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In numerical mathematics, the Gauss–Kronrod quadrature formula is a method for numerical integration (calculating approximate values of integrals). Gauss–Kronrod quadrature is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order (the latter called an embedded rule). The difference between these two approximations is used to estimate the calculational error of the integration.
These formulas are named after Alexander Kronrod, who invented them in the 1960s, and Carl Friedrich Gauss. Gauss–Kronrod quadrature is used in the QUADPACK library, the GNU Scientific Library, the NAG Numerical Libraries and R.[1]
Description
The problem in numerical integration is to approximate definite integrals of the form
Such integrals can be approximated, for example, by n-point Gaussian quadrature
where wi, xi are the weights and points at which to evaluate the function f(x).
If the interval [a, b] is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at the midpoint for odd numbers of evaluation points), and thus the integrand must be evaluated at every point. Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by adding points to an -point rule in such a way that the resulting rule is of order . These extra points are the zeros of Stieltjes polynomials. This allows for computing higher-order estimates while reusing the function values of a lower-order estimate. The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error.
Example
A popular example combines a 7-point Gauss rule with a 15-point Kronrod rule Template:Harv. Because the Gauss points are incorporated into the Kronrod points, a total of only 15 function evaluations yields both a quadrature estimate and an error estimate.
(G7,K15) on [−1,1] Gauss nodes Weights ±0.94910 79123 42759 ∗ 0.12948 49661 68870 ±0.74153 11855 99394 ∗ 0.27970 53914 89277 ±0.40584 51513 77397 ∗ 0.38183 00505 05119 0.00000 00000 00000 ∗ 0.41795 91836 73469 Kronrod nodes Weights ±0.99145 53711 20813 0.02293 53220 10529 ±0.94910 79123 42759 ∗ 0.06309 20926 29979 ±0.86486 44233 59769 0.10479 00103 22250 ±0.74153 11855 99394 ∗ 0.14065 32597 15525 ±0.58608 72354 67691 0.16900 47266 39267 ±0.40584 51513 77397 ∗ 0.19035 05780 64785 ±0.20778 49550 07898 0.20443 29400 75298 0.00000 00000 00000 ∗ 0.20948 21410 84728
The recommended error estimate is .
Template:Harvtxt showed how to find further extensions of this type.
See also
- Clenshaw–Curtis quadrature, another nested quadrature rule with similar accuracy
Notes
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References
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (Reference guide for QUADPACK) - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. Erratum in Math. Comput. 23: 892.
External links
- QUADPACK (part of SLATEC), source code [1]. QUADPACK is a collection of algorithms, in Fortran, for numerical integration based on Gauss-Kronrod rules. SLATEC (at Netlib) is a large public domain library for numerical computing.
- ALGLIB source code in C#, C++, Delphi & Visual Basic