求方程2的X次方加上3X=7的近似解

Question

求方程2的X次方加上3X=7的近似解

Answer 1

f(x)=2^x+3x-7
f(1)=-2
f(2)=3
f(x)在实数域严格单增,故方程f(x)=0有且仅有一个实根,且该根在区间(1,2)上.
利用迭代法：
f(x)=0解得
x=(7-2^x)/3
则迭代式
x(n+1)=[7-2^x(n)]/3
取初值1.5,利用EXCELL表下拉计算得：
1.390524292
1.459420199
1.416673923
1.443435621
1.426774145
1.437183641
1.430694257
1.434745299
1.432218546
1.433795388
1.43281167
1.433425492
1.433042528
1.433281479
1.433132392
1.433225413
1.433167375
1.433203587
1.433180993
1.43319509
1.433186295
1.433191782
1.433188358
1.433190495
1.433189162
1.433189994
1.433189475
1.433189798
1.433189596
1.433189722
1.433189644
1.433189693
1.433189662
1.433189681
1.433189669
1.433189677
1.433189672
1.433189675
1.433189673
1.433189674
1.433189674
1.433189674
1.433189674
1.433189674
1.433189674
可见,近似解为x=1.433189674
精度可达小数点后8位.