北航數(shù)值分析第三次大作業(yè)

1、數(shù)值分析第三次大作業(yè)一、 算法的設(shè)計方案：（一）、總體方案設(shè)計：（1）解非線性方程組。將給定的當作已知量代入題目給定的非線性方程組，求得與相對應的數(shù)組tij,uij。（2）分片二次代數(shù)插值。通過分片二次代數(shù)插值運算，得到與數(shù)組t1121,u1121對應的數(shù)組z1121，得到二元函數(shù)z=。（3）曲面擬合。利用xi,yj,z1121建立二維函數(shù)表，再根據(jù)精度的要求選擇適當k值，并得到曲面擬合的系數(shù)矩陣Crs。（4）觀察和的逼近效果。觀察逼近效果只需要重復上面（1）和（2）的過程，得到與新的插值節(jié)點對應的，再與對應的比較即可，這里求解可以直接使用（3）中的Crs和k。（二）具體算法設(shè)計：（1）解非線

2、性方程組牛頓法解方程組的解，可采用如下算法：1）在附近選取，給定精度水平和最大迭代次數(shù)M。2）對于執(zhí)行 計算和。 求解關(guān)于的線性方程組 若，則取，并停止計算；否則轉(zhuǎn)。 計算。 若，則繼續(xù)，否則，輸出M次迭代不成功的信息，并停止計算。（2）分片雙二次插值給定已知數(shù)表以及需要插值的節(jié)點，進行分片二次插值的算法：設(shè)已知數(shù)表中的點為： ，需要插值的節(jié)點為。1 根據(jù)選擇插值節(jié)點：若或，插值節(jié)點對應取或，若或，插值節(jié)點對應取或。若則選擇為插值節(jié)點。2）計算插值多項式的公式為：注：本步進行插值運算的是，利用與的對應關(guān)系就可以得到與的對應關(guān)系。（3）曲面擬合根據(jù)插值得到的數(shù)表進行曲面擬合的過程：1） 根據(jù)擬合

3、節(jié)點和基底函數(shù)寫出矩陣B和G： 2） 計算 。在這里，為了簡化計算和編程、避免矩陣求逆，記：，對上面兩式進行變形，得到如下兩個線性方程組：，通過解上述兩個線性方程組，則有：3） 對于每一個， 。4） 擬合需要達到的精度條件為：。其中對應著插值得到的數(shù)表中的值。5） 讓k逐步增加，每一次重復執(zhí)行以上幾步，直到 成立。此時的k值就是要求解最小的k。二、 源程序：#include #include #include #include #include #include #define Epsilon1 1e-12 /*解線性方程組時近似解向量的精度*/#define M 200 /*解線性方程組時的

4、最大迭代次數(shù)*/#define N 10 /*求解迭代次數(shù)時假設(shè)的k的最大值，用于定義包含k的存儲空間*/void Newton(; /*牛頓法求解非線性方程組子程序*/void fpeccz(; /*分片二次代數(shù)插值子程序*/void qmnh(; /*曲面擬合子程序*/void duibi(; /*對比𝑓和p逼近效果的子程序*/double x11,y21,t1121,u1121;/*定義全局變量*/double z1121,C1010;double kz;void Newton(double x11,double y21/*牛頓法求解非線性方程組子程序*/ double

5、X4,dx4,F4,dF44,temp,m,fx,fX;int i,j,k,l,p,ik,n;for(i=0;i<=10;i+ for(j=0;j<=20;j+ X0=1; /*選取迭代初始向量，四個分別代表t，u，v，w*/X1=1;X2=1;X3=1;n=0;loop1: F0=0.5*cos(X0+X1+X2+X3-xi-2.67; F1=X0+0.5*sin(X1+X2+X3-yj-1.07;F2=0.5*X0+X1+cos(X2+X3-xi-3.74;F3=X0+0.5*X1+X2+sin(X3-yj-0.79;/*求解F（x）*/dF00=-0.5*sin(X0; /*

6、求解F'（x）*/dF01=1;dF02=1;dF03=1;dF10=1;dF11=0.5*cos(X1;dF12=1;dF13=1;dF20=0.5;dF21=1;dF22=-sin(X2;dF23=1;dF30=1;dF31=0.5;dF32=1;dF33=cos(X3;/*高斯選主元消去法求解x*/for(k=0;k<3;k+ik=k;for(l=k;l<=3;l+if(dFikk ik=l; /*選主元*/temp=0;temp=Fik;Fik=Fk;Fk=temp;for(l=k;l<=3;l+ temp=0;temp=dFikl;dFikl=dFkl;dF

