數值分析B第二次上機作業課后題第8題關于x,y,t,u,v,w的方程組(A.3)(A.3)以及關于z,t,u的二維數表（見表A-1）確定了一個二元函數z=f(x,y)。表A-1二維數表tzu00.40.81.21.620-0.5-0.340.140.942.063.50.2-0.42-0.5-0.260.31.182.380.4-0.18-0.5-0.5-0.180.461.420.60.22-0.34-0.58-0.5-0.10.620.80.78-0.02-0.5-0.66-0.5-0.021.01.50.46-0.26-0.66-0.74-0.5試用數值方法求出f(x,y)在區域上的近似表達式要求p(x,y)以最小的k值達到以下的精度其中。計算(i=1,2,…,8;j=1,2,…,5)的值，以觀察p(x,y)逼近f(x,y)的效果，其中。算法設計：1.題目要求對f(x,y)進行擬合，可選用乘積型最小二乘擬合。與的數表由方程組與表A-1得到。2.與1使用相同方法求得，由計算得出的p(x,y)直接帶入求得。算法實現1.與的數表的獲得對區域上的f(x,y)值可由方程組及二維數表得到。將區域D上的分別回代于方程組(A.3)，成為關于t,u,v,w的4元非線性方程組，解出每個對應的t,u。再通過表A-1進行插值近似，得到相應的z值。對應的z即為D區域上對應的。從而得到與的數表。(1)4元非線性方程組求解代入(A.3)后，原方程組變為關于t,u,v,w的4元非線性方程組。觀察到方程組中方程形式較為簡單，易于對變量t,u,v,w求偏導數，故而選用Newton法對方程組求解。計算方程組矩陣為：計算方程組偏導數矩陣為：迭代公式為：，k=0,1,2,…,n其中為線性方程組的解。取為迭代終止條件。由表A-1觀察到t,u基本在（0,2）上，于是選取為迭代初值。通過以上方法求得與對應的。分片二元雙二次代數插值為保證代數插值的收斂性，應采用分片低次插值。故此使用分片雙二次代數插值。給定如滿足如下關系式：，，則選擇為插值節點，相應插值多項式為其中如果，則上式取m=1或m=4；如果或，則取n=1或n=4。得到表達式后，直接帶入，得到的值即為與對應的。2.乘積型最小二乘曲面擬合使用乘積型最小二乘擬合，根據k值不用，有基函數矩陣如下：，數表矩陣如下：記C=[]，則系數的表達式矩陣為：通過求解如下線性方程，即可得到系數矩陣C。計算(i=1,2,…,8;j=1,2,…,5)的值的計算與相同。將代入原方程組，求解響應進行分片雙二次插值求得。的計算則可以直接將代入所求p(x,y)。C程序計算結果二元二次分片插值得到數表：xi，yj，fxi,yj，i,,,1;j,,,20。xi=0.00yi=1.50f(xi,yi)=-6.149885466094e-001xi=0.08yi=0.50f(xi,yi)=6.380152265102e-001xi=0.08yi=0.55f(xi,yi)=5.066117551462e-001xi=0.08yi=0.60f(xi,yi)=3.821763692772e-001xi=0.08yi=0.65f(xi,yi)=2.648634911536e-001xi=0.08yi=0.70f(xi,yi)=1.547802002848e-001xi=0.08yi=0.75f(xi,yi)=5.199268349093e-002xi=0.08yi=0.80f(xi,yi)=-4.346804020491e-002xi=0.08yi=0.85f(xi,yi)=-1.316010567885e-001xi=0.08yi=0.90f(xi,yi)=-2.124310883088e-001xi=0.08yi=0.95f(xi,yi)=-2.860045510580e-001xi=0.08yi=1.00f(xi,yi)=-3.523860789794e-001xi=0.08yi=1.05f(xi,yi)=-4.116554565222e-001xi=0.08yi=1.10f(xi,yi)=-4.639049115188e-001xi=0.08yi=1.15f(xi,yi)=-5.092367247005e-001xi=0.08yi=1.20f(xi,yi)=-5.477611179623e-001xi=0.08yi=1.25f(xi,yi)=-5.795943883391e-001xi=0.08yi=1.30f(xi,yi)=-6.048572588895e-001xi=0.08yi=1.35f(xi,yi)=-6.236734213318e-001xi=0.08yi=1.40f(xi,yi)=-6.361682484133e-001xi=0.08yi=1.45f(xi,yi)=-6.424676566901e-001xi=0.08yi=1.50f(xi,yi)=-6.426971026996e-001xi=0.16yi=0.50f(xi,yi)=8.400813957651e-001xi=0.16yi=0.55f(xi,yi)=6.997641656726e-001xi=0.16yi=0.60f(xi,yi)=5.660614423514e-001xi=0.16yi=0.65f(xi,yi)=4.391716081175e-001xi=0.16yi=0.70f(xi,yi)=3.192421380408e-001xi=0.16yi=0.75f(xi,yi)=2.063761923874e-001xi=0.16yi=0.80f(xi,yi)=1.006385238914e-001xi=0.16yi=0.85f(xi,yi)=2.060740067836e-003xi=0.16yi=0.90f(xi,yi)=-8.935402476698e-002xi=0.16yi=0.95f(xi,yi)=-1.736269688649e-