Algebraic Error Propagation
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equations in this document used the SYMBOL.TTF font. Not all computers and browsers supported that font, so this has been re-edited to make it more browser friendly. If any errors remain, please let me know. One of the standard notations for expressing a quantity propagation of error division with error is x ± Δx. In some cases I find it more convenient to use error propagation formula physics upper case letters for measured quantities, and lower case for their errors: A ± a. The notation
How To Calculate Fractional Error
measured quantity? We can think of it as the value we'd measure if we somehow eliminated all error from instruments and procedure. This is a natural enough concept, and a useful one, even though at this point in the discussion it may sound like circular logic. We can improve the measurement process, of course, but since we can never eliminate measurement errors entirely, we can never hope to measure true values. We have only introduced the concept of true value for purposes of discussion. When we specify the "error" in a quantity or result, we are giving an estimate of how much that measurement is likely to deviate from the true value of the quantity. This estimate is far more than a guess, for it is founded on a physical analysis of the measurement process and a mathematical analysis of the equations which apply to the instruments and to the physical process being studied. A measurement or experimental result is of little use if nothing is known about the probable size of its error. We know nothing about the reliability of a result unless we can estimate the probable sizes of the errors and uncertainties in the data which were used to obtain that result. That is why it is important for students to learn how to determine quantitative estimates of the nature and size of experimental errors and to predict how these errors affect the
x, y, or z leads to an error in the determination of u. This is simply the multi-dimensional definition of slope. It
Error Propagation Quotient
describes how changes in u depend on changes in x, y, error propagation chemistry and z. Example: A miscalibrated ruler results in a systematic error in length measurements. The values of adding errors in quadrature r and h must be changed by +0.1 cm. 3. Random Errors Random errors in the measurement of x, y, or z also lead to error in the https://www.lhup.edu/~dsimanek/errors.htm determination of u. However, since random errors can be both positive and negative, one should examine (du)2 rather than du. If the measured variables are independent (non-correlated), then the cross-terms average to zero as dx, dy, and dz each take on both positive and negative values. Thus, Equating standard deviation with differential, i.e., results in the famous error http://www.chem.hope.edu/~polik/Chem345-2000/errorpropagation.htm propagation formula This expression will be used in the Uncertainty Analysis section of every Physical Chemistry laboratory report! Example: There is 0.1 cm uncertainty in the ruler used to measure r and h. Thus, the expected uncertainty in V is ±39 cm3. 4. Purpose of Error Propagation · Quantifies precision of results Example: V = 1131 ± 39 cm3 · Identifies principle source of error and suggests improvement Example: Determine r better (not h!) · Justifies observed standard deviation If sobserved » scalculated then the observed standard deviation is accounted for If sobserved differs significantly from scalculated then perhaps unrealistic values were chosen for sx, sy, and sz. · Identifies type of error If ½uobserrved - uliterature½ £ scalculated then error is random error If ½uobserrved - uliterature½ >> scalculated then error is systematic error 5. Calculating and Reporting Values when using Error Propagation Use full precision (keep extra significant figures and do not round) until the end of a calculation. Then keep two significant
measurands based on more complicated functions can be done using basic propagation of errors principles. For example, suppose we want to compute the uncertainty of http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc553.htm the discharge coefficient for fluid flow (Whetstone et al.). The measurement http://www.sciencedirect.com/science/article/pii/0168927489900251 equation is $$ C_d = \frac{\dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$ where $$ \begin{eqnarray*} C_d &=& \mbox{discharge coefficient} \\ \dot{m} &=& \mbox{mass flow rate} \\ d &=& \mbox{orifice diameter} \\ D &=& \mbox{pipe diameter} \\ \rho &=& \mbox{fluid density} error propagation \\ \Delta P &=& \mbox{differential pressure} \\ K &=& \mbox{constant} \\ F &=& \mbox{thermal expansion factor (constant)} \\ \end{eqnarray*} $$ Assuming the variables in the equation are uncorrelated, the squared uncertainty of the discharge coefficient is $$ s^2_{Cd} = \left[ \frac{\partial C_d}{\partial \dot{m}} \right]^2 s^2_{\dot m} + \left[ \frac{\partial C_d}{\partial d} \right]^2 s^2_d + \left[ \frac{\partial algebraic error propagation C_d}{\partial D} \right]^2 s^2_D + \left[ \frac{\partial C_d}{\partial \rho} \right]^2 s^2_{\rho} + \left[ \frac{\partial C_d}{\partial \Delta P} \right]^2 s^2_{\Delta P} $$ and the partial derivatives are the following. $$ \frac{\partial C_d}{\partial \dot{m}} = \frac{\sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial d} = \frac{-2\dot{m} d}{K F D^4 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} - \frac{2 \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^3 F \sqrt{\rho} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial D} = \frac{2 \dot{m} d^2}{K F D^5 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} $$ $$ \frac{\partial C_d}{\partial \rho} = \frac{- \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{2 K d^2 F \rho^{\frac{3}{2}} \sqrt{\Delta P}} $$ $$ \frac{\partial C_d}{\partial \Delta P} = \frac{- \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{2 K d^2 F \sqrt{\rho} (\Delta P)^{\frac{3}{2}}} $$ Software can simplify propagation of error Propagation of error for more complicated functions can be done reliably with software capable of symbolic computations or algebraic representations. Symbolic computation software can also be used to combine the partia
institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Purchase Loading... Export You have selected 1 citation for export. Help Direct export Save to Mendeley Save to RefWorks Export file Format RIS (for EndNote, ReferenceManager, ProCite) BibTeX Text Content Citation Only Citation and Abstract Export Advanced search Close This document does not have an outline. JavaScript is disabled on your browser. Please enable JavaScript to use all the features on this page. Applied Numerical Mathematics Volume 5, Issues 1–2, February 1989, Pages 71-87 Algebraic stability and error propagation in Runge-Kutta methods Author links open the overlay panel. Numbers correspond to the affiliation list which can be exposed by using the show more link. Opens overlay J.F.B.M. Kraaijevanger, Opens overlay M.N. Spijker Department of Mathematics and Computer Science, University of Leiden, 2300 RA Leiden, The Netherlands Available online 12 August 2002 Show more Choose an option to locate/access this article: Check if you have access through your login credentials or your institution. Check access Purchase Sign in using your ScienceDirect credentials Username: Password: Remember me Not Registered? Forgotten username or password? OpenAthens login Login via your institution Other institution login doi:10.1016/0168-9274(89)90025-1 Get rights and content AbstractThis paper is concerned with Runge-Kutta methods for the numerical solution of initial value problems in ordinary differential equations. For these methods we review the fundamental concept of algebraic stability (introduced in 1979 independently by Burrage and Butcher [1] and by Crouzeix [2]). We prove a new theorem implying that algebraic stability is a necessary and sufficient condition for a stable propagation of numerical errors.Dahlquist and Jeltsch