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AN EXAMPLE OF GAUSSIAN QUADRATURE In this example we work out the one- and two-point Gaussian quad-rature formulas for improper integrals of the form Z 1 0 f(x) Example: Gaussian Quadrature Three-point formula Four-point formula 4967.106689 (4.79%) (8589.142689 ) 9 5 (218.3926001) 9 8 (2.221191545 ) 9 5 (4 4 0.6 )e 9 5 (4 )e 9

Examples. First five Legendre polynomials: ํ ตํฑํ ตํฑ 0 (ํ ตํฑฅํ ตํฑฅ) = 1 , ํ ตํฑํ ตํฑ 1 (ํ ตํฑฅํ ตํฑฅ) = ํ ตํฑฅํ ตํฑฅ, ํ ตํฑํ ตํฑ 2 (ํ ตํฑฅํ ตํฑฅ) = ํ ตํฑฅํ ตํฑฅ 2 โ1/3, ํ ตํฑํ ตํฑ 3 (ํ ตํฑฅํ ตํฑฅ) = ํ ตํฑฅํ ตํฑฅ 3 โ Here, we will discuss the Gauss quadrature rule of approximating integrals of the form = โซ ( ) b a I. f x. dx. where . f (x) is called the integrand, a = lower limit of Beispiel: Gauร-Tschebysche๏ฌ-Quadratur. โข Integrationsintervall: I= [โ1,1] โข Gewichtsfunktion: w(x) = 1/ โ 1โx2. โข Knoten: Nullstellen xi = cos 2i+1 2n+2 ฯ fยจur 0โค iโค n We can use a Gauss quadrature using only N=2 in this example, because is a polynomial function of degree less than 3 in each variable. N=2; Similarly, if the even derivatives of gare negative, then the quadrature rule gives a lower bound. Example We will use Gaussian quadrature to approximate the

Compute the integral value according Gauss formula. evalF=F (ptGaussDomain (:,1),ptGaussDomain (:,2)); suma=0; for i=1:size (ptGaussDomain,1) suma=suma+w (i)*evalF Example of Two-Point Gauss Quadrature Rule Use two-point Gauss quadrature rule to approximate the distance in meters covered by a rocket from t = 8 s To watch more videos on Higher Mathematics, download AllyLearn android app - https://play.google.com/store/apps/details?id=com.allylearn.app&hl=en_US&gl=USUs..

Gaussian Quadrature Weights and Abscissae. This page is a tabulation of weights and abscissae for use in performing Legendre-Gauss quadrature integral Learn via example how to apply the Gauss quadrature formula to estimate definite integrals. For more videos and resources on this topic, please visit http://..

### 2D Gaussian quadratures: Triangular exampl

1. ed as a and b, but as unknowns
2. Gaussian Quadrature over a rectangle. We are now going to use the work on 1-dimensional quadrature to develop a rule for 2-dimensional quadrature. First recall
3. f ( x ) {\displaystyle f (x)} , but by applying an n-point Gauss-Legendre quadrature rule, as described here, for example. The input values should be an function f
4. Example using Numerical Integration. An alternative approach to Silvester's method using differentiation matrices and analytical integration is to use numerical

1. Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of
2. Examples include Trapezoidal, Boole's and Gaussian quadratures. Gaussian quadrature is probably the most popular method in practice today. This article focuses on
3. Examples >>> from scipy import integrate >>> f = lambda x : x ** 8 >>> integrate . quadrature ( f , 0.0 , 1.0 ) (0.11111111111111106, 4.163336342344337e-17) >>>
4. Gauss-Legendre quadrature is optimal in a very narrow sense for computing integrals of a function f over [โ1, 1], since no other quadrature rule integrates all
5. Gauss-Legendre quadrature on [-1,1] [x,w]=GaussLegendre(n) computes quadrature nodes and weights such that. Example: Use n=2,4,6,8 nodes to approximate. Note
6. โ Gaussian quadrature formulas use n points and are exact of degree 2nโ1 8. Theorem 1 Suppose that โข Present Value Example โ Use Gauss-Laguerre quadrature

Hopefully after you analyze the above example to see how it operates you will be able to write your own, much better, code. Share. Cite. Follow answered May 23 '16 at 15:41.. Gaussian Quadrature: Example in 1D 5/6/2015 Adrian Egger | FEM I | FS 2015 27. Gaussian Quadrature: Reduced Integration Reduced integration entails using fewer integration points than required by (full) conventional Gaussian quadrature. This has the effect that only a lower degree of polynomial effect can be captured in the integration process. This can be beneficial when encountering shear.

