The general idea is to find a root of the polynomial and then apply Horner\'s method to remove
the corresponding factor according to the Ruffini rule.
This iterative scheme is numerically unstable; the approximation errors accumulate during the
successive factorizations, so that the last roots are determined with a polynomial that deviates
widely from a factor of the original polynomial. To reduce this error, it is advisable to find the
roots in increasing order of magnitude.
Wilkinson\'s polynomial illustrates that high precision may be necessary when computing the
roots of a polynomial given its coefficients: the problem of finding the roots from the
coefficients is in general ill-conditioned.
The most simple method to find a single root fast is Newton\'s method. One can use Horner\'s
method twice to efficiently evaluate the value of the polynomial function and its first derivative;
this combination is called Birge–Vieta\'s method. This method provides quadratic convergence
for simple roots at the cost of two polynomial evaluations per step.
Closely related to Newton\'s method are Halley\'s method and Laguerre\'s method. Using one
additional Horner evaluation, the value of the second derivative is used to obtain methods of
cubic convergence order for simple roots. If one starts from a point x close to a root and uses the
same complexity of 6 function evaluations, these methods perform two steps with a residual of
O(|f(x)|9), compared to three steps of Newton\'s method with a reduction O(|f(x)|8), giving a
slight advantage to these methods.
When applying these methods to polynomials with real coefficients and real starting points,
Newton\'s and Halley\'s method stay inside the real number line. One has to choose complex
starting points to find complex roots. In contrast, the Laguerre method with a square root in its
evaluation will leave the real axis of its own accord.
Another class of methods is based on translating the problem of finding polynomial roots to the
problem of finding eigenvalues of the companion matrix of the polynomial. In principle, one can
use any eigenvalue algorithm to find the roots of the polynomial. However, for efficiency
reasons one prefers methods that employ the structure of the matrix, that is, can be implemented
in matrix-free form. Among these methods are the power method, whose application to the
transpose of the companion matrix is the classical Bernoulli\'s method to find the root of greatest
modulus. The inverse power method with shifts, which finds some smallest root first, is what
drives the complex (cpoly) variant of the Jenkins–Traub method and gives it its numerical
stability. Additionally, it is insensitive to multiple roots and has fast convergence with order
1
+
2.6
{\\displaystyle 1+\\Phi \\approx 2.6}
even in the presence of clustered roots. This fast convergence comes with a cost of three Horner
evaluations per step, resulting in a residual of O(|f(x)|2+3), which is slower than three steps of
New.
This document discusses roots of equations and methods to find them. It provides a brief history of solving quadratic equations from ancient Indian, Babylonian, Chinese, Greek and Indian mathematicians. It then describes three main methods to find roots: the bisection method, Regula-Falsi method, and Newton's Rapshon method. It explains each method in 1-2 sentences and provides examples of applications for the bisection method in shot detection in video content and locating periodic orbits in molecular systems. It also provides a brief example of applying the Regula-Falsi method to predict trace quantities of air pollutants from combustion reactions.
A Structured Approach To Solve The Inverse Eigenvalue Problem For A Beam With...Wendy Berg
This research article presents a method for solving the inverse eigenvalue problem of imposing two desired natural frequencies on a dynamical system consisting of an Euler-Bernoulli beam with a single attached mass. The authors develop equations of motion using the assumed modes method to model the beam and mass vibrations. Their method determines the magnitude and mounting position of the attached mass required to achieve the two target natural frequencies.
This document discusses numerical methods for finding the roots of nonlinear equations. It covers the bisection method, Newton-Raphson method, and their applications. The bisection method uses binary search to bracket the root within intervals that are repeatedly bisected until a solution is found. The Newton-Raphson method approximates the function as a linear equation to rapidly converge on roots. Examples and real-world applications are provided for both methods.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
Finite element method have many techniques that are used to design the structural elements like automobiles and building materials as well. we use different design software to get our simulated results at ansys, pro-e and matlab.we use these results for our real value problems.
The document discusses and compares several numerical methods for finding the root of a function, including the bisection method. The bisection method works by repeatedly bisecting an interval containing a root until the interval is smaller than a threshold. It converges linearly but is simple and guaranteed. Faster methods like Newton-Raphson require calculating derivatives but may not converge for all functions. The regula falsi and secant methods are alternatives that converge faster than bisection but more slowly than Newton-Raphson. Key factors in choosing a method include the nature of the function and desired precision versus speed of convergence.
This presentation summarizes Newton's method, an iterative method for finding approximations of the roots of a function. It is given by six group members and discusses the motivation, background, applicability, and advantages of the method. An example problem is provided to demonstrate the iterative process of applying Newton's method to find the root of a function. Programming and applications are briefly discussed along with potential failures if the starting point has a derivative of zero.
