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If all you need is the sparse array of the Jacobian matrix of $f$ evaluated at a vector, you can do the following:

Clear[x];
xVariables = Map[x, Range[10000]];
Evaluate[xVariables] = RandomInteger[{1, 20}, 10000];
sparse = SparseArray[{{10000, 10000} -> x[1], {10000, 1} -> x[10000], {i_,i_} -> x[i + 1], ({i_, j_} /; j - i == 1) -> x[i]}, {10000, 10000}]

Here I used x[i] as dummy variable following J.M's suggestion, and xVariables is set to {x[1],x[2],...,x[10000]}. The next line sets x[i] to the i-th element of the random vector. The last line is where I use the fact that the Jacobian matrix of $f$ has a really nice pattern. You can check that sparse does have the correct values, say by evaluating sparse[[1,2]] and sparse[[1,3]].

Here is a a function which produces the Jacobian matrix at a given vector:

sparse[v_] := (
  Clear[x];
  xVariables = Map[x, Range[10000]];
  Evaluate[xVariables] = v;
  SparseArray[{{10000, 10000} -> x[1], {10000, 1} -> x[10000], {i_, i_} -> x[i + 1], ({i_, j_} /; j - i == 1) -> x[i]}, {10000, 10000}]
 )

Now you should be able to use it in a loop:

In[2]:= sparseTest1=sparse[RandomInteger[{1,20},10000]];
        sparseTest2=sparse[RandomInteger[{1,20},10000]];

In[3]:= sparseTest1[[1,2]]
Out[3]= 11

In[4]:= sparseTest2[[1,2]]
Out[4]= 8

From what I understand, you want to compute the Jacobian matrix of a function $f$ at $(x_1,x_2,\ldots,x_n)$ and evaluate it at a given point, say $(2,2,\dots,2)$.

This seems to be the function you are interested in:

f[{X__}] := {X} RotateLeft[{X}, 1] - 1;

In[2]:= f[{x1,x2,x3}]
Out[2]= {-1+x1 x2,-1+x2 x3,-1+x1 x3}

Since the dimension may change, we want to use SetDelay or := so that the Jacobian function recomputes the matrix every time it is called.

J[{X__}] := D[f[{X}], {{X}}]

In[4]:= J[{x1,x2,x3}]
Out[4]= {{x2,x1,0},{0,x3,x2},{x3,0,x1}}

If you want the Jacobian matrix of $f$ at the point $(2,3,4)$, you can do the substitution

In[5]:= J[{x1,x2,x3}] /. Thread[{x1,x2,x3}->{2,3,4}]
Out[5]= {{3,2,0},{0,4,3},{4,0,2}}

I would also recommend that you avoid using subscripts. Here is one way to generate a vector of variables such as {x1,x2,x3,x4,x5} programatically:

In[6]:= ToExpression[Table["x"~~ToString[k],{k,1,5}]]
Out[6]= {x1,x2,x3,x4,x5}

variableVector[n_] := ToExpression[Table["x" ~~ ToString[k], {k, 1, n}]];

Now we can write a function which computes the Jacobian at a given point as follows:

Jvalue[{Y__}] := With[
  {X=variableVector[Length[{Y}]]},
  J[X] /. Thread[X->{Y}]
 ]

In[9]:= Jvalue[{2,3,4}]
Out[9]= {{3,2,0},{0,4,3},{4,0,2}}

From what I understand, you want to compute the Jacobian matrix of a function $f$ at $(x_1,x_2,\ldots,x_n)$ and evaluate it at a given point, say $(2,2,\dots,2)$.

This seems to be the function you are interested in:

f[{X__}] := {X} RotateLeft[{X}, 1] - 1;

In[2]:= f[{x1,x2,x3}]
Out[2]= {-1+x1 x2,-1+x2 x3,-1+x1 x3}

Since the dimension may change, we want to use SetDelay or := so that the Jacobian function recomputes the matrix every time it is called.

J[{X__}] := D[f[{X}], {{X}}]

In[4]:= J[{x1,x2,x3}]
Out[4]= {{x2,x1,0},{0,x3,x2},{x3,0,x1}}

If you want the Jacobian matrix of $f$ at the point $(2,3,4)$, you can do the substitution

In[5]:= J[{x1,x2,x3}] /. Thread[{x1,x2,x3}->{2,3,4}]
Out[5]= {{3,2,0},{0,4,3},{4,0,2}}

I would also recommend that you avoid using subscripts. Here is one way to generate a vector of variables such as {x1,x2,x3,x4,x5} programatically:

In[6]:= ToExpression[Table["x"~~ToString[k],{k,1,5}]]
Out[6]= {x1,x2,x3,x4,x5}

variableVector[n_] := ToExpression[Table["x" ~~ ToString[k], {k, 1, n}]];

