
1 2 3[1,] 0.58967559 0.71092333 0.3049628
[2,] -0.07288125 -0.11174051 0.2422385
[3,] -0.07288125 -0.11174051 0.2422385
[4,] 0.05963012 -0.15968259 0.0224123
[5,] 0.19214148 -0.10138725 -0.2628119
[6,] 0.19214148 -0.10138725 -0.2628119
[7,] 0.19214148 -0.10138725 -0.2628119
[8,] -0.07288125 -0.11174051 0.2422385
[9,] -0.20539262 0.04243901 0.2150832
[10,] -0.07288125 -0.11174051 0.2422385
[11,] 0.05963012 -0.15968259 0.0224123
[12,] -0.33790399 0.30285595 -0.2406372
[13,] -0.07288125 -0.11174051 0.2422385
[14,] -0.33790399 0.30285595 -0.2406372
[15,] 0.19214148 -0.10138725 -0.2628119
[16,] 0.05963012 -0.15968259 0.0224123
[17,] -0.33790399 0.30285595 -0.2406372
[18,] -0.20539262 0.04243901 0.2150832
[19,] 0.05963012 -0.15968259 0.0224123
[20,] 0.19214148 -0.10138725 -0.2628119
attr(,"coefs")
attr(,"coefs")$alpha
[1] 4.550000 5.352546 6.177915
attr(,"coefs")$norm2
[1] 1.0000 20.0000 56.9500 354.4091 1091.8162
attr(,"degree")
[1] 1 2 3
attr(,"class")
[1] "poly" "matrix"
范例5
> orthogonal_y_degree5<-poly(y,degree=5)> orthogonal_y_degree5
输出结果
1 2 3 4 5[1,] 0.58967559 0.71092333 0.3049628 0.06126607 0.01071962
[2,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433
[3,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433
[4,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992
[5,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731
[6,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731
[7,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731
[8,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433
[9,] -0.20539262 0.04243901 0.2150832 -0.46796052 0.37518656
[10,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433
[11,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992
[12,] -0.33790399 0.30285595 -0.2406372 0.12904848 -0.05359808
[13,] -0.07288125 -0.11174051 0.2422385 0.02345881 -0.27013433
[14,] -0.33790399 0.30285595 -0.2406372 0.12904848 -0.05359808
[15,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731
[16,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992
[17,] -0.33790399 0.30285595 -0.2406372 0.12904848 -0.05359808
[18,] -0.20539262 0.04243901 0.2150832 -0.46796052 0.37518656
[19,] 0.05963012 -0.15968259 0.0224123 0.30209786 0.28138992
[20,] 0.19214148 -0.10138725 -0.2628119 -0.16763520 -0.07503731
attr(,"coefs")
attr(,"coefs")$alpha
[1] 4.550000 5.352546 6.177915 4.717763 4.126941
attr(,"coefs")$norm2
[1] 1.0000 20.0000 56.9500 354.4091 1091.8162 982.6517 729.7255
attr(,"degree")
[1] 1 2 3 4 5
attr(,"class")
[1] "poly" "matrix"
以上是 如何在R中生成正交多项式？ 的全部内容， 来源链接： utcz.com/z/321567.html
