# XLPack Solver – System of Nonlinear Equations

Solve the system of nonlinear equations (find the real solution).

Let’s solve the following example.

f1(x1,x2) = 4*x1^2 + x2^2 – 16
f2(x1,x2) = x1^2 + x2^2 – 9

The solver program will write the value of x1 and x2 into the variable value cells (B8 and B9 in this case). The function value cells (D8 and D9 in this case) must contain the formula to compute f1(x1,x2) and f2(x1,x2) from x1 and x2. To solve this example, the formula for D8 is =4*B8^2+B9^2-16 and that for D9 is =B8^2+B9^2-9.

Jacobian cells contain the formulas of Jacobian matrix. These are necessary only when Hybrj1 is selected as the solver program.

Initial value cells contain the initial values of x1 and x2. If these are not specified (range remains blank), initial values are assumed to be contained in the variable value cells.

The obtained solutions are output to the output cell range. If it is not specified (range remains blank), the solutions will be output to the variable value cells.

Hybrd1, Hybrj1 or Sos can be selected as the solver program.

The standard value of tolerance is 1.0e-8. The tolerance value will be set to XTol of Hybrd1 and Hybrj1, and RTolx and ATolx of Sos.

Click “Compute”. Then the solutions will be computed and output.

The cell ranges can be specified as larger than required. In that case, only the necessary range will be used from left upper corner.

The system of nonlinear equations may have several solutions. The different solution may be obtained depending on the initial values. The different solution may be obtained if the solver program is different even if the initial values are same. It is therefore recommended to give the initial values nearest to the desired solution.

If the solution is not obtained, #NUM! error will be output. The major suspected causes in that case are as follows: The algorithm was not converged due to the bad initial values; The given system of equations has no real solution. Please refer to here for “Save/Restore” button.

When “Help” button is clicked, this page will be displayed if the network connection is available.

The “?” button of right upper corner will not work correctly. Please use “Help” button.

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