001xi=0.16yi=1.00f(xi,yi)=-2.507999561599e-001xi=0.16yi=1.05f(xi,yi)=-3.209322694446e-001xi=0.16yi=1.10f(xi,yi)=-3.840977350046e-001xi=0.16yi=1.15f(xi,yi)=-4.403821754175e-001xi=0.16yi=1.20f(xi,yi)=-4.898811523126e-001xi=0.16yi=1.25f(xi,yi)=-5.326979655338e-001xi=0.16yi=1.30f(xi,yi)=-5.689418792921e-001xi=0.16yi=1.35f(xi,yi)=-5.987265495151e-001xi=0.16yi=1.40f(xi,yi)=-6.221686297503e-001xi=0.16yi=1.45f(xi,yi)=-6.393865356972e-001xi=0.16yi=1.50f(xi,yi)=-6.504993507878e-001xi=0.24yi=0.50f(xi,yi)=1.051515091801e+000xi=0.24yi=0.55f(xi,yi)=9.029274308302e-001xi=0.24yi=0.60f(xi,yi)=7.605802668593e-001xi=0.24yi=0.65f(xi,yi)=6.247151981455e-001xi=0.24yi=0.70f(xi,yi)=4.955197560009e-001xi=0.24yi=0.75f(xi,yi)=3.731340427746e-001xi=0.24yi=0.80f(xi,yi)=2.576567488723e-001xi=0.24yi=0.85f(xi,yi)=1.491505594102e-001xi=0.24yi=0.90f(xi,yi)=4.764698677337e-002xi=0.24yi=0.95f(xi,yi)=-4.684932320146e-002xi=0.24yi=1.00f(xi,yi)=-1.343567603849e-001xi=0.24yi=1.05f(xi,yi)=-2.149133449274e-001xi=0.24yi=1.10f(xi,yi)=-2.885737006348e-001xi=0.24yi=1.15f(xi,yi)=-3.554063647857e-001xi=0.24yi=1.20f(xi,yi)=-4.154913964886e-001xi=0.24yi=1.25f(xi,yi)=-4.689182499695e-001xi=0.24yi=1.30f(xi,yi)=-5.157838831247e-001xi=0.24yi=1.35f(xi,yi)=-5.561910752001e-001xi=0.24yi=1.40f(xi,yi)=-5.902469305629e-001xi=0.24yi=1.45f(xi,yi)=-6.180615482412e-001xi=0.24yi=1.50f(xi,yi)=-6.397468392579e-001xi=0.32yi=0.50f(xi,yi)=1.271246751481e+000xi=0.32yi=0.55f(xi,yi)=1.115002018146e+000xi=0.32yi=0.60f(xi,yi)=9.646077272154e-001xi=0.32yi=0.65f(xi,yi)=8.203473694749e-001xi=0.32yi=0.70f(xi,yi)=6.824476781794e-001xi=0.32yi=0.75f(xi,yi)=5.510852085975e-001xi=0.32yi=0.80f(xi,yi)=4.263923859018e-001xi=0.32yi=0.85f(xi,yi)=3.084629956332e-001xi=0.32yi=0.90f(xi,yi)=1.973571296919e-001xi=0.32yi=0.95f(xi,yi)=9.310562085941e-002xi=0.32yi=1.00f(xi,yi)=-4.285992234034e-003xi=0.32yi=1.05f(xi,yi)=-9.483392529689e-002xi=0.32yi=1.10f(xi,yi)=-1.785729903640e-001xi=0.32yi=1.15f(xi,yi)=-2.555537790546e-001xi=0.32yi=1.20f(xi,yi)=-3.258401501575e-001xi=0.32yi=1.25f(xi,yi)=-3.895069883634e-001xi=0.32yi=1.30f(xi,yi)=-4.466382045995e-001xi=0.32yi=1.35f(xi,yi)=-4.973249517677e-001xi=0.32yi=1.40f(xi,yi)=-5.416640326994e-001xi=0.32yi=1.45f(xi,yi)=-5.797564797951e-001xi=0.32yi=1.50f(xi,yi)=-6.117062881476e-001xi=0.40yi=0.50f(xi,yi)=1.498321052481e+000xi=0.40yi=0.55f(xi,yi)=1.334998632066e+000xi=0.40yi=0.60f(xi,yi)=1.177125123739e+000xi=0.40yi=0.65f(xi,yi)=1.025024055020e+000xi=0.40yi=0.70f(xi,yi)=8.789600231743e-001xi=0.40yi=0.75f(xi,yi)