a[x_]:=Normal[SparseArray[{
{i_,i_}->x^(i-1),
{i_,j_}/;j==i+1->x^(i-1)
},{10,10}]]
a[x]//MatrixForm
ar[x_]:=Normal[SparseArray[{
{i_,i_}->x^(10-i),
{i_,j_}/;j==i+1->x^(9-i)
},{10,10}]];
ar[x]//MatrixForm
b[x_]:=Normal[SparseArray[{
{i_,i_}->x^Abs[10-i],
{i_,j_}/;j==i+1->x^Abs[10-i]
},{20,20}]];
b[x]//MatrixForm

{U,S,V}=SingularValueDecomposition[N[a[10^10],100]];
Diagonal[S]

Chop[U.S.V\[Transpose]]//MatrixForm

{U,S,V}=SingularValueDecomposition[N[ar[10^10],100]];
Diagonal[S]

Chop[U.S.V\[Transpose]]//MatrixForm

{U,S,V}=SingularValueDecomposition[N[b[10^10],100]];
Diagonal[S]
N[%,18]//CForm