2. 高斯型求积公式的求积节点和求积系数表

[高斯求积公式]

                 

式中为勒让德多项式的根.

n

求积节点

求积系数

2

0.57735 02692

1

3

0

0.77459 66692

            0.88888 88889

            0.55555 55556

4

0.33998 10436

0.86113 63116

            0.65214 51549

            0.34785 48451

5

0

0.53846 93101

0.90617 98459

0.56888 88889

0.47862 86705

0.23692 68851

6

0.23861 91861

0.66120 93865

0.93246 95142

0.46791 39346

0.36076 15731

0.17132 44924

7

0

0.40584 51514

0.74153 11856

0.94910 79123

0.41795 91837

0.38183 00505

0.27970 53915

0.12948 49662

8

0.18343 46425

0.52553 24099

0.79666 64774

0.96028 98565

0.36268 37834

0.31370 66459

0.22238 10345

0.10122 85363

n

求积节点

求积系数

9

0

0.32425 34234

0.61337 14327

0.83603 11073

0.96816 02395

0.33023 93550

0.31234 70770

0.26061 06964

0.18064 81607

0.08127 43884

10

0.14887 43390

0.43339 53941

0.67940 95683

0.86506 33667

0.97390 65285

0.29552 42247

0.26926 67193

0.21908 63625

0.14945 13492

0.06667 13443

[勒贝陶求积公式]

             

式中为的根.

n

求积节点

求积系数

3

     1

      0

0.33333 333

1.33333 333

4

      1

0.44721 360

0.16666 667

0.83333 333

5

     1

0.65465 367

       0

0.10000 000

0.54444 444

0.71111 111

6

     1

0.76505 532

0.28523 152

0.06666 667

0.37847 496

0.55485 838

7

     1

0.83022 390

0.46884 879

      0

0.04761 904

0.27682 604

0.43174 538

0.48761 904

8

     1

0.87174 015

0.59170 018

0.20929 922

0.03571 428

0.21070 422

0.34112 270

0.41245 880

9

     1

0.87174 015

0.67718 62795

0.36311 74638

      0

0.02777 77778

0.16549 53616

0.27453 87126

0.34642 85110

0.37151 92744

10

     1

0.91953 39082

0.73877 38651

0.47792 49498

0.16527 89577

0.02222 22222

0.13330 59908

0.22488 93420

0.29204 26836

0.32753 97612

       [拉盖尔求积公式]

             

             

式中为拉盖尔多项式（见第十二章，§2，四）的根.

