Numerical Methods

• Note 1
  • Hello World
    Output: "Hello World"
  • Sums
    Output:
    S_up = S_down = 14.39273 (single precision)
    S_up = S_down = 14.3927267228657 (double precision)
    The apparent lack of difference between S_up and S_down is due to FORTRAN 90's urge to turn backward loops around. The backward loop should be less precise because large numbers are added to small numbers which results in inaccuracy on the last digits.

• Note 2
  • OPEN24
    
  • DUMBINT
    Output: The value of the integral is 3.041675. Dividing into 259 intervals gives a relative error of 9.97E-05, just below the limit (1E-04). The external function is then called 2072 times.

• Note 3
  • ADAPT
    Output: To obtain the same value of the integral as DUMBINT (J=3.041675) with the required absolute accuracy (0.0001), we need a relative accuracy of 0.00020. Using these accuracies, the external function is called 288 times - a little more than a tenth the number of calls used by DUMBINT.

• Note 4
  • Main program
    
  • QR decomposition by the Gram-Schmidt method
    
  • QR back-substitution
    
  • Determinant of a matrix
    
  • Output
    

• Note 5
  • Linear fit - main program
    
  • Inverse of a matrix
    
  • Output
    

• Note 6
  • Spring constant - Main program
    
  • Spring constant - Output
    
  • QR algorithm - Main program
    
  • QR algorithm with shifts and deflation
    
  • QR algorithm - Output
    

• Note 7
  • Main program
    
  • Lanczo's tridiagonalisation
    
  • Output
    

• Note 8
  • Runge-Kutta - Main program
    
  • Runge-Kutta subroutine
    
  • Runge-Kutta step-size routine
    
  • k1 subroutine
    
  • k1/2 subroutine
    
  • Runge-Kutta output
    
  • Predictor-Corrector - Main program
    
  • Predictor-Corrector subroutine
    
  • Predictor-Corrector step-size routine
    
  • Predictor-Corrector output
    
  • Plots and results
    

• Note 9
  • Runge-Kutta - Main program - programs and routines now for systems of differential equations.
    
  • Runge-Kutta subroutine
    
  • Runge-Kutta step-size routine
    
  • k1 subroutine
    
  • k1/2 subroutine
    
  • Runge-Kutta plot
    
  • Predictor-Corrector - Main program
    
  • Predictor-Corrector subroutine
    
  • Predictor-Corrector step-size routine
    
  • Predictor-Corrector output
    
  • Predictor-Corrector plot
    

    Corrected Runge-Kutta

  • New RK main
    
  • RK routine
    
  • RKSTEP routine
    
  • New Runge-Kutta plot
    
• Note 10
  • Modified Newton's method - main program
    
  • Newton subroutine
    
  • QR decomposition routine (from Note 4)
    
  • QR back-substitution routine (from Note 4)
    
  • Newton test function
    
  • Newton output
    
  • Downhill Simplex method - main program
    
  • Simplex subroutine
    
  • Sorting routine for finding the highest and lowest point of the Simplex
    
  • Simplex test function
    
  • Simplex output
    

• Note 11
  • Monte Carlo - main program
    
  • Plain Monte Carlo routine (random sampling)
    
  • Random point routine
    
  • Test function
    
  • Monte Carlo output
    

• Note 12
  • Monte Carlo - main program
    
  • Quasi-Random Monte Carlo routine
    
  • "Quasi-Random" point routine (not at all random!)
    
  • Quasi-Random Monte Carlo output
    The program uses the same test function as the plain Monte Carlo program.

• Note 13
  • Interpolation - Main program
    
  • Locate routine
    
  • Lagrange interpolation routine
    
  • Spline interpolation routine
    
  • Interpolation output
    
  • Plot of tabulated and interpolated points
    

Module with nice home made subroutines: My stuff


Project: QR decomposition