=7.391451087035e-001xi=0.40yi=0.80f(xi,yi)=6.057448714871e-001xi=0.40yi=0.85f(xi,yi)=4.788838610666e-001xi=0.40yi=0.90f(xi,yi)=3.586506258818e-001xi=0.40yi=0.95f(xi,yi)=2.451022361964e-001xi=0.40yi=1.00f(xi,yi)=1.382683509285e-001xi=0.40yi=1.05f(xi,yi)=3.815486540699e-002xi=0.40yi=1.10f(xi,yi)=-5.525282116814e-002xi=0.40yi=1.15f(xi,yi)=-1.419868808137e-001xi=0.40yi=1.20f(xi,yi)=-2.220944390959e-001xi=0.40yi=1.25f(xi,yi)=-2.956352324598e-001xi=0.40yi=1.30f(xi,yi)=-3.626795115028e-001xi=0.40yi=1.35f(xi,yi)=-4.233061642240e-001xi=0.40yi=1.40f(xi,yi)=-4.776010361325e-001xi=0.40yi=1.45f(xi,yi)=-5.256554266672e-001xi=0.40yi=1.50f(xi,yi)=-5.675647436551e-001xi=0.48yi=0.50f(xi,yi)=1.731892740382e+000xi=0.48yi=0.55f(xi,yi)=1.562034577208e+000xi=0.48yi=0.60f(xi,yi)=1.397216918208e+000xi=0.48yi=0.65f(xi,yi)=1.237801006739e+000xi=0.48yi=0.70f(xi,yi)=1.084087532678e+000xi=0.48yi=0.75f(xi,yi)=9.363227723149e-001xi=0.48yi=0.80f(xi,yi)=7.947044490537e-001xi=0.48yi=0.85f(xi,yi)=6.593871980282e-001xi=0.48yi=0.90f(xi,yi)=5.304875868399e-001xi=0.48yi=0.95f(xi,yi)=4.080886854542e-001xi=0.48yi=1.00f(xi,yi)=2.922442012295e-001xi=0.48yi=1.05f(xi,yi)=1.829822068535e-001xi=0.48yi=1.10f(xi,yi)=8.030849403543e-002xi=0.48yi=1.15f(xi,yi)=-1.579041305164e-002xi=0.48yi=1.20f(xi,yi)=-1.053445516210e-001xi=0.48yi=1.25f(xi,yi)=-1.883980906096e-001xi=0.48yi=1.30f(xi,yi)=-2.650071493189e-001xi=0.48yi=1.35f(xi,yi)=-3.352378389040e-001xi=0.48yi=1.40f(xi,yi)=-3.991645038868e-001xi=0.48yi=1.45f(xi,yi)=-4.568681433016e-001xi=0.48yi=1.50f(xi,yi)=-5.084349932782e-001xi=0.56yi=0.50f(xi,yi)=1.971221786400e+000xi=0.56yi=0.55f(xi,yi)=1.795329599501e+000xi=0.56yi=0.60f(xi,yi)=1.624067113228e+000xi=0.56yi=0.65f(xi,yi)=1.457830582708e+000xi=0.56yi=0.70f(xi,yi)=1.296954649752e+000xi=0.56yi=0.75f(xi,yi)=1.141718105447e+000xi=0.56yi=0.80f(xi,yi)=9.923495333243e-001xi=0.56yi=0.85f(xi,yi)=8.490326633294e-001xi=0.56yi=0.90f(xi,yi)=7.119113522641e-001xi=0.56yi=0.95f(xi,yi)=5.810941589219e-001xi=0.56yi=1.00f(xi,yi)=4.566585132334e-001xi=0.56yi=1.05f(xi,yi)=3.386544961394e-001xi=0.56yi=1.10f(xi,yi)=2.271082557696e-001xi=0.56yi=1.15f(xi,yi)=1.220250891932e-001xi=0.56yi=1.20f(xi,yi)=2.339221963760e-002xi=0.56yi=1.25f(xi,yi)=-6.881870197104e-002xi=0.56yi=1.30f(xi,yi)=-1.546493442129e-001xi=0.56yi=1.35f(xi,yi)=-2.341526664587e-001xi=0.56yi=1.40f(xi,yi)=-3.073910919133e-001xi=0.56yi=1.45f(xi,yi)=-3.744348623481e-001xi=0.56yi=1.50f(xi,yi)=-4.353605565359e-001xi=0.64yi=0.50f(xi,yi)=2.215667863688e+000xi=0.64yi=0.55f(xi,yi)=2.034201133607e+000xi=0.64yi=0.60f(xi,yi)=1.856955143619e+000xi=0.64yi=0.65f(xi,yi)=1.684358164161e+000xi=0.64yi=0.70f(xi,yi)=1.516776352400e+000xi=0.64yi=0.75f(xi,yi)=1.354519041151e+000xi=0.64yi=0.80f(xi,yi)=1.197844086673e+000xi=0.64yi=0.85f(xi,yi)=1.046963049419e+000xi=0.64yi=0.90f(xi,yi)=9.020460838023e-001xi=0.64yi=0.95f(xi,yi)=7.632264776629e-001xi=0.64yi=1.00f(xi,yi)=6.306048219543e-001xi=0.64yi=1.05f(xi,yi)=5.042528145972e-001xi=0.64yi=1.10f(xi,yi)=3.842167155457e-001xi=0.64yi=1.15f(xi,yi)=2.705204766410e-001xi=0.64yi=1.20f(xi,yi)=1.631685723996e-001xi=0.64yi=1.25f(xi,yi)=6.214855811676e-002xi=0.64yi=1.30f(xi,yi)=-3.256661939682e-002xi=0.64yi=1.35f(xi,yi)=-1.210165348444e-001xi=0.64yi=1.40f(xi,yi)=-2.032513996228e-001xi=0.64yi=1.45f(xi,yi)=-2.793303595584e-001xi=0.64yi=1.50f(xi,yi)=-3.493199575400e-001xi=0.72yi=0.50f(xi,yi)=2.464684222660e+000xi=0.72yi=0.55f(xi,yi)=2.278058979399e+000xi=0.72yi=0.60f(xi,yi)=2.095251250840e+000xi=0.72yi=0.65f(xi,yi)=1.916718127997e+000xi=0.72yi=0.70f(xi,yi)=1.742854628776e+000xi=0.72yi=0.75f(xi,yi)=1.573998427334e+000xi=0.72yi=0.80f(xi,yi)=1.410434835231e+000xi=0.72yi=0.85f(xi,yi)=1.252401750608e+000xi=0.72yi=0.90f(xi,yi)=1.100094409628e+000xi=0.72yi=0.95f(xi,yi)=9.536698512613e-001xi=0.72yi=1.00f(xi,yi)=8.132510552489e-001xi=0.72yi=1.05f(xi,yi)=6.789307429659e-001xi=0.72yi=1.10f(xi,yi)=5.507748485043e-001xi=0.72yi=1.15f(xi,yi)=4.288256769731e-001xi=0.72yi=1.20f(xi,yi)=3.131047717398e-001xi=0.72yi=1.25f(xi,yi)=2.036155140327e-001xi=0.72yi=1.30f(xi,yi)=1.003454782409e-001xi=0.72yi=1.35f(xi,yi)=3.268565186572e-003xi=0.72yi=1.40f(xi,yi)=-8.765306591329e-002xi=0.72yi=1.45f(xi,yi)=-1.724672478188e-001xi=0.72yi=1.50f(xi,yi)=-2.512302207523e-001xi=0.80yi=0.50f(xi,yi)=2.717811109469e+000xi=0.80yi=0.55f(xi,yi)=2.526399501256e+000xi=0.80yi=0.60f(xi,yi)=2.338411386860e+000xi=0.80yi=0.65f(xi,yi)=2.154329377280e+000xi=0.80yi=0.70f(xi,yi)=1.974574556652e+000xi=0.80yi=0.75f(xi,yi)=1.799510579099e+000xi=0.80yi=0.80f(xi,yi)=1.629448220554e+000xi=0.80yi=0.85f(xi,yi)=1.464650043751e+000xi=0.80yi=0.90f(xi,yi)=1.305334967651e+000xi=0.80yi=0.95f(xi,yi)=1.151682621307e+000xi=0.80yi=1.00f(xi,yi)=1.003837419906e+000xi=0.80yi=1.05f(xi,yi)=8.619123372279e-001xi=0.80yi=1.10f(xi,yi)=7.259923711112e-001xi=0.80yi=1.15f(xi,yi)=5.961377115201e-001xi=0.80yi=1.20f(xi,yi)=4.723866279136e-001xi=0.80yi=1.25f(xi,yi)=3.547580958979e-001xi=0.80yi=1.30f(xi,yi)=2.432541841813e-001xi=0.80yi=1.35f(xi,yi)=1.378622225247e-001xi=0.80yi=1.40f(xi,yi)=3.855677032640e-002xi=0.80yi=1.45f(xi,yi)=-5.469859593446e-002xi=0.80yi=1.50f(xi,yi)=-1.419496597088e-001二元擬合選擇過程的k，值。