# BERNSTEIN_POLYNOMIAL The Bernstein Polynomials

BERNSTEIN_POLYNOMIAL is a FORTRAN77 library which evaluates the Bernstein polynomials.

A Bernstein polynomial BP(n,x) of degree n is a linear combination of the (n+1) Bernstein basis polynomials B(n,x) of degree n:

```        BP(n,x) = sum ( 0 <= k <= n ) CP(n,k) * B(n,k)(x).
```

For 0 <= k <= n, the k-th Bernstein basis polynomial of degree n is:

```        B(n,k)(x) = C(n,k) * (1-x)^(n-k) * x^k
```
where C(n,k) is the combinatorial function "N choose K" defined by
```        C(n,k) = n! / k! / ( n - k )!
```

For an arbitrary value of n, the set B(n,k) forms a basis for the space of polynomials of degree n or less.

Every basis polynomial B(n,k) is nonnegative in [0,1], and may be zero only at the endpoints.

Except for the case n = 0, the basis polynomial B(n,k)(x) has a unique maximum value at

```        x = k/n.
```

For any point x, (including points outside [0,1]), the basis polynomials for an arbitrary value of n sum to 1:

```        sum ( 1 <= k <= n ) B(n,k)(x) = 1
```

For 0 < n, the Bernstein basis polynomial can be written as a combination of two lower degree basis polynomials:

```        B(n,k)(x) = ( 1 - x ) * B(n-1,k)(x) + x * B(n-1,k-1)(x) +
```
where, if k is 0, the factor B(n-1,k-1)(x) is taken to be 0, and if k is n, the factor B(n-1,k)(x) is taken to be 0.

A Bernstein basis polynomial can be written as a combination of two higher degree basis polynomials:

```        B(n,k)(x) = ( (n+1-k) * B(n+1,k)(x) + (k+1) * B(n+1,k+1)(x) ) / ( n + 1 )
```

The derivative of B(n,k)(x) can be written as:

```        d/dx B(n,k)(x) = n * B(n-1,k-1)(x) - B(n-1,k)(x)
```

A Bernstein polynomial can be written in terms of the standard power basis:

```        B(n,k)(x) = sum ( k <= i <= n ) (-1)^(i-k) * C(n,k) * C(i,k) * x^i
```

A power basis monomial can be written in terms of the Bernstein basis of degree n where k <= n:

```        x^k = sum ( k-1 <= i <= n-1 ) C(i,k) * B(n,k)(x) / C(n,k)
```

Over the interval [0,1], the n-th degree Bernstein approximation polynomial to a function f(x) is defined by

```        BA(n,f)(x) = sum ( 0 <= k <= n ) f(k/n) * B(n,k)(x)
```
As a function of n, the Bernstein approximation polynomials form a sequence that slowly, but uniformly, converges to f(x) over [0,1].

By a simple linear process, the Bernstein basis polynomials can be shifted to an arbitrary interval [a,b], retaining their properties.

### Languages:

BERNSTEIN_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

CHEBYSHEV_POLYNOMIAL, a FORTRAN77 library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

DIVDIF, a FORTRAN77 library which uses divided differences to interpolate data.

HERMITE, a FORTRAN77 library which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

HERMITE_POLYNOMIAL, a FORTRAN77 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

INTERP, a FORTRAN90 library which can be used for parameterizing and interpolating data;

JACOBI_POLYNOMIAL, a FORTRAN77 library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Legendre polynomial and associated functions.

NMS, a FORTRAN77 library which includes a wide variety of numerical software, including solvers for linear systems of equations, interpolation of data, numerical quadrature, linear least squares data fitting, the solution of nonlinear equations, ordinary differential equations, optimization and nonlinear least squares, simulation and random numbers, trigonometric approximation and Fast Fourier Transforms.

PPPACK, a FORTRAN77 library which implements piecewise polynomial functions, including, in particular, cubic splines, by Carl deBoor.

SPLINE, a FORTRAN77 library which constructs and evaluates spline interpolants and approximants.

TEST_APPROX, a FORTRAN77 library which defines a number of test problems for approximation and interpolation.

TEST_INTERP_1D, a FORTRAN77 library which defines test problems for interpolation of data y(x), depending on a 1D argument.

TOMS446, a FORTRAN77 library which manipulates Chebyshev series for interpolation and approximation;
this is a version of ACM TOMS algorithm 446, by Roger Broucke.

### Reference:

1. Kenneth Joy,
"Bernstein Polynomials",
On-Line Geometric Modeling Notes,
idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf
2. David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0-13-627258-4,
LC: TA345.K34.
3. Josef Reinkenhof,
Differentiation and integration using Bernstein's polynomials,
International Journal of Numerical Methods in Engineering,
Volume 11, Number 10, 1977, pages 1627-1630.

### List of Routines:

• BERNSTEIN_MATRIX returns the Bernstein matrix.
• BERNSTEIN_MATRIX_INVERSE returns the inverse Bernstein matrix.
• BERNSTEIN_POLY_01 evaluates the Bernstein polynomials based in [0,1].
• BERNSTEIN_POLY_01_VALUES returns some values of the Bernstein polynomials.
• BERNSTEIN_POLY_AB evaluates the Bernstein polynomials based in [A,B].
• BERNSTEIN_POLY_AB_APPROX: Bernstein approximant to F(X) on [A,B].
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_MOP returns the I-th power of -1 as an R8 value.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_IS_IDENTITY determines if an R8MAT is the identity.
• R8MAT_MM multiplies two R8MAT's.
• R8MAT_NORM_FRO returns the Frobenius norm of an R8MAT.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8VEC_DOT_PRODUCT finds the dot product of a pair of R8VEC's.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN77 source codes.

Last revised on 13 May 2013.