7、kl=temp; for(l=k+1;l<=3;l+m=dFlk/dFkk;Fl=Fl-m*Fk;for(p=k+1;p<=3;p+dFlp=dFlp-m*dFkp; /*消去過程*/dx3=-F3/dF33;for(k=2;k>=0;k-temp=0;for(l=k+1;l<=3;l+temp=temp+dFkl*dxl/dFkk;dxk=-Fk/dFkk-temp; temp=0;for(l=0;l<=3;l+ /*求解矩陣范數(shù)，用無窮范數(shù)*/ if(temp temp=fabs(dxl;fx=temp;temp=0;for(l=0;l<=3;l+ if

8、(temp temp=fabs(Xl;fX=temp;if(fabs(fx/fX 判斷 是否成立 */ tij=X0;uij=X1;goto loop4;else for(l=0;l<=3;l+ Xl=Xl+dxl;n=n+1;goto loop3;loop3:if(n 判斷是否超出規(guī)定迭代次數(shù) */ goto loop1;else printf("迭代不成功n"goto loop4; loop4:continue; void fpeccz(double t1121,double u1121/*分片二次代數(shù)插值子程序*/int s1121,r1121;int i,j,

9、i1,j1,m;double z066=-0.5,-0.34,0.14,0.94,2.06,3.5,-0.42,-0.5,-0.26,0.3,1.18,2.38,-0.18,-0.5,-0.5,-0.18,0.46,1.42,0.22,-0.34,-0.58,-0.5,-0.1,0.62,0.78,-0.02,-0.5,-0.66,-0.5,-0.02,1.5,0.46,-0.26,-0.66,-0.74,-0.5; double t06=0,0.2,0.4,0.6,0.8,1.0;double u06=0,0.4,0.8,1.2,1.6,2.0; double temp1,temp2;for

10、(i=0;i<=10;i+ /*選取插值節(jié)點*/for(j=0;j<=20;j+if(tij<=0.3 sij=1; else if(tij>0.7 sij=4;elsefor(m=2;m<=3;m+if(tij>0.2*m-0.1&&(tij<=0.2*m+0.1sij=m;for(i=0;i<=10;i+ for(j=0;j<=20;j+if(uij<=0.6 rij=1;else if(uij>1.4 rij=4; elsefor(m=2;m<=3;m+if(uij>0.4*m-0.2&

11、&(uij<=0.4*m+0.2rij=m; for(i=0;i<=10;i+ /*插值運算*/for(j=0;j<=20;j+zij=0; for(i1=sij-1;i1<=sij+1;i1+for(j1=rij-1;j1<=rij+1;j1+temp1=1.0;for(m=sij-1;m<=sij+1;m+ if(m!=i1 temp1*=(tij-t0m/(t0i1-t0m; temp2=1.0;for(m=rij-1;m<=rij+1;m+ if(m!=j1 temp2*=(uij-u0m/(u0j1-u0m; zij+=temp1*t

12、emp2*z0i1j1; void qmnh( /*曲面擬合子程序*/ int i,j,k,m,l,i1,j1,ik; double AN21,DN21,B11N,BtN11,BtBNN,BtUN21,BtB1NN;double G21N,GtN21,GtGNN,GtG1NN;double sigma,p1121,temp,q;printf("選擇過程的k和sigma值:n"k=0;sigma=1; /*選取初值，使循環(huán)開始*/*求解系數(shù)矩陣Crs*/while(sigma>1e-7 for(i=0;i<=10;i+for(j=0;j<=k;j+Bij=p

13、ow(xi,j;Btji=Bij;for(i=0;i<=k;i+for(j=0;j<=k;j+temp=0;for(l=0;l<=10;l+ temp+=Btil*Blj; BtBij=temp;for(i=0;i<=k;i+for(j=0;j<=20;j+temp=0;for(l=0;l<=10;l+ temp+=Btil*zlj; BtUij=temp;for(l=0;l<=20;l+for(i=0;i<=k;i+for(j=0;j<=k;j+ BtB1ij=BtBij; for(m=0;m<=k-1;m+ ik=m;for(i=

