ClearAll["Global`*"]
x=1;
FindRoot[((Max[{Exp[x],1}])/((n+1)!)) (x^(n+1))==(10^(-16)),{n,10}]
N[Sum[((x^k)/(k!)),{k,0,Ceiling[n/.%]}],30]
N[Exp[x],30]

ClearAll["Global`*"]
f[x_]:=(x^3)-2(x^2)+7x+4;
x0=-1;
x1=1;
a={{x0,f[x0]},{x1,f[x1]}};
For[i=1,i<100,len=Length[a];m=(a[[len,2]]*a[[len-1,1]]-a[[len-1,2]]*a[[len,1]])/(a[[len,2]]-a[[len-1,2]]);
n=f[m];
a=Join[a,{{m,n}}];
If[Abs[n]<(10^(-16)),Break[]]
]
N[a,10]

ClearAll["Global`*"]
f[x_]:=Piecewise[{{Sin[x],-100<=x<=-20},{(x^2),-20<x<20},{Cos[x],20<=x<=100}}]
a=RandomReal[{-100,100},30];
ListPlot[Table[{a[[k]],f[a[[k]]]},{k,1,30}]]

ClearAll["Global`*"]
A=RandomReal[{-10,10},{4,4}];
A//MatrixForm
Eigensystem[A]//TableForm

ClearAll["Global`*"]
A=RandomReal[{-10,10},{4,4}];
A//MatrixForm
Sqrt[Max[Abs[Eigenvalues[Transpose[A].A]]]]

ClearAll["Global`*"]
f[n_]:=Module[{A},A=ConstantArray["*",{n,n}];
For[i=2,i<=n-1,i++,
A[[2,i]]=0;
A[[i,2]]=0;
A[[n-1,i]]=0;
A[[i,n-1]]=0;
];A]
f[5]//MatrixForm
f[6]//MatrixForm

BeginPackage["Global`"];

Norm1::usage="计算矩阵的1范数";

NormInf::usage="计算矩阵的∞范数";

Norm1[x_]:=Module[{n},n=Dimensions[x][[2]];
Max[Table[Total[Abs[x[[All,j]]]],{j,1,n}]]];

NormInf[x_]:=Module[{m},m=Dimensions[x][[1]];
Max[Table[Total[Abs[x[[i,All]]]],{i,1,m}]]];

EndPackage[];

Clear[A];
A=RandomReal[{-10,10},{4,4}];
A//MatrixForm
{Norm1[A],NormInf[A]}