k=1delt=3.220908973633e+000k=2delt=4.659960033245e-003k=3delt=1.721175379304e-004k=4delt=3.309534300965e-006k=5delt=2.541377726903e-0083、達到精度要求的k，的值及px,y中的系數crsr,,,k;s,,,kk=5delt=2.541377726903e-008c00=2.021230518450e+000c01=-3.668426794957e+000c02=7.092486168036e-001c03=8.486054057126e-001c04=-4.158974374886e-001c05=6.743199534489e-002c10=3.191909002842e+000c11=-7.411103426492e-001c12=-2.697124675595e+000c13=1.631183480281e+000c14=-4.847200129551e-001c15=6.061428925366e-002c20=2.568898167761e-001c21=1.579918650153e+000c22=-4.634081038207e-001c23=-8.134442647695e-002c24=1.020942593968e-001c25=-2.101523522901e-002c30=-2.692603588795e-001c31=-7.302475370891e-001c32=1.076144819072e+000c33=-8.070125477467e-001c34=3.028726690098e-001c35=-4.597260945236e-002c40=2.174597893315e-001c41=-1.783725755708e-001c42=-7.240538548997e-002c43=2.433300406299e-001c44=-1.413345241120e-001c45=2.651019960902e-002c50=-5.590328377840e-002c51=1.431993440001e-001c52=-1.362705851344e-001c53=4.071984198418e-002c54=3.774922834692e-003c55=-2.667680026547e-003f(0.1,0.7)=1.94720407918e-001p(0.1,0.7)=1.94730357534e-001f(0.1,0.9)=-1.83037079189e-001p(0.1,0.9)=-1.83041838756e-001f(0.1,1.1)=-4.45497646915e-001p(0.1,1.1)=-4.45500041806e-001f(0.1,1.3)=-5.97566707641e-001p(0.1,1.3)=-5.97558856937e-001f(0.1,1.5)=-6.46459593901e-001p(0.1,1.5)=-6.46446110815e-001f(0.2,0.7)=4.05979189288e-001p(0.2,0.7)=4.05989539815e-001f(0.2,0.9)=-2.25159583746e-002p(0.2,0.9)=-2.25211161201e-002f(0.2,1.1)=-3.38220816040e-001p(0.2,1.1)=-3.38224022551e-001f(0.2,1.3)=-5.44437831522e-001p(0.2,1.3)=-5.44430450907e-001f(0.2,1.5)=-6.47361338568e-001p(0.2,1.5)=-6.47348010856e-001f(0.3,0.7)=6.34777195151e-001p(0.3,0.7)=6.34787452928e-001f(0.3,0.9)=1.58801168839e-001p(0.3,0.9)=1.58796295997e-001f(0.3,1.1)=-2.07365694171e-001p(0.3,1.1)=-2.07368580368e-001f(0.3,1.3)=-4.65357906898e-001p(0.3,1.3)=-4.65349923294e-001f(0.3,1.5)=-6.20270953075e-001p(0.3,1.5)=-6.20257138940e-001f(0.4,0.7)=8.78960023174e-001p(0.4,0.7)=8.78969865341e-001f(0.4,0.9)=3.58650625882e-001p(0.4,0.9)=3.58646043341e-001f(0.4,1.1)=-5.52528211681e-002p(0.4,1.1)=-5.52554368602e-002f(0.4,1.3)=-3.62679511503e-001p(0.4,1.3)=-3.62671062974e-001f(0.4,1.5)=-5.67564743655e-001p(0.4,1.5)=-5.67550582797e-001f(0.5,0.7)=1+000p(0.5,0.7)=1+000f(0.5,0.9)=5.74980340948e-001p(0.5,0.9)=5.74975842722e-001f(0.5,1.1)=1-001p(0.5,1.1)=1-001f(0.5,1.3)=-2.38568304012e-