14、m;i<=k-1;i+if(fabs(BtB1im ik=i+1;else ;if(ik!=m for(i=m;i<=k;i+temp=BtB1mi;BtB1mi=BtB1iki;BtB1iki=temp;temp=BtUml;BtUml=BtUikl;BtUikl=temp;for(i=m+1;i<=k;i+q=BtB1im/BtB1mm;for(j=m;j<=k;j+ BtB1ij=BtB1ij-q*BtB1mj; BtUil=BtUil-q*BtUml;Akl=BtUkl/BtB1kk;for(m=k-1;m>=0;m-temp=0;for(i=m+1;i&

15、lt;=k;i+ temp+=Ail*BtB1mi; Aml=(BtUml-temp/BtB1mm;for(i=0;i<=20;i+for(j=0;j<=k;j+Gij=pow(yi,j;Gtji=Gij;for(i=0;i<=k;i+for(j=0;j<=k;j+temp=0;for(m=0;m<=20;m+ temp+=Gtim*Gmj; GtGij=temp;for(l=0;l<=20;l+for(i=0;i<=k;i+for(j=0;j<=k;j+ GtG1ij=GtGij; for(m=0;m<=k-1;m+ ik=m;for(i

16、=m;i<=k-1;i+if(fabs(GtG1im ik=i+1;else ;if(ik!=m for(i=m;i<=k;i+temp=GtG1mi;GtG1mi=GtG1iki;GtG1iki=temp;temp=Gtml;Gtml=Gtikl;Gtikl=temp;for(i=m+1;i<=k;i+q=GtG1im/GtG1mm;for(j=m;j<=k;j+ GtG1ij=GtG1ij-q*GtG1mj; Gtil=Gtil-q*Gtml;Dkl=Gtkl/GtG1kk;for(m=k-1;m>=0;m-temp=0;for(i=m+1;i<=k;i

17、+ temp+=Dil*GtG1mi; Dml=(Gtml-temp/GtG1mm;for(i=0;i<=k;i+for(j=0;j<=k;j+temp=0;for(m=0;m<=20;m+ temp+=Aim*Djm; Cij=temp;sigma=0;/*歸零，開始計算sigma值*/for(i=0;i<=10;i+for(j=0;j<=20;j+pij=0;for(i1=0;i1<=k;i1+for(j1=0;j1<=k;j1+ pij+=Ci1j1*pow(xi,i1*pow(yj,j1; sigma+=pow(pij-zij,2; print

18、f("k=%d sigma=%.12en",k,sigma; k=k+1; k-;printf("達到精度要求時的k和sigma值：n"printf("k=%d sigma=%.12en",k,sigma; kz=k; /*定義為全局變量，便于duibi(調(diào)用*/void duibi( /*對比𝑓和p逼近效果的子程序*/ int i,j,i1,j1;double p85;for(i=0;i<=7;i+ for(j=0;j<=4;j+xi=0.1*(i+1;yj=0.5+0.2*(j+1; /*重新輸入節(jié)點*

19、/Newton(x,y;fpeccz(t,u; /*求解𝑓（x*，y*）*/for(i=0;i<=7;i+ /*求解p（x*，y*）*/for(j=0;j<=4;j+pij=0;for(i1=0;i1<=kz;i1+ for(j1=0;j1<=kz;j1+ pij+=Ci1j1*pow(xi,i1*pow(yj,j1; printf("x%d=%.6f y%d=%.6fn",i+1,xi,j+1,yj;printf("f(x%d,y%d=%.12en",i+1,j+1,zij;printf("p(x%d,

20、y%d=%.12en",i+1,j+1,pij;printf("=%.12en",zij-pij; /*數(shù)表xi*,yi*,𝑓 (xi*,yi*,p（xi*,yi*）*/ void main( int i,j;for(i=0;i<=10;i+ for(j=0;j<=20;j+xi=0.08*i;yj=0.5+0.05*j; /*輸入節(jié)點*/Newton(x,y;fpeccz(t,u;for(i=0;i<=10;i+ for(j=0;j<=20;j+printf("x%d=%.6f y%d=%.6f z%d%d=%.