Now we can write a function which computes the Jacobian at a given point as follows:

Jvalue[{Y__}] := With[
  {X=variableVector[Length[{Y}]]},
  J[X] /. Thread[X->{Y}]
 ]

In[9]:= Jvalue[{2,3,4}]
Out[9]= {{3,2,0},{0,4,3},{4,0,2}}

Update

If all you need is the sparse array of the Jacobian matrix of $f$ evaluated at a vector, you can do the following:

Clear[x];
xVariables = Map[x, Range[10000]];
Evaluate[xVariables] = RandomInteger[{1, 20}, 10000];
sparse = SparseArray[{{10000, 10000} -> x[1], {10000, 1} -> x[10000], {i_,i_} -> x[i + 1], ({i_, j_} /; j - i == 1) -> x[i]}, {10000, 10000}]

Here I used x[i] as dummy variable following J.M's suggestion, and xVariables is set to {x[1],x[2],...,x[10000]}. The next line sets x[i] to the i-th element of the random vector. The last line is where I use the fact that the Jacobian matrix of $f$ has a really nice pattern. You can check that sparse does have the correct values, say by evaluating sparse[[1,2]] and sparse[[1,3]].

From what I understand, you want to compute the Jacobian matrix of a function $f$ at $(x_1,x_2,\ldots,x_n)$ and evaluate it at a given point, say $(2,2,\dots,2)$.

This seems to be the function you are interested in:

f[{X__}] := {X} RotateLeft[{X}, 1] - 1;

In[2]:= f[{x1,x2,x3}]
Out[2]= {-1+x1 x2,-1+x2 x3,-1+x1 x3}

Since the dimension may change, we want to use SetDelay or := so that the Jacobian function recomputes the matrix every time it is called.

J[{X__}] := D[f[{X}], {{X}}]

In[4]:= J[{x1,x2,x3}]
Out[4]= {{x2,x1,0},{0,x3,x2},{x3,0,x1}}

If you want the Jacobian matrix of $f$ at the point $(2,3,4)$, you can do the substitution

In[5]:= J[{x1,x2,x3}] /. Thread[{x1,x2,x3}->{2,3,4}]
Out[5]= {{3,2,0},{0,4,3},{4,0,2}}

I would also recommend that you avoid using subscripts. Here is one way to generate a vector of variables such as {x1,x2,x3,x4,x5} programatically:

In[6]:= ToExpression[Table["x"~~ToString[k],{k,1,5}]]
Out[6]= {x1,x2,x3,x4,x5}

Now we can write a function which computes the Jacobian at a given point as follows:

Jvalue[{Y__}]:=With[
  {X=variableVector[Length[{Y}]]},
  J[X] /. Thread[X->{Y}]
 ]

In[8]:= Jvalue[{2,3,4}]
Out[8]= {{3,2,0},{0,4,3},{4,0,2}}

From what I understand, you want to compute the Jacobian matrix of a function $f$ at $(x_1,x_2,\ldots,x_n)$ and evaluate it at a given point, say $(2,2,\dots,2)$.

This seems to be the function you are interested in:

f[{X__}] := {X} RotateLeft[{X}, 1] - 1;

In[2]:= f[{x1,x2,x3}]
Out[2]= {-1+x1 x2,-1+x2 x3,-1+x1 x3}

Since the dimension may change, we want to use SetDelay or := so that the Jacobian function recomputes the matrix every time it is called.

J[{X__}] := D[f[{X}], {{X}}]

In[4]:= J[{x1,x2,x3}]
Out[4]= {{x2,x1,0},{0,x3,x2},{x3,0,x1}}

If you want the Jacobian matrix of $f$ at the point $(2,3,4)$, you can do the substitution

In[5]:= J[{x1,x2,x3}] /. Thread[{x1,x2,x3}->{2,3,4}]
Out[5]= {{3,2,0},{0,4,3},{4,0,2}}

I would also recommend that you avoid using subscripts. Here is one way to generate a vector of variables such as {x1,x2,x3,x4,x5} programatically:

In[6]:= ToExpression[Table["x"~~ToString[k],{k,1,5}]]
Out[6]= {x1,x2,x3,x4,x5} 

variableVector[n_] := ToExpression[Table["x" ~~ ToString[k], {k, 1, n}]];

Now we can write a function which computes the Jacobian at a given point as follows:

Jvalue[{Y__}] := With[
  {X=variableVector[Length[{Y}]]},
  J[X] /. Thread[X->{Y}]
 ]

In[9]:= Jvalue[{2,3,4}]
Out[9]= {{3,2,0},{0,4,3},{4,0,2}}