001p(0.5,1.3)=-2.38560419243e-001f(0.5,1.5)=-4.91434393656e-001p(0.5,1.5)=-4.91420900529e-001f(0.6,0.7)=1.40604179891e+000p(0.6,0.7)=1.40605068705e+000f(0.6,0.9)=8.05941494063e-001p(0.6,0.9)=8.05937301858e-001f(0.6,1.1)=3.04429221045e-001p(0.6,1.1)=3.04425832104e-001f(0.6,1.3)=-9.50161300996e-002p(0.6,1.3)=-9.50089457510e-002f(0.6,1.5)=-3.93902307746e-001p(0.6,1.5)=-3.93889837520e-001f(0.7,0.7)=1.68578351531e+000p(0.7,0.7)=1.68579121728e+000f(0.7,0.9)=1.04988115306e+000p(0.7,0.9)=1.04987773922e+000f(0.7,1.1)=5.08293783940e-001p(0.7,1.1)=5.08291044786e-001f(0.7,1.3)=6.61487967065e-002p(0.7,1.3)=6.61563555471e-002f(0.7,1.5)=-2.76834341778e-001p(0.7,1.5)=-2.76822043364e-001f(0.8,0.7)=1.97457455665e+000p(0.8,0.7)=1.97458126127e+000f(0.8,0.9)=1.30533496765e+000p(0.8,0.9)=1.30533200389e+000f(0.8,1.1)=7.25992371111e-001p(0.8,1.1)=7.25989310611e-001f(0.8,1.3)=2.43254184181e-001p(0.8,1.3)=2.43260790457e-001f(0.8,1.5)=-1.41949659709e-001p(0.8,1.5)=-1.41938788779e-001C程序代碼#include<iostream.h>#include<stdio.h>#include<math.h>doublefabsmax(doublex[4]);intfabsmax_flag(doublex[4],intk);voidgauss(doublea_0[4][4],doubleb[4],doubledeltx[4]);voidfunction(doublet_1[11][21],doubleu_1[11][21],doublex_0[4]);voidchazhi(doublet_1[11][21],doubleu_1[11][21],doublez[11][21],doublet_0[6],doubleu_0[6],doublez_0[6][6]);voidf(doublet_1,doubleu_1,intflag[2]);intnihe(doubleu[11][21],doublec[11][11],doubledelt[11]);voidgauss_1(doubleb[11][11],doubleu[11][21],doublea[11][21],intl);voidgauss_2(doubleg[21][21],doubled[21][21],intl);intfabsmax_flag_1(doublex[11],intk,intl);void f_p(doublf_xy[8][5],double g_xy[8][5],double x_0[4],doubleu_0[6],doublet_0[6],doublez_0[6][6],doublec[11][11]);/*-main函數：在給定區域上作二元擬合得到滿足精度要求的P(X,Y)，為確定擬合節點(x,y,z)需要解非線性方程組和作二元分片插值。*/main(){doubleu_0[6]={0,0.4,0.8,1.2,1.6,2};doublet_0[6]={0,0.2,0.4,0.6,0.8,1.0};doublez_0[6][6]={{-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}};inti,j,k;doubleu_1[11][21],t_1[11][21],z[11][21],x_0[4]={0.5,0.5,0.5,1};doubledelt[11],c[11][11];doublef_xy[8][5],g_xy[8][5];function(t_1,u_1,x_0);chazhi(t_1,u_1,z,t_0,u_0,z_0);k=nihe(z,c,delt);for(i=0;i<=10;i++)for(j=0;j<=20;j++){printf("\n");printf("\t");printf("xi=%3.2f yi=%3.2ff(xi,yi)=%14.12e",0.08*i,(0.5+0.05*j),z[i][j]);}printf("\n");for(i=1;i<=5;i++){printf("\t");printf("k=%d delt=%14.12e",i,delt[i-1]);printf("\n");}printf("\n");printf("\t");printf("k=5 