21、12e n",i,xi,j,yj,i,j,zij; /*數(shù)表：xi,yi, 𝑓 (xi,yi*/qmnh(;for(i=0;i<=kz;i+ for(j=0;j<=kz;j+printf("C%d%d=%.12e n",i,j,Cij;/*數(shù)表：Crs*/duibi(;三、運行結(jié)果輸出1、數(shù)表數(shù)表：xi,yi,𝑓(xi,yix0=0.000000 y0=0.500000 z00=4.465040184807e-001x0=0.000000 y1=0.550000 z01=3.246832629277e-001x0=0.0

22、00000 y3=0.650000 z03=1.030436083160e-001x0=0.000000 y4=0.700000 z04=3.401895562675e-003x0=0.000000 y5=0.750000 z05=-8.873581363800e-002x0=0.000000 y6=0.800000 z06=-1.733716327497e-001x0=0.000000 y7=0.850000 z07=-2.505346114666e-001x0=0.000000 y8=0.900000 z08=-3.202765063876e-001x0=0.000000 y9=0.950

23、000 z09=-3.826680661097e-001x0=0.000000 y10=1.000000 z010=-4.377957667384e-001x0=0.000000 y11=1.050000 z011=-4.857589414438e-001x0=0.000000 y12=1.100000 z012=-5.266672548835e-001x0=0.000000 y13=1.150000 z013=-5.606384797965e-001x0=0.000000 y14=1.200000 z014=-5.877965387677e-001x0=0.000000 y15=1.2500

24、00 z015=-6.082697790899e-001x0=0.000000 y16=1.300000 z016=-6.221894528764e-001x0=0.000000 y17=1.350000 z017=-6.296883781856e-001x0=0.000000 y18=1.400000 z018=-6.308997600028e-001x0=0.000000 y19=1.450000 z019=-6.259561525454e-001x1=0.080000 y0=0.500000 z10=6.380152265113e-001x1=0.080000 y1=0.550000 z

25、11=5.066117551467e-001x1=0.080000 y2=0.600000 z12=3.821763692774e-001x1=0.080000 y3=0.650000 z13=2.648634911537e-001x1=0.080000 y4=0.700000 z14=1.547802002848e-001x1=0.080000 y6=0.800000 z16=-4.346804020490e-002x1=0.080000 y9=0.950000 z19=-2.860045510580e-001x1=0.080000 y10=1.000000 z110=-3.52386078

26、9794e-001x1=0.080000 y12=1.100000 z112=-4.639049115188e-001x1=0.080000 y13=1.150000 z113=-5.092367247005e-001x1=0.080000 y14=1.200000 z114=-5.477611179623e-001x1=0.080000 y15=1.250000 z115=-5.795943883391e-001x1=0.080000 y16=1.300000 z116=-6.048572588895e-001x1=0.080000 y17=1.350000 z117=-6.23673421

27、3318e-001x1=0.080000 y18=1.400000 z118=-6.361682484133e-001x1=0.080000 y19=1.450000 z119=-6.424676566901e-001x1=0.080000 y20=1.500000 z120=-6.426971026996e-001x2=0.160000 y0=0.500000 z20=8.400813957666e-001x2=0.160000 y1=0.550000 z21=6.997641656732e-001x2=0.160000 y2=0.600000 z22=5.660614423517e-001

28、x2=0.160000 y3=0.650000 z23=4.391716081176e-001x2=0.160000 y5=0.750000 z25=2.063761923874e-001x2=0.160000 y6=0.800000 z26=1.006385238914e-001x2=0.160000 y7=0.850000 z27=2.060740067837e-003x2=0.160000 y8=0.900000 z28=-8.935402476698e-002x2=0.160000 y9=0.950000 z29=-1.736269688648e-001x2=0.160000 y10=

29、1.000000 z210=-2.507999561599e-001x2=0.160000 y11=1.050000 z211=-3.209322694446e-001x2=0.160000 y12=1.100000 z212=-3.840977350046e-001x2=0.160000 y13=1.150000 z213=-4.403821754175e-001x2=0.160000 y14=1.200000 z214=-4.898811523126e-001x2=0.160000 y15=1.250000 z215=-5.326979655338e-001x2=0.160000 y16=

