DIVDIF - Divided Differences

DIVDIF contains routines to create, print and manipulate divided difference polynomials. Divided difference polynomials are a systematic method of computing polynomial approximations to scattered data. The representations are compact, and may easily be updated with new data, rebased at zero, or analyzed to produce the standard form polynomial, integral or derivative polynomials.

Other routines are available to convert the divided difference representation to standard polynomial format. This is a natural way to determine the coefficients of the polynomial that interpolates a given set of data, for instance.

One surprisingly simple but useful routine is available to take a set of roots and compute the divided difference or standard form polynomial that passes through those roots.

Finally, the Newton-Cotes quadrature formulas can be derived using divided difference methods, so a few routines are given which can compute the weights and abscissas of open or closed rules for an arbitrary number of nodes.

Files you may copy include:

• divdif.f90, the source code;
• divdif_prb.f90, the calling program;
• divdif_prb.out, the sample output.

The list of routines includes:

• DATA_TO_DIF sets up a divided difference table from raw data.
• DATA_TO_DIF_DISPLAY computes a divided difference table and shows how.
• DATA_TO_RPOLY computes the coefficients of a polynomial interpolating data.
• DIF_ANTIDERIV computes the antiderivative of a divided difference polynomial.
• DIF_APPEND adds a pair of data values to a divided difference table.
• DIF_BASIS computes all Lagrange basis polynomials in divided difference form.
• DIF_BASIS_I computes the I-th Lagrange basis polynomial in divided difference form.
• DIF_DERIV computes the derivative of a polynomial in divided difference form.
• DIF_PRINT prints the polynomial represented by a divided difference table.
• DIF_ROOT seeks a zero of F(X) using divided difference techniques.
• DIF_SHIFT_X replaces one abscissa of a divided difference table with a new one.
• DIF_SHIFT_ZERO shifts a divided difference table so that all abscissas are zero.
• DIF_TO_RPOLY converts a divided difference polynomial to standard form.
• DIF_VAL evaluates a divided difference polynomial at a point.
• NC_RULE computes the coefficients of a Newton-Cotes quadrature rule.
• NCC_RULE computes the coefficients of a Newton-Cotes closed quadrature rule.
• NCO_RULE computes the coefficients of a Newton-Cotes open quadrature rule.
• R_SWAP swaps two real values.
• ROOTS_TO_DIF sets up a divided difference table for a polynomial from its roots.
• ROOTS_TO_RPOLY converts polynomial roots to polynomial coefficients.
• RPOLY_ANT_COF integrates a polynomial in standard form.
• RPOLY_ANT_VAL evaluates the antiderivative of a polynomial in standard form.
• RPOLY_BASIS computes all Lagrange basis polynomial in standard form.
• RPOLY_BASIS_1 computes the I-th Lagrange basis polynomial in standard form.
• RPOLY_DEGREE returns the degree of a polynomial.
• RPOLY_DER_COF computes the coefficients of the derivative of a polynomial.
• RPOLY_DER_VAL evaluates the derivative of a polynomial in standard form.
• RPOLY_PRINT prints out a polynomial.
• RPOLY_SHIFT adjusts the coefficients of a polynomial for a new argument.
• RPOLY_VAL evaluates a polynomial in standard form.
• RVEC_DISTINCT is true if the entries in a real vector are distinct.
• RVEC_IDENTITY sets a real vector to the identity vector A(I)=I.
• RVEC_PRINT prints a real vector.

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