delt=%14.12e",delt[4]);printf("\n");printf("\n");for(i=0;i<=k;i++)for(j=0;j<=k;j++){printf("\t");printf("c%d%d=%14.12e",i,j,c[i][j]);printf("\n");}printf("\n");f_p(f_xy,g_xy,x_0,u_0,t_0,z_0,c);printf("\n");for(i=1;i<=8;i++)for(j=1;j<=5;j++){printf("\n");printf("f(%2.1f,%2.1f)=%12.11ep(%2.1f,%2.1f)=%12.11e",0.1*i,(0.5+0.2*j),f_xy[i-1][j-1],0.1*i,(0.5+0.2*j),g_xy[i-1][j-1]);}printf("\n");printf("\n");}/*fabsmax函數：求無窮范數*/doublefabsmax(doublex[4]){doubleb=0;inti;b=fabs(x[0]);for(i=1;i<=3;i++){if(fabs(x[i])>b)b=fabs(x[i]);}returnb;}/*fabsmax_flag 函數：返回數組 x[4]中最大元素所在的位置*/intfabsmax_flag(doublex[4],intk){doubleb=0;inti,flag=k;b=fabs(x[k]);for(i=k+1;i<=3;i++)if(fabs(x[i])>b){flag=i;b=fabs(x[i]);}returnflag;}/*fabsmax_flag_1函數：比較數據絕對值的大小，并返回按模最大值所在的行號*/intfabsmax_flag_1(doublex[11],intk,intl){doubleb=0;inti,flag=k;b=fabs(x[k]);for(i=k+1;i<=l;i++)if(fabs(x[i])>b){flag=i;b=fabs(x[i]);}returnflag;}/*gauss 函 數 ： 列 主 元 素 Gauss 法 解 線 性 方 程 組*/voidgauss(doublea_0[4][4],doubleb[4],doubledeltx[4]){doublea[4][4],x[4],aa;inti,j,k,flag;for(i=0;i<=3;i++)for(j=0;j<=3;j++)a[i][j]=a_0[i][j];for(k=0;k<=2;k++){for(i=k;i<=3;i++)x[i]=a[i][k];flag=fabsmax_flag(x,k);if(flag!=k){for(j=k;j<=3;j++){aa=a[k][j];a[k][j]=a[flag][j];a[flag][j]=aa;}aa=b[k];b[k]=b[flag];b[flag]=aa;}for(i=k+1;i<=3;i++){aa=a[i][k]/a[k][k];for(j=k+1;j<=3;j++)a[i][j]-=aa*a[k][j];b[i]-=aa*b[k];}}deltx[3]=b[3]/a[3][3];for(k=2;k>=0;k--){for(j=k+1;j<=3;j++)b[k]-=a[k][j]*deltx[j];deltx[k]=b[k]/a[k][k];}}/*function 函 數 ： Newton 法 解 非 線 性 方 程 組*/voidfunction(doublet_1[11][21],doubleu_1[11][21],doublex_0[4]){doublea[4][4]={{0,1,1,1},{1,0,1,1},{0.5,1,0,1},{1,0.5,1,0}};doubleb[4],deltx[4],x[4];inti,j,k;for(i=0;i<=10;i++)for(j=0;j<=20;j++){for(k=0;k<=3;k++)x[k]=x_0[k];for(;;){a[0][0]=-0.5*sin(x[0]);a[1][1]=0.5*cos(x[1]);a[2][2]=-sin(x[2]);a[3][3]=cos(x[3]);b[0]=-(0.5*cos(x[0])+x[1]+x[2]+x[3]-0.08*i-2.67);b[1]=-(x[0]+0.5*sin(x[1])+x[2]+x[3]-(0.5+0.05*j)-1.07);b[2]=-(0.5*x[0]+x[1]+cos(x[2])+x[3]-0.08*i-3.74);b[3]=-(x[0]+0.5*x[1]+x[2]+sin(x[3])-(0.5+0.05*j)-0.79);gauss(a,b,deltx);for(k=0;k<=3;k++)x[k]+=deltx[k];if((fabsmax(deltx)/fabsmax(x))<=1e-12){t_1[i][j]=x[0];u_1[i][j]=x[1];break;}}}}/*f函數：作二元分片插值時用到的函數，用于確定給定的（t，u）所在的插值區域*/voidf(doublet_1,doubleu_1,intflag[2]){intflag1,flag2;flag1=(int)(t_1/0.1);flag2=(int)(u_1/0.2);if(flag1>=0&&flag1<3)flag1=0;elseif(flag1>=3&&flag1<5)flag1=1;elseif(flag1>=5&&flag1<7)flag1=2;elseflag1=3;if(flag2>=0&&flag2<3)flag2=0;elseif(flag2>=3&&flag2<5)flag2=1;elseif(flag2>=5&&flag2<7)flag2=2;elseflag2=3;flag[0]=flag1;flag[1]=flag2;}/*chazhi 函 數 ： 分 片 二 元 插 