A summary of the subroutines and functions in this collection that are based on the originals in the book Computer Methods for Mathematical Computations by Forsythe, Malcolm and Moler.

The original procedure names and lists of dummy arguments have been modified to be more in line with modern Fortran style. Click on a procedure name to see a list of the dummy arguments.

name | description |
---|---|

Decomp | LU-decomposition of a square matrix |

Solve | Solves a system of linear equations. Use after Decomp. |

FMMspline | Fit a cubic spline to data. FMM end conditions. |

NaturalSpline | Same as FMMspline, but with zero second derivatives at endpoints |

Seval | Evaluate a cubic spline at a given point. |

Seval3 | Evaluate a cubic spline at a given point. Returns value of spline plus 1st,2nd,3rd derivatives. |

Quanc8 | Numerical integration of a function. |

Rkf45 | Solves a system of ordinary differential equations as an initial value problem. |

Zeroin | Find a zero of a function. |

BrentZero | Same as Zeroin, but with additional dummy arguments |

Fmin | Find the minimum of a function. |

BrentMin | Same as Fmin, but with additional dummy arguments. |

SVD | Singular Value Decomposition of a matrix |

All of the vector and matrix arguments are now assumed-shape arrays, a feature introduced with Fortran 90. With this feature, a procedure can determine the size of the input arrays without requiring separate arguments for the size and shape of the array. If you use the technique of dimensioning for the largest imaginable size, then you must be careful to call the procedure with the appropriate size. For example, if you define the matrix a as a(100,100) and you wish to call Decomp for a matrix of order n stored in a, you would use the following function call:

CALL Decomp(a(1:n,1:n), ipvt(1:n), errCode, cond)

CALL Decomp(a, ipvt, errCode, cond) | |||

var | intent | dim | def |
---|---|---|---|

a | in out | : , : | matrix to be decomposed |

ipvt | out | : | index of pivot rows |

errCode | out | - | error code |

cond | out | - | condition number |

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CALL Solve(a, b, ipvt) | |||

var | intent | dim | def |
---|---|---|---|

a | in | : , : | decomposed matrix (from Decomp) |

b | in out | : | right-hand side; replaced with solution |

ipvt | in | : | record of row interchanges (from Decomp) |

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CALL FMMspline(x, y, b, c, d) | |||

var | intent | dim | def |
---|---|---|---|

x | in | : | abscissas of knots |

y | in | : | ordinates of knots |

b | out | : | linear coefficients |

c | out | : | quadratic coefficients |

d | out | : | cubic coefficients |

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CALL NaturalSpline(x, y, b, c, d) | |||

var | intent | dim | def |
---|---|---|---|

x | in | : | abscissas of knots |

y | in | : | ordinates of knots |

b | out | : | linear coefficients |

c | out | : | quadratic coefficients |

d | out | : | cubic coefficients |

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Seval(u, x, y, b, c, d) | |||

var | intent | dim | def |
---|---|---|---|

u | in | - | abscissa where spline is to be evaluated |

x | in | : | abscissas of knots |

y | in | : | ordinates of knots |

b | in | : | linear coefficients |

c | in | : | quadratic coefficients |

d | in | : | cubic coefficients |

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CALL Seval3(u, x,y, b, c, d, f, fp, fpp, fppp) | |||

var | intent | dim | def |
---|---|---|---|

u | in | - | abscissa where spline is to be evaluated |

x | in | : | abscissas of knots |

y | in | : | ordinates of knots |

b | in | : | linear coefficients |

c | in | : | quadratic coefficients |

d | in | : | cubic coefficients |

f | out | - | value of spline at u |

fp | out | - | value of 1st derivative of spline at u |

fpp | out | - | value of 2nd derivative of spline at u |

fppp | out | - | value of 3rd derivative of spline at u |

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CALL Quanc8(F, a, b, abserr, relerr, result, errest, nofun, flag) | |||