### 19. Gaussian Quadrature Formula - Derivation and Examples ..

• Another very signi cant example is the Gauss quadrature formula, which is de ned by choosing the nodes optimally in the sense of maximizing the degree of polynomials that (1) can integrate exactly. Since there are n + 1 nodes, the attainable degree is n + 1 order higher. Thus the (n+1)-point Gauss formula integrates exactly polynomials of degree 2n+1, what makes it the most accurate. \] WDS'11.
• Example of Gaussian Quadrature Formula 1.305f(x)dx 1.305 9 11.75 9 3.75 9 8 1 -1 11. Example of Gaussian Quadrature Formula Formula.hree Pointdx with T2xxEvaluate 4 2 2 3tx 2)(4 2 1 t24 2 1 x 42, ba ab 2 1 tab 2 1 Here x 12
• Gauss-Legendre quadrature on [-1,1] [x,w]=GaussLegendre(n) computes quadrature nodes and weights such that. Example: Use n=2,4,6,8 nodes to approximate. Note that the function goes to infinity at the left endpoint -1, so it is not surprising that the errors are fairly large. They converge to zero as
• To prepare it for Gaussian quadrature, we can let $$\theta = \frac{\pi}{4}(x+1)$$, as we did in example 2, so that the integral becomes ,where $$\frac{\pi}{4} \int_{-1}^1 \frac{c^3 (c^2+1)}{e^c -1}dx,$$ where $$c = \cot \frac{\pi}{4} (x+1)$$. Using 16- point Gaussian quadrature, I got 6.48. Thus we would need to extend our table of constants for the Gaussian method to much higher order in. ### Gaussian Quadrature Weights and Abscissa

• J. M. Pimbley, Hermite Polynomials and Gauss Quadrature, example, if n = 1, we see by inspection that y 2x solves equation (1). Of course, y 2x is a polynomial of order 1. More generally, let's call the solution for each non-negative integer n y x Hn x . We've already seen that H1 x 2. We list the first few Hermite polynomials below: H0 x 1 H 2 x 2 4x 2 H x 8x3 12x 3 16 4 48 2 12 H4.
• Because the abscissas of the N-point Gaussian quadrature formulas (4.5.1) and (4.5.4)with weightingfunctionW(x)in the interval(a,b)are precisely the roots of the orthogonal polynomial p N(x) for the same interval and weighting function. This is the fundamental theorem of Gaussian quadratures, and lets you ๏ฌnd the abscissas for any particular.
• โ Gaussian quadrature formulas use n points and are exact of degree 2nโ1 8. Theorem 1 Suppose that โข Present Value Example โ Use Gauss-Laguerre quadrature to compute present values. โ Suppose discounted pro๏ฌts equal ฮท ฮท โ1 ฮท ฮทโ1 โ 0 eโrtm(t)1โฮท dt. โ Errors: Table 7.7 r = .05 r = .10 r = .05 ฮป = .05 ฮป = .05 ฮป = .20 Truth: 49.7472 20.3923 74.4005 Errors: GLag.
• Keywords: Interval Gaussian quadrature rule; Hermite weight function; Nodes; Weights 1. Introduction By the Gaussian interval quadrature formula for the positive weight function w, we assume a quadrature formula of the following form Z b a fwdx โ Xn k=1 ยตk 2hk Z x k+hk x kโh fwdx, (1.1) which integrates exactly all polynomials of degree less than 2n, and where the subintervals (xk โ hk.
• yes, that's because you're passing a function to gauss1(f,n), with the aim of creating a Gauss-Legendre quadrature approximating it. You should then by calling f(x) inside the function to create the quadrature
• Gaussian quadrature as above will only produce accurate results if the function f(x) is well approximated by a polynomial function within the range [-1,1]. The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as , where g(x) is approximately polynomial, and W(x) is known, then there are alternative weights such that.

