# Tutorial

## 2. Systems of linear equations

Examples of solving systems of linear equations using worksheet function WDgesv and VBA subroutine Dgesv.

## 3. Eigenvalue problems

Examples of solving eigenvalue problems of symmetric matrices using worksheet function WDsyev and VBA subroutine Dsyev.

## 4. Linear least squares problems

Examples of solving linear least squares problems using worksheet function WDgels and VBA subroutine Dgels.

## 5. Interpolation

Examples of computing interpolation by cubic spline function using worksheet functions WPchse and Wpchfe and VBA subroutines Pchse and Pchfe.

## 6. Algebraic equations (roots of polynomials)

Examples of solving algebraic equations (finding roots of polynomials) using worksheet function WRpzero2 and VBA subroutine Rpzero2.

## 7. Solution of nonlinear equation

Examples of solving a single variable nonlinear equation (finding a root of arbitrary one-dimensional nonlinear function) using VBA subroutine Dfzero. The example using the reverse communication routine Dfzero_r is also described.

## 8. System of nonlinear equations

Examples of obtaining real solutions of a system of general nonlinear equations using VBA subroutine Hybrd1. The example using the reverse communication routine Hybrd1_r is also described.

## 9. Nonlinear optimization of univariate function

Examples of solving a nonlinear optimization of univariate function (finding a minimum point of general one-dimensional nonlinear function) using VBA subroutine Dfmin. The example using the reverse communication routine Dfmin_r is also described.

## 10. Nonlinear optimization of multivariate function

Examples of solving a nonlinear optimization of multivariate function (finding a minimum point of general multidimensional nonlinear function) using VBA subroutine Optif0. The example using the reverse communication routine Optif0_r is also described.

## 11. Numerical integration

There are two types of quadrature routines. Function input type computes an integral using user defined function which can compute a function value at any points within integral range. Data input type computes an integral using given data at some points including integral range.

Example of the former using VBA subroutines Qag and Qk15 is shown. For the latter, example using worksheet functions WPchse and Wpchia is shown. The example using the reverse communication routine Qag_r and Qk15_r is also shown.

## 12. Ordinary Differential Equations

Example of solving an initial value problem of ordinary differential equations using VBA subroutine Derkf. The example using the reverse communication routine Derkf_r is also described.

## 13. Fast Fourier transform (FFT)

Example of fast Fourier transform (FFT) computations using worksheet functions WRfft1f and WRfft1b is described. Example of power spectrum analysis using FFT is also described.

## 14. Nonlinear least squares problems

Example of solving nonlinear least squares problem using VBA subroutine Lmdif1 is described. The example using the reverse communication routine Lmdif1_r is also described.