var | intent | dim | def |
---|---|---|---|

F | - | - | function to be integrated |

a | in | - | lower limit of integration |

b | in | - | upper limit of integration |

abserr | in | - | absolute error tolerance |

relerr | in | - | relative error tolerance |

result | out | - | approximate value of the integral |

errest | out | - | estimate of actual error |

nofun | out | - | number of function evaluations |

flag | out | - | reliability indicator |

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CALL Rkf45 (F, y, t, tout, relerr, abserr, iflag, work, iwork) | |||

var | intent | dim | def |
---|---|---|---|

F | - | - | subroutine that computes derivatives |

y | in out | : | solution vector at t |

t | in out | - | independent variable |

tout | in out | - | output point at which solution is desired |

relerr | in out | - | relative error tolerance |

abserr | in | - | absolute error tolerance |

iflag | in out | - | indicator for status of work |

work | in out | : | work array |

iwork | in out | : | work array |

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Zeroin(ax, bx, F, tol) | |||

var | intent | dim | def |
---|---|---|---|

ax | in | - | lower endpoint of interval |

bx | in | - | upper endpoint of interval |

F | in | - | function to be investigated |

tol | in | - | desired interval of uncertainity |

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CALL BrentZero(ax, bx, F, tol, maxIter, neval, xZero, fZero) | |||

var | intent | dim | def |
---|---|---|---|

ax | in | - | left-hand limit on x-coor |

bx | in | - | right-hand limit on x-coor |

F | in | - | the function to be investigated |

tol | in | - | user-specified tolerance |

maxIter | in | - | user specified limit on the number of iterations |

neval | out | - | number of function evaluations required to find the zero |

xZero | out | - | x-coor of the zero |

fZero | out | - | last evaluation of the function. Should be very small.) |

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Fmin(ax, bx, F, tol) | |||

var | intent | dim | def |
---|---|---|---|

ax | in | - | lower endpoint of initial interval |

bx | in | - | upper endpoint of initial interval |

F | in | - | function to be investigated |

tol | in | - | desired interval of uncertainity |

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CALL BrentMin(ax, bx, F, tol, maxIter, neval, errCode, xZero, fZero) | |||

var | intent | dim | def |
---|---|---|---|

ax | in | - | lower endpoint of initial interval |

bx | in | - | upper endpoint of initial interval |

F | in | - | function to be investigated |

tol | in | - | desired interval of uncertainity |

maxIter | in | - | maximum number of iterations allowed |

neval | out | - | number of function evaluations |

errCode | out | - | errorCode; =0 OK; =1 too many iter |

xZero | out | - | x-coor of the minimum point |

fZero | out | - | f(xZero) |

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CALL SVD(a, w, matu, u, matv, v, ierr) | |||

var | intent | dim | def |
---|---|---|---|

a | in | : , : | matrix to be decomposed. On output, a is unaltered (unless overwritten by u or v). |

w | out | : | w contains the n (non-negative) singular values of a (the diagonal elements of s). They are unordered. If an error exit is made, the singular values should be correct for indices ierr+1,ierr+2,...,n. |

matu | in | - | matu should be set to .TRUE. if the u matrix in the decomposition is desired, and to .FALSE. otherwise. |

u | out | : , : | u contains the matrix u of orthogonal column vectors of the decomposition if matu has been set to .TRUE. Otherwise, u is used as a temporary array. u may coincide with a. If an error exit is made, the columns of u corresponding to indices of correct singular values should be correct. |

matv | in | - | matv should be set to .TRUE. if the v matrix in the decomposition is desired, and to .FALSE. otherwise. |

v | out | : , : | v contains the matrix v (orthogonal) of the decomposition if matv has been set to .TRUE. Otherwise v is not referenced. v may also coincide with a if u is not needed. If an error exit is made, the columns of v corresponding to indices of correct singular values should be correct. |

ierr | out | - | zero for normal return, k if the k-th singular value has not been determined after 30 iterations. |