Gauss quadrature, numerical integration, Hermite polynomials, equilibrum mea-sures, Riemann{Hilbert problems AMS subject classi cations. 65D32, 33C45, 35Q15 1. Introduction. Numerical quadrature is the approximation of a de nite inte- gral of a continuous function fby a weighted linear combination of function evalua-tions, i.e., Zb a f(x)dxห Xn k=1 wkf(xk); 1 a<b 1; where fxkg n k=1 are the. Clenshaw-Curtis quadrature also converges geometrically for analytic functions. In some circumstances Gauss converges up to twice as fast as C-C, with respect to Npts, but as this example suggests, the two formulas are often closer than that. The computer time is often faster with C-C. For details of the cmoparison, see  and Chapter 19 of 

The following example performs a Gauร-Christoffel quadrature for a Gaussian with variance 2 centered around x = 0. An extended version of this example is gauss.c. Compiling requires linking against a blas and lapack Examples and Tests: kronrod_prb.f, a sample calling program. kronrod_prb.sh, commands to compile and run the sample program. kronrod_prb_output.txt, the output file. List of Routines: KRONROD adds N+1 points to an N-point Gaussian rule. ABWE1 calculates a Kronrod abscissa and weight PROGRAM gaussian_quad IMPLICIT NONE INTEGER::i,n REAL,DIMENSION(100)::x,c REAL::a,b,f,s,t,ev OPEN(1,FILE='input.txt') OPEN(2,FILE='output.txt') READ(1,*)a,b,n. QUADRATURE_RULES_CHEBYSHEV2 is a dataset directory which contains examples of quadrature rules of Gauss-Chebyshev type 2.. The Gauss-Chebyshev type 2 quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form sqrt (1-x^2).. Gauss-Chebyshev type 2 quadrature assumes that the integrand we are considering has a form like

SYMMETRICAL GAUSSIAN QUADRATURE RULES 1131 Substitution yields By successive application of one-dimensional rules The Gaussian points and weights in the u or a direction are ui and wi, and in the v or fl direction are uj and wj.The number of points m and n in each direction may be different, but the lower value will control polyval (coefs,0.5) ans =. 0.08984375. Finally, in order to use them as polynomials for Gaussian quadrature, you will need the derivative polynomials too. polyder (coefs) ans =. 39.375 0 -26.25 0 1.875. Its been a while since I had to derive the Gaussian quadrature but you need some roots too. Again, trivial Gaussian quadrature approximates the value of an integral as a linear combination of values of the integrand evaluated at optimal abscissas. x i. : . The abscissas are optimal in the sense that the quadrature formula is exact for all polynomials up to degree 2n - 1. The precision argument acts similarly to the WorkingPrecision option used in. to use of the Gauss-Hermite quadrature (7). The sample size n was taken as 300. Note that the quadratures are applied for each binary observation. The accuracy of the Laplace approximation seen here does not decrease for small n, for example n = 10; on the contrary adequacy of dealin ### Gauss Quadrature Rule: Example - YouTub

Compute the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth degree Hermite polynomial, $$H_n(x)$$. These sample points and weights correctly integrate polynomials of degree $$2n - 1$$ or less over the interval $$[-\infty, \infty]$$ with weight function $$w(x) = e^{-x^2}$$. See 22.2.14 in for details. Parameters n int. quadrature order. mu bool. (Chebfun example quad/GaussQuad.m). Guass quadrature. Gauss (or more specifically Gauss-Legendre) quadrature  provides an approximation to the integral of a function f over the interval [-1, 1] (which may be trivially scaled to any finite interval [a, b]) by evaluating f at a set of n nodes x = {x_j} and summing with some specified weights w = {w_j} This study examined the characteristics of a variable three-point Gauss quadrature using a variable set of weighting factors and corresponding optimal sampling points. The major findings were as follows. The one-point, two-point, and three-point Gauss quadratures that adopt the Legendre sampling points and the well-known Simpson's 1/3 rule were found to be special cases of the variable three. Calculates the integral of the given function f(x) over the interval (a,b) using Gaussian quadrature. (i.e. Gauss-Legendre, Gauss-Chebyshev 1st, Gauss-Chebyshev 2nd, Gauss-Laguerre, Gauss-Hermite, Gauss-Jacobi, Gauss-Lobatto and Gauss-Kronrod It compares the trapezoidal rule and Gaussian quadrature with the exact result from symbolic python SYMPY up to 1000 integration points for the integral $$I = 2 = \int_0^{\infty} x^2 \exp{-x} dx.$$ For the trapezoidal rule the results will vary strongly depending on how the infinity limit is approximated. Try to run the code below for different finite approximations to $$\infty$$ The Legendre-Gauss Quadrature formula or Gauss-Legendre quadrature is the numeric approximation of a definite integral, . It is possible to choose quadrature points x i and weights w i, so that polynomial functions of degree smaller than 2N are integrated exactly by equation (1).. The Legendre-Gauss quadrature formula is a special case of Gaussian quadratures which allow efficient.