值*/voidchazhi(doublet_1[11][21],doubleu_1[11][21],doublez[11][21],doublet_0[6],doubleu_0[6],doublez_0[6][6]){doublel_t[3],l_u[3];inti,j,k,l,flag[2];for(i=0;i<=10;i++)for(j=0;j<=20;j++){z[i][j]=0;f(t_1[i][j],u_1[i][j],flag);l_t[0]=(t_1[i][j]-t_0[flag[0]+1])*(t_1[i][j]-t_0[flag[0]+2])/0.08;l_t[1]=(t_1[i][j]-t_0[flag[0]])*(t_1[i][j]-t_0[flag[0]+2])/(-0.04);l_t[2]=(t_1[i][j]-t_0[flag[0]])*(t_1[i][j]-t_0[flag[0]+1])/0.08;l_u[0]=(u_1[i][j]-u_0[flag[1]+1])*(u_1[i][j]-u_0[flag[1]+2])/0.32;l_u[1]=(u_1[i][j]-u_0[flag[1]])*(u_1[i][j]-u_0[flag[1]+2])/(-0.16);l_u[2]=(u_1[i][j]-u_0[flag[1]])*(u_1[i][j]-u_0[flag[1]+1])/0.32;for(k=0;k<=2;k++)for(l=0;l<=2;l++)z[i][j]+=z_0[flag[0]+k][flag[1]+l]*l_t[k]*l_u[l];}}/*gauss_1函數：列主元素Gauss法解線性方程組*/voidgauss_1(doubleb[11][11],doubleu[11][21],doublea[11][21],intl){doubleb_1[11][11],b_2[11][11],b_3[11][21],x[11],aa;inti,j,k,flag;for(i=0;i<=10;i++)for(j=0;j<=l;j++)b_2[j][i]=b[i][j];for(i=0;i<=l;i++)for(j=0;j<=20;j++){b_3[i][j]=0;for(k=0;k<=10;k++)b_3[i][j]+=b_2[i][k]*u[k][j];}for(i=0;i<=l;i++)for(j=0;j<=l;j++){b_1[i][j]=0;for(k=0;k<=10;k++)b_1[i][j]+=b_2[i][k]*b[k][j];}for(k=0;k<l;k++){for(i=k;i<=l;i++)x[i]=b_1[i][k];flag=fabsmax_flag_1(x,k,l);if(flag!=k){for(j=k;j<=l;j++){aa=b_1[k][j];b_1[k][j]=b_1[flag][j];b_1[flag][j]=aa;}for(j=0;j<=20;j++){aa=b_3[k][j];b_3[k][j]=b_3[flag][j];b_3[flag][j]=aa;}}for(i=k+1;i<=l;i++){aa=b_1[i][k]/b_1[k][k];for(j=k+1;j<=l;j++)b_1[i][j]-=aa*b_1[k][j];for(j=0;j<=20;j++)b_3[i][j]-=aa*b_3[k][j];}}for(i=0;i<=20;i++){a[l][i]=b_3[l][i]/b_1[l][l];for(k=l-1;k>=0;k--){for(j=k+1;j<=l;j++)b_3[k][i]-=b_1[k][j]*a[j][i];a[k][i]=b_3[k][i]/b_1[k][k];}}}/*gauss_2函數：列主元素Gauss法解線性方程組*/voidgauss_2(doubleg[21][21],doubled[21][21],intl){doubleg_1[21][21],g_2[21][21],d_0[21][21],x[11],aa;inti,j,k,flag;for(i=0;i<=20;i++)for(j=0;j<=l;j++)g_2[j][i]=g[i][j];for(i=0;i<=l;i++)for(j=0;j<=l;j++){g_1[i][j]=0;for(k=0;k<=20;k++)g_1[i][j]+=g_2[i][k]*g[k][j];}for(k=0;k<l;k++){for(i=k;i<=l;i++)x[i]=g_1[i][k];flag=fabsmax_flag_1(x,k,l);if(flag!=k){for(j=k;j<=l;j++){aa=g_1[k][j];g_1[k][j]=g_1[flag][j];g_1[flag][j]=aa;}for(j=0;j<=20;j++){aa=g_2[k][j];g_2[k][j]=g_2[flag][j];g_2[flag][j]=aa;}}for(i=k+1;i<=l;i++){aa=g_1[i][k]/g_1[k][k];for(j=k+1;j<=l;j++)g_1[i][j]-=aa*g_1[k][j];for(j=0;j<=20;j++)g_2[i][j]-=aa*g_2[k][j];}}for(i=0;i<=20;i++){d_0[l][i]=g_2[l][i]/g_1[l][l];for(k=l-1;k>=0;k--){for(j=k+1;j<=l;j++)g_2[k][i]-=g_1[k][j]*d_0[j][i];d_0[k][i]=g_2[k][i]/g_1[k][k];}}for(i=0;i<=l;i++)for(j=0;j<=20;j++)d[j][i]=d_0[i][j];}/*nihe 函 數 ： 二 元 擬 合*/intnihe(doubleu[11][21],doublec[11][11],doubledelt[11]){doublea[11