n

求积节点

求积系数

2

0.58578 64376

3.41421 35624

(-1)8.53553 39059*

(-1)1.46446 60941

1.53332 60331

4.45095 73351

3

0.41577 45568

2.29428 03603

6.28994 50829

(-1)7.11093 00993

(-1)2.78517 73357

(-1)1.03892 56502

1.07769 28593

2.76214 29619

5.60109 46254

4

0.32254 76896

1.74576 11012

4.53662 02969

9.39507 09123

(-1)6.03154 10434

(-1)3.57418 69244

(-2)3.88879 08515

(-4)5.39294 70556

0.83273 91238

2.04810 24385

3.63114 63058

6.48714 50844

5

0.26356 03197

1.41340 30591

3.59642 57710

7.08581 00059

12.64080 08443

(-1)5.21755 61058

(-1)3.98666 81108

(-2)7.59424 49582

(-3)3.61175 86799

(-5)2.33699 72386

0.67909 40422

1.63848 78736

2.76944 32424

4.31565 69009

7.21918 63544

6

0.22284 66042

1.18893 21017

2.99273 63261

5.77514 35691

9.8374674184

15.98287 39806

(-1) 4.58964 67395

(-1)4.17000 83077

(-1)1.13373 38207

(-2)1.03991 97453

(-4)2.61017 20282

(-7)8.98547 90643

0.57353 55074

1.36925 25907

2.26068 45934

3.35052 45824

4.88682 68002

7.84901 59456

7

0.19304 36766

1.02666 48953

2.56787 67450

4.90035 30845

8.18215 34446

12.73418 02918

19.39572 78623

(-1)4.09318 95170

(-1)4.21831 27786

(-1)1.47126 34866

(-2)2.06335 14469

(-3)1.07401 01433

(-5)1.58654 64349

(-8)3.17031 54790

0.49647 75975

1.17764 30609

1.91824 97817

2.77184 86362

3.84124 91225

5.38067 82079

8.40543 24868

8

0.17027 96323

0.90370 17768

2.25108 66299

4.26670 01703

7.04560 54024

10.75851 60102

15.74067 86413

22.86313 17369

(-1)3.69188 58934

(-1)4.18786 78081

(-1)1.75794 98664

(-2)3.33434 92261

(-3)2.79453 62352

(-5)9.07650 87734

(-7)8.48574 67163

(-9)1.04800 11749

0.43772 34105

1.03386 93477

1.66970 97657

2.37692 47018

3.20854 09134

4.26857 55108

5.81808 33687

8.90622 62153

9

0.15232 22277

0.80722 00227

2.00513 51556

3.78347 39733

6.20495 67779

9.37298 52517

13.46623 69111

18.83359 77890

26.37407 18909

(-1)3.36126 42180

(-1)4.11213 98042

(-1)1.99287 52537

(-2)4.74605 62766

(-3)5.59962 66108

(-4)3.05249 76709

(-6)6.59212 30261

(-8)4.11076 93304

(-11)3.29087 40304

0.39143 11243

0.92180 50285

1.48012 79099

2.08677 08076

2.77292 13897

3.59162 60681

4.64876 60021

6.21227 54198

9.36321 82377

*表示数，其他类同，.

       [埃尔米特求积公式]

             

             

式中为埃尔米特多项式（见第十二章，§2，五）的根.

n

求积节点

求积系数

2

0.70710 67812

(-1)8.86226 92545*

1.46114 11827

3

    0

1.22474 48714

(0)1.18163 59006

(-1)2.95408 97515

1.18163 59006

1.32393 11752

4

0.52464 76233

1.65068 01239

(-1)8.04914 09001

(-2)8.13128 35447

1.05996 44829

1.24022 58177

5

     0

0.95857 24646

2.02018 28705

(-1)9.45308 72048

(-1)3.93619 32315

(-2)1.99532 42059

0.94530 87205

0.98658 09968

1.18148 86255

6

0.43607 74119

1.33584 90740

2.35060 49737

(-1)7.24629 59522

(-1)1.57067 32032

(-3)4.53000 99055

0.87640 13344

0.93558 05576

1.13690 83327

7

    0

0.81628 78829

1.67355 16288

2.65196 13568

(-1)8.10264 61756

(-1)4.25607 25261

(-2)5.45155 82819

(-4)9.7178124510

0.81026 46176

0.82868 73033

0.89718 46002

1.10133 07296

8

0.38118 69902

1.15719 37125

1.98165 67567

2.93063 74203

(-1)6.61147 01256

(-1)2.07802 32582

(-2)1.70779 83007

(-4)1.99604 07221

0.76454 41287

0.79289 00484

0.86675 26066

1.07193 01443

9

    0

0.72355 10188

1.46855 32892

2.26658 05845

3.19099 32018

(-1)7.20235 21561

(-1)4.32651 55900

(-2)8.84745 27394

(-3)4.94362 42755

(-5)3.96069 77263

0.72023 52156

0.73030 24528

0.76460 81251

0.84175 27015

1.04700 35810

10

0.34290 13272

1.03661 08298

1.75668 36493

2.53273 16742

3.43615 91188

(-1)6.10862 63374

(-1)2.40138 61108

(-2)3.38743 94456

(-3)1.34364 57468

(-6)7.64043 28552

0.68708 18540

0.70329 63231

0.74144 19319

0.82066 61264

1.02545 16914

* 表示数其他类同.