### Numerical integration/Gauss-Legendre Quadrature - Rosetta Cod

1. On Gauss quadrature and partial cross validation. Computational Statistics & Data Analysis, 2004. Andrzej Kozek. Download PDF. Download Full PDF Package. This paper. A short summary of this paper. 37 Full PDFs related to this paper. Read Paper. On Gauss quadrature and partial cross validation. Download . Related Papers. A bootstrap approach to goodness of fit for linear models under length.
2. Computes the sample points and weights for Gauss-Legendre quadrature. These sample points and weights will correctly integrate polynomials of degree or less over the interval with the weight function ; Gauss-Legendre quadrature Graphs of Legendre polynomials (up to n = 5) For the simplest integration problem stated above, i.e. with, the associated polynomials are Legendre polynomials, P n (x.
3. Gaussian Quadrature Example Edit. I realized that there was insufficient information after the derived and solved sample on Gaussian Quadrature thus i took the pain to edit this wikiversity page by adding a solved example to the information already on there and below is what i factored in. Find the constants C_0, C_1, and x_1 so that the quadrature formula () = + (). has the highest possible.
4. Table :n-point Gaussian Quadrature rule. Numerical Analysis: Gaussian Numerical Integration Natasha S. Sharma, PhD Example Example Apply the 2 and 3 points Gaussian numerical integration formula to obtain an approximation I n(f) for I = R1 1 exdx. Use the nodes and weights provided in Table. Proof. I 2(f) =1 e( 0:57735) + 1 e(0:57735): I 3(f) =0:555 e( 0:33998) + 0:8888 e(0)+ 0:555 e(0:33998.
5. Example 1 Approximate โซ ( ) using Gaussian quadrature with n = 3. Gaussian quadrature on arbitrary intervals Use substitution or transformation to transform โซ ( ) into an integral defined over . Let ( ) ( ) , with Then Example 2 )Consider โซ ( . Compare results from the closed Newton-Cotes formula wit

### ECE 632: Quadratic Triangle Example with Numerical Integratio

1. For example, we speak of three-point'' Gauss. The following two exercises involve writing m-files analogous to nc_single.m and nc_quad.m. Exercise 8: Download the file gl_weight.m. This file returns both the node points and weights for Gauss-Legendre quadrature for points. Write a routine called gl_single.m with the signature function quad = gl_single ( func, a, b, N ) % quad = gl_single.
2. For example, consider evaluating Z 1 0 (log x) f(x) dx for arbitrary continuous functions f(x). P. Sam Johnson (NITK) Romberg Integration and Gaussian Quadrature October 17, 2014 8 / 19. Most numerical integration formulas are based on de ning f n(x) in I n(f) = Z b a f n(x) dx = I(f n) by using polynomials or piecewise polynomial interpolation. Formulas using such interpolation with evenly.
3. Gaussian Quadrature Numerical Analysis and Computing Lecture Notes #08 โ Numerical Di๏ฌerentiation and Integration โ Composite Numerical Integration; Romberg Integration Adaptive Quadrature / Gaussian Quadrature Joe Maha๏ฌy, hmahaffy@math.sdsu.edui Department of Mathematics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-7720.
4. Chapter 2 : Gaussian Quadrature and Eigenvalue Problems The main aim of this chapter is to see which numerical integration method is better one. For this purpose a model problem of de nite integral is selected. Model Problem Evaluate the integral I= Z 1 0 sinxdx (1.1) Exact value (analytical solution) of problem calculated by simple inte- gration rules is calculated equal to 4:59769769413186e.
5. An example from the Lake Michigan coast shows that the total internal geometrical area is found to be preserved during the course of the progressive deformation. Displacement monitoring of the slope shows that it became less stable over a period of eleven years due to progressive failure. The Gaussian quadrature technique allows representation and manipulation of geometrical models in a.

