Thus if we want to use this formula to write a polynomial expression for the sum of the fifth powers, then we must first have computed polynomial expressions for the sums of the integers, the squares, the cubes, and the fourth powers. His final result was. Again, in his actual justification, Pascal used the numerical example above with just four terms, but with an arithmetic progression more complicated than 1, 2, 3, 4. Summing these n equations gives the equations shown below the line, which can be solved for the sum of the cubes:. Exercise Give A. Solutions to these exercises can be found by clicking here.
Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M. Author s :. From the result obtained subtract the following: 1 st. NumLib defines the type complex as a Turbo Pascal object. Real and imaginary parts are denoted as xreal and imag , respectively. Methods for basic operations are provided:. The real and imaginary parts of a complex number can also be retrieved by these stand-alone functions:.
It is calculated as the sum of the element-by-element products of the vectors. Both vectors must have the same lengths. The product of a m x n matrix A and a n x p matrix B is a m x p matrix C in which the elements are calculated as the inner product of the row vectors of A and the column vectors of B see here for more details :.
A matrix A of size m x n multiplied with a vector b of length n yields another vector of length m.
The elements of the result vector c are the dot products of the matrix rows with b. Note that the length of vector b must be equal to the number of columns of matrix A. The transpose matrix A T of matrix A is obtained by flipping rows and columns:. A norm is a function which assigns a positive "length" to a vector a or a matrix M. Unit det provides a series of routines for calculation of the determinant of a square matrix.
Besides a procedure for general matrices there are also routines optimized for special matrix types tridiagonal, symmetric. In order to avoid possible overflows and underflows a factor may be split off during calculation, and the determinant is returned as a floating point number f times a multiplier 8 k. The term "standard matrix" means here a matrix with is stored in the standard NumLib storage as described here , i.
The 2D array has at least n rows and n columns, the 1D array is dimensioned to contain at least ' 2 elements. Without making any assumptions on the properties of the matrix the determinant can be calculated by the procedure detgen. More efficient algorithms can be used, however, if the matrix has these particular properties:. The matrix is positive definite because all elements are positive and the product b T A b cannot become zero or negative for any non-zero vector b.
While the routine detgpd is the best for this matrix type the code below contains also commented-out calls to the other routines. If a matrix has the shape of a band matrix then its determinant is best calculated by using the procedure detgba. If a matrix has the shape of a band matrix and is symmetric positive definite then detgpb is the dedicated routine for calculation of its determinant.
Calculation of the determinant of a tridiagonal matrix is done by the procedure detgtr :. Suppose a square matrix A. Then the matrix A -1 is the inverse of A if the product A -1 A is the identity matrix I all elements zero, except for the diagonal elements which are 1. NumLib provides various routines for calculation of the inverse matrix. Which algorithm is used depends on the properties of the matrix A. All routines discussed here have in common that the matrix is stored in the standard NumLib way, i. No special requirements are made for the matrix passed to invgen , except that it is square.
The calculation in this routine is based on LU decomposition with partial pivoting. If the matrix is symmetric, it can be processed by invgsy which uses a reduction to tridiagonal form internally. A symmetric positive definite matrix can be passed to invgpd where the inverse is calculated by Cholesky decomposition. Therefore, invgpd is best-suited for this task although the other routines can be used as well uncomment their calls below.
A system of linear equations or linear system is a collection of two or more linear equations involving the same set of variables, x NumLib offers several procedures to solve the matrix equation depending on the properties of the matrix. Here we describe several routines in which the matrix of the equation system is square, i. This matrix is assumed to be passed to the procedures in the standard NumLib way as the first element of a 2D or 1D array.
The 2D array must be dimensioned to contain at least n rows and n columns, the 1D array must contain at least n 2 elements. The most general procedure is slegen and applied to any square matrix A.
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The system is solved using the Gauss elimination procedure with partial pivoting. If the matrix A is symmetric then the reduction to tridiagonal form generally is more stable. This is done in routine slegsy. If A , furthermore, is a symmetric positive definite matrix, then the Cholesky method is best-suited.
This is applied in routine slegpd. This matrix is symmetric positive definite. Therefore, all routines presented here can be applied.
Uncomment the requested line in the code below. As a test the result vector x is multiplied to matrix A , and the result must equal the constant vector b. A special solution, slegba , is implemented for a band matrix , i. The optimized solution method for a symmetric positive band matrix is called slegpb.
The matrix of the band matrix example is symmetrical positive definite.
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Therefore, procedure slegpb can be applied as well:. NumLib offers two specialized procedures for solving a system of linear equations based on a tridiagonal matrix A. Both rely on the memory-optimized storage of tridiagonal matrices as described here. They differ in the internal calculation algorithm and in the behavior with critical matrices. However, this procedure does not provide the parameter ca from which the accuracy of the determined solution can be evaluated. If this is needed the less stable procedure slegtr must be used.
Unlike the other routines in unit sle which require a square matrix A , slegls can solve linear systems described by a rectangular matrix which has more rows than colums, or speaking in terms of equations, has more equations than unknowns. Such a system generally is not solvable exactly.
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But an approximate solution can be found such that the sum of squared residuals of each equation, i. The most prominent application of this technique is fitting of equations to data regression analysis. The method is based on reduction of the array A to upper triangle shape through Householder transformations.
A vector is an eigenvector if it does not change its direction after a linear transformation has been applied to it.
NumLib offers a variety of routines for calculation of eigenvectors and eigenvalues. They are optimized for the matrix type.
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Every matrix type will be supported by two or four procedures of various degree of complexity; they are identified by appended numbers 1 and 2, or 1 to The routines discussed here assume that the n x n matrix for which the eigenvalues are to be calculated is stored in the conventional way as a 2D or 1D array of ArbFloat values. Additionally, in case of procedure eigge3 , eigge4 , eiggg3 or eiggg4 :.
Additionally, in case of procedures eiggs2 , eiggs4 , eiggg2 and eiggg4 :. NumLib provides four routines to calculate eigenvalues and eigenvectors of symmetric band matrices : eigbs The datatype complex has been described in section complex numbers. These roots can be calculated by the function roopol :. The expected zero points are:. It always has two, not necessarily different, complex roots. These solutions can be determined by the procedure rooqua. Note that rooqua assumes that the coefficient of the quadratic term has been normalized to be 1.
It has n complex solutions which can be calculated by the procedure roobin :. In the bisection method two x values a and b are estimated to be around the expected root such that the function values have opposite signs at a and b. The center point of the interval is determined, and the subinterval for which the function has opposite signs at its endpoints is selected for a new iteration.
The process ends when a given precision, i. The following program determines the square root of 2. The NumLib function int1fr calculates the integral of a given function between limits a and b with a specified absolute accuracy ae :. The algorithm is adaptive and is based on an explicit one-step Runge-Kutta method of order five.
The accuracy can not be guaranteed in all cases.
In so-called unstable problems, e. Call the procedure with different values of ae to investigate this case.