30、1.300000 z216=-5.689418792921e-001x2=0.160000 y17=1.350000 z217=-5.987265495151e-001x2=0.160000 y18=1.400000 z218=-6.221686297503e-001x2=0.160000 y19=1.450000 z219=-6.393865356972e-001x2=0.160000 y20=1.500000 z220=-6.504993507878e-001x3=0.240000 y0=0.500000 z30=1.051515091803e+000x3=0.240000 y1=0.55

31、0000 z31=9.029274308310e-001x3=0.240000 y2=0.600000 z32=7.605802668596e-001x3=0.240000 y3=0.650000 z33=6.247151981456e-001x3=0.240000 y4=0.700000 z34=4.955197560009e-001x3=0.240000 y5=0.750000 z35=3.731340427746e-001x3=0.240000 y6=0.800000 z36=2.576567488723e-001x3=0.240000 y7=0.850000 z37=1.4915055

32、94102e-001x3=0.240000 y8=0.900000 z38=4.764698677337e-002x3=0.240000 y9=0.950000 z39=-4.684932320146e-002x3=0.240000 y10=1.000000 z310=-1.343567603849e-001x3=0.240000 y12=1.100000 z312=-2.885737006348e-001x3=0.240000 y13=1.150000 z313=-3.554063647857e-001x3=0.240000 y15=1.250000 z315=-4.689182499695

33、e-001x3=0.240000 y17=1.350000 z317=-5.561910752001e-001x3=0.240000 y18=1.400000 z318=-5.902469305629e-001x3=0.240000 y20=1.500000 z320=-6.397468392579e-001x4=0.320000 y0=0.500000 z40=1.271246751483e+000x4=0.320000 y2=0.600000 z42=9.646077272157e-001x4=0.320000 y3=0.650000 z43=8.203473694751e-001x4=0

34、.320000 y4=0.700000 z44=6.824476781795e-001x4=0.320000 y6=0.800000 z46=4.263923859018e-001x4=0.320000 y7=0.850000 z47=3.084629956332e-001x4=0.320000 y8=0.900000 z48=1.973571296919e-001x4=0.320000 y10=1.000000 z410=-4.285992234033e-003x4=0.320000 y11=1.050000 z411=-9.483392529689e-002x4=0.320000 y12=

35、1.100000 z412=-1.785729903640e-001x4=0.320000 y13=1.150000 z413=-2.555537790546e-001x4=0.320000 y14=1.200000 z414=-3.258401501575e-001x4=0.320000 y15=1.250000 z415=-3.895069883634e-001x4=0.320000 y16=1.300000 z416=-4.466382045995e-001x4=0.320000 y17=1.350000 z417=-4.973249517677e-001x4=0.320000 y19=

36、1.450000 z419=-5.797564797951e-001x5=0.400000 y0=0.500000 z50=1.498321052482e+000x5=0.400000 y1=0.550000 z51=1.334998632066e+000x5=0.400000 y3=0.650000 z53=1.025024055020e+000x5=0.400000 y4=0.700000 z54=8.789600231744e-001x5=0.400000 y5=0.750000 z55=7.391451087035e-001x5=0.400000 y6=0.800000 z56=6.0

37、57448714871e-001x5=0.400000 y7=0.850000 z57=4.788838610666e-001x5=0.400000 y8=0.900000 z58=3.586506258818e-001x5=0.400000 y9=0.950000 z59=2.451022361964e-001x5=0.400000 y10=1.000000 z510=1.382683509285e-001x5=0.400000 y12=1.100000 z512=-5.525282116814e-002x5=0.400000 y14=1.200000 z514=-2.22094439095

38、9e-001x5=0.400000 y15=1.250000 z515=-2.956352324598e-001x5=0.400000 y16=1.300000 z516=-3.626795115028e-001x5=0.400000 y17=1.350000 z517=-4.233061642240e-001x5=0.400000 y18=1.400000 z518=-4.776010361325e-001x5=0.400000 y19=1.450000 z519=-5.256554266672e-001x5=0.400000 y20=1.500000 z520=-5.67564743655

39、1e-001x6=0.480000 y0=0.500000 z60=1.731892740383e+000x6=0.480000 y1=0.550000 z61=1.562034577209e+000x6=0.480000 y2=0.600000 z62=1.397216918208e+000x6=0.480000 y3=0.650000 z63=1.237801006739e+000x6=0.480000 y4=0.700000 z64=1.084087532678e+000x6=0.480000 y5=0.750000 z65=9.363227723149e-001x6=0.480000