### Delta Quants - Gaussian Quadrature - Gauss Legendre

1. The following Matlab project contains the source code and Matlab examples used for gauss-hermite quadrature using tabulated abscissas and weights . Numerical integration from -Inf to Inf of an integrand in the form f(x)*exp(-x^2) f(x) must be a slowly varying function fn_val = GaussHermite(func, npt, varargin) func is a handle for f(x) npt is the number of points at which f(x) is to be.
2. available Gaussian quadrature for square domain integrals to evaluate the triangular domain integrals. Finally, it presents n n points and n(n+1) 2 1 points (for n > 1) Gaussian quadrature formulae for triangle utilizing n-point one-dimensional Gaussian quadrature. By use of simple but straightforward algorithms, Gaussian points and corresponding weights are calculated and presented for.
3. numerical integration methods such as Gaussian quadrature [6, 33] and sparse grids . 2. Unlike many other kernels used in machine learning, such as the Gaussian kernel, the sparse ANOVA kernel allows us to encode prior information about the relationships among the input variables into the kernel itself. Sparse ANOVA kernels have been shown  to work well for many classi๏ฌcation tasks.
4. example in signal processing, Gauss-Hermite rules have been successfully applied in a variety of applications of stochastic ๏ฌltering, often with remarkable performance . Particularly, the Quadrature Kalman ๏ฌlter (QKF) ,  and its variants for high-dimensional systems ,  showed improved per
5. The theory of strong moment problems has provided Gaussian quadrature rules for approximate integration with respect to strong distributions. In Hagler (Ph.D. Thesis, University of Colorado, Boulder, 1997) and Hagler et al. (Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, in press), a transformation of the form v(x)=(1/ฮป)(xโฮณ/x), ฮป,ฮณ>0, is used to obtain strong.
6. Gauss-Hermite Quadrature in Financial Risk Analysis Joe Pimbley* Introduction Financial risk analysis often focuses on calculating the probability of loss or expected loss of a given risky transaction or portfolio of transactions. In structured finance applications, these calculations may include the presence of a loss buffer (otherwise known as equity) when the calculations pertain to a.
7. Gauss Quadrature Example Free PDF eBooks. Posted on May 22, 2015. Gaussian Quadratures Gaussian Quadratures. โข Newton-Cotes Formulae. - use evenly-spaced functional values. - Did not use the flexibility we have to select the quadrature points. Lecture16_integration.pdf. Read/Download File Report Abuse. Gaussian Quadrature In Lecture 26, we learned that a Newton-Cotes quadrature rule with.

Recently, I got a request how one can find the quadrature and weights of a Gauss-Legendre quadrature rule for large n. It seems that the internet has these points available free of charge only up to n=12. Below is the MATLAB program that finds these values for any n. I tried the program for n=2 Computes the definite integral of a function using the Gauss quadrature for 3 points. HOME LIBRARY PRODUCTS FORUMS CART Tel: +44 (0) 20 7193 9303 Email Us Join CodeCog For a more detailed explanation of these methods, including step-by-step examples, see the main article on Riemann sums. Gaussian Quadrature. Gaussian quadrature gives exact solutions for a polynomial of degree not higher than 2n - 1. The basic idea is that you use a Gaussian curve as the superimposing shape, instead of rectangles or a trapezoid In order to understand why Gaussian quadrature works so well, we first need to understand some properties of polynomials in general, and of Legendre polynomials in particular. We also need to remind ourselves of the use of Lagrange polynomials for approximating an arbitrary function. First, a statement concerning polynomials in general: Let $$P$$ be a polynomial of degree $$n$$, and let \(S. Computational Physics Lectures: Numerical integration, from Newton-Cotes quadrature to Gaussian quadrature. Morten Hjorth-Jensen [1, 2]  Department of Physics, University of Oslo  Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University Aug 23, 201 This example shows how to solve polynomial equations and systems of equations, and work with the results using Symbolic Math Toolboxโข. Gaussian quadrature rules approximate an integral by sums โซ a b f (t) w (t) d t โ โ i = 1 n f (x i) ฮฑ i. Here, the x i and ฮฑ i are parameters of the method, depending on n but not on f rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of. degree 2n -1 or less by a suitable choice of the points xiand weights wifor i. Calculates the integral of the given function f(x) over the interval (a,b) using Gauss-Legendre quadrature

The other example is concerned with numerical solution for the Poisson equation in 2D. It is observed in the calculation of components of element matrices by using the Gauss quadrature rule (e.g. Three-Point Gaussian Quadrature Rule Consider a function f(x) over interval [-1,1] with sampling point and The two point formula is, The formula of Three point Gaussian quadrature rule, )()()()( 3322 1 1 11 xfwxfwxfwdxxf 5 3 9 5 )0( 9 8 5 3 - 9 5 )( 1 1 fffdxxf 321 xxx 321 www 7. Example 1 Evaluate by one point , two point & Three point Gaussian quadrature. Here, Using one point. ### scipy.integrate.quadrature โ SciPy v1.7.1 Manua

The package approximates these integrals using the adaptive Gauss-Hermite quadrature rule. Multiple random effects terms can be included for the grouping factor (e.g., random intercepts, random linear slopes, random quadratic slopes), but currently only a single grouping factor is allowed. Basic Features . The package contains a single model-fitting function named mixed_model() with four. We investigate the structure of the matrices for Gauss-Laurent and associated anti-Gauss-Laurent quadrature rules and discuss properties of these quadrature rules. Computed examples show that Gauss-Laurent rules may give higher accuracy than standard Gauss rules with the same number of nodes. Moreover, they illustrate that pairs of Gauss-Laurent and anti-Gauss-Laurent rules provide. The Legendre-Gauss Quadrature formula or Gauss-Legendre quadrature is the approximation of the integral (1) with special choice of nodes and weights , characterised in that, if the function is polynomial of degree smaller than , then the exact equality takes place in equation (1).. The Legendre-Gauss quadrature formula is a special case of Gaussian quadratures which allow efficient. EXAMPLES: We verify that \(\int_0^1 n x^{n-1} \, Compute the integration nodes and weights for the Gauss-Legendre quadrature scheme. We use the recurrence relations for Legendre polynomials to compute their values. This is a version of the algorithm that in [Neu2018] is called the REC algorithm. INPUT: degree - integer. The number of nodes. Must be 3 or even. prec - integer (minimal.

The Gaussian quadrature is targeted to approximate an integral by taking the weighted sum of integrand values sampled at special points (called abscissas). The abscissas and weights are calculated in a special way so that the rule provides a precise answer for all polynomials up to certain degree. The most common case, the Gauss-Legendre. The Gaussian quadrature for a square domain ([-1, 1] x [-1, 1]) can be conducted by a similar manner to 1D integration For example, if two Gauss points (i.e. integration points) are taken per direction (see Figure 1), the coordinates and weights are given in Table 1. Table 1. Coordinates (in reference square) and weights of the Gauss points . Point $\xi$ $\eta$ weight: 1 (i = 1, j = 1. $\begingroup$ @FedericoPoloni, The Runge function seems to be an example for equidistant case instead of Gauss quadrature. $\endgroup$ - gondolier Mar 24 '14 at 21:11 | Show 1 more comment 2 Answers Two numerical integration rules based on composition of Gauss-Legendre formulas for solving integration of fuzzy numbers-valued functions are investigated in this paper. The methods' constructions are presented and the corresponding convergence theorems are shown in detail. Two numerical examples are given to illustrate the proposed algorithms finally Translations in context of Gauss-Legendre quadrature in English-French from Reverso Context: Herein the combination of acoustic technology and Gauss-Legendre quadrature (GLq) integration is investigated as an alternative method of measuring discharge 