40、y6=0.800000 z66=7.947044490537e-001x6=0.480000 y7=0.850000 z67=6.593871980282e-001x6=0.480000 y8=0.900000 z68=5.304875868400e-001x6=0.480000 y9=0.950000 z69=4.080886854542e-001x6=0.480000 y10=1.000000 z610=2.922442012295e-001x6=0.480000 y11=1.050000 z611=1.829822068536e-001x6=0.480000 y12=1.100000 z

41、612=8.030849403543e-002x6=0.480000 y13=1.150000 z613=-1.579041305164e-002x6=0.480000 y14=1.200000 z614=-1.053445516210e-001x6=0.480000 y15=1.250000 z615=-1.883980906096e-001x6=0.480000 y16=1.300000 z616=-2.650071493189e-001x6=0.480000 y17=1.350000 z617=-3.352378389040e-001x6=0.480000 y18=1.400000 z6

42、18=-3.991645038868e-001x6=0.480000 y19=1.450000 z619=-4.568681433016e-001x6=0.480000 y20=1.500000 z620=-5.084349932782e-001x7=0.560000 y0=0.500000 z70=1.971221786400e+000x7=0.560000 y1=0.550000 z71=1.795329599501e+000x7=0.560000 y2=0.600000 z72=1.624067113228e+000x7=0.560000 y3=0.650000 z73=1.457830

43、582708e+000x7=0.560000 y4=0.700000 z74=1.296954649752e+000x7=0.560000 y6=0.800000 z76=9.923495333243e-001x7=0.560000 y7=0.850000 z77=8.490326633294e-001x7=0.560000 y10=1.000000 z710=4.566585132334e-001x7=0.560000 y11=1.050000 z711=3.386544961394e-001x7=0.560000 y12=1.100000 z712=2.271082557696e-001x

44、7=0.560000 y13=1.150000 z713=1.220250891932e-001x7=0.560000 y14=1.200000 z714=2.339221963760e-002x7=0.560000 y15=1.250000 z715=-6.881870197104e-002x7=0.560000 y16=1.300000 z716=-1.546493442129e-001x7=0.560000 y17=1.350000 z717=-2.341526664587e-001x7=0.560000 y18=1.400000 z718=-3.073910919133e-001x7=

45、0.560000 y19=1.450000 z719=-3.744348623481e-001x7=0.560000 y20=1.500000 z720=-4.353605565359e-001x8=0.640000 y1=0.550000 z81=2.034201133607e+000x8=0.640000 y2=0.600000 z82=1.856955143619e+000x8=0.640000 y3=0.650000 z83=1.684358164161e+000x8=0.640000 y5=0.750000 z85=1.354519041151e+000x8=0.640000 y7=

46、0.850000 z87=1.046963049419e+000x8=0.640000 y8=0.900000 z88=9.020460838023e-001x8=0.640000 y9=0.950000 z89=7.632264776629e-001x8=0.640000 y10=1.000000 z810=6.306048219543e-001x8=0.640000 y11=1.050000 z811=5.042528145972e-001x8=0.640000 y12=1.100000 z812=3.842167155457e-001x8=0.640000 y13=1.150000 z8

47、13=2.705204766410e-001x8=0.640000 y14=1.200000 z814=1.631685723996e-001x8=0.640000 y16=1.300000 z816=-3.256661939682e-002x8=0.640000 y18=1.400000 z818=-2.032513996228e-001x8=0.640000 y19=1.450000 z819=-2.793303595584e-001x8=0.640000 y20=1.500000 z820=-3.493199575400e-001x9=0.720000 y0=0.500000 z90=2

48、.464684222659e+000x9=0.720000 y1=0.550000 z91=2.278058979398e+000x9=0.720000 y2=0.600000 z92=2.095251250840e+000x9=0.720000 y4=0.700000 z94=1.742854628776e+000x9=0.720000 y5=0.750000 z95=1.573998427334e+000x9=0.720000 y7=0.850000 z97=1.252401750608e+000x9=0.720000 y9=0.950000 z99=9.536698512613e-001x9=0.720000 y11=1.050000 z911=6.789307429659e-001x9=0.720000 y12=1.100000 z912=5.507748485043e-001x9=0.720000 y13=1.150000 z913=4.288256769731e-