A Comparison Of Iterative Methods For The Solution Of Non-Linear Systems Of E...Stephen Faucher
This document compares iterative methods for solving nonlinear systems of equations, including Gauss-Seidel method. It describes Bisection method, Newton-Raphson method, Secant method, and False Position method. Bisection method converges linearly but is simple. Newton-Raphson converges faster if the initial guess is close to the root but may diverge otherwise. Secant method does not require derivatives but can fail if the function is flat. False Position combines features of Bisection and Secant methods. The document analyzes the algorithms and convergence properties of each method.
This document discusses roots of equations and methods to find them. It provides a brief history of solving quadratic equations from ancient Indian, Babylonian, Chinese, Greek and Indian mathematicians. It then describes three main methods to find roots: the bisection method, Regula-Falsi method, and Newton's Rapshon method. It explains each method in 1-2 sentences and provides examples of applications for the bisection method in shot detection in video content and locating periodic orbits in molecular systems. It also provides a brief example of applying the Regula-Falsi method to predict trace quantities of air pollutants from combustion reactions.
A Structured Approach To Solve The Inverse Eigenvalue Problem For A Beam With...Wendy Berg
This research article presents a method for solving the inverse eigenvalue problem of imposing two desired natural frequencies on a dynamical system consisting of an Euler-Bernoulli beam with a single attached mass. The authors develop equations of motion using the assumed modes method to model the beam and mass vibrations. Their method determines the magnitude and mounting position of the attached mass required to achieve the two target natural frequencies.
This document discusses numerical methods for finding the roots of nonlinear equations. It covers the bisection method, Newton-Raphson method, and their applications. The bisection method uses binary search to bracket the root within intervals that are repeatedly bisected until a solution is found. The Newton-Raphson method approximates the function as a linear equation to rapidly converge on roots. Examples and real-world applications are provided for both methods.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
Finite element method have many techniques that are used to design the structural elements like automobiles and building materials as well. we use different design software to get our simulated results at ansys, pro-e and matlab.we use these results for our real value problems.
The document discusses and compares several numerical methods for finding the root of a function, including the bisection method. The bisection method works by repeatedly bisecting an interval containing a root until the interval is smaller than a threshold. It converges linearly but is simple and guaranteed. Faster methods like Newton-Raphson require calculating derivatives but may not converge for all functions. The regula falsi and secant methods are alternatives that converge faster than bisection but more slowly than Newton-Raphson. Key factors in choosing a method include the nature of the function and desired precision versus speed of convergence.
This presentation summarizes Newton's method, an iterative method for finding approximations of the roots of a function. It is given by six group members and discusses the motivation, background, applicability, and advantages of the method. An example problem is provided to demonstrate the iterative process of applying Newton's method to find the root of a function. Programming and applications are briefly discussed along with potential failures if the starting point has a derivative of zero.
A Comparison Of Iterative Methods For The Solution Of Non-Linear Systems Of E...Stephen Faucher
This document compares iterative methods for solving nonlinear systems of equations, including Gauss-Seidel method. It describes Bisection method, Newton-Raphson method, Secant method, and False Position method. Bisection method converges linearly but is simple. Newton-Raphson converges faster if the initial guess is close to the root but may diverge otherwise. Secant method does not require derivatives but can fail if the function is flat. False Position combines features of Bisection and Secant methods. The document analyzes the algorithms and convergence properties of each method.
The document discusses various numerical methods for finding the roots or zeros of equations, where the root is the value of x that makes the equation f(x) equal to zero. It begins by defining roots and describing graphical and incremental search methods. It then covers closed methods like bisection and false position that evaluate the function within an interval known to contain the root. Open methods like Newton-Raphson, secant method, and fixed-point iteration are also discussed. The document provides examples of applying these root-finding methods and notes situations where roots could be missed. It concludes by listing references for further information.
This document discusses various methods for finding the roots of equations, including bracketing methods like bisection and false position, open methods like fixed point iteration and Newton-Raphson, and the secant method. It provides formulas and explanations of how each method works to successively approximate a root through iterative calculations. Examples are given of applying the methods to solve engineering problems involving equations of state.
The document discusses the Newton Raphson method for finding roots of equations. It describes how Isaac Newton and Joseph Raphson discovered the method in the 17th century. The method works by taking the derivative of the function and using it to calculate successive approximations that converge on a root. The document provides an example of using the method to find the root of a function and discusses advantages like fast convergence and requiring only an initial guess, as well as potential drawbacks such as failure to converge or slow convergence for roots with multiplicity greater than one.
This document discusses accurate quantum chemical models for large molecules. It describes composite approaches that combine results from smaller calculations to model larger systems. These include composite models that extrapolate results from smaller basis sets to the basis set limit for medium molecules. It also describes fragment-based methods that divide large molecules into smaller fragments and recombine the results to model intermolecular interactions, reducing computational cost. Several specific fragment-based methods are outlined, and the document concludes by comparing top-down and bottom-up fragmentation approaches.
The document discusses the Newton-Raphson method for finding the roots of functions. It describes how the method works by taking the tangent line at a current point and finding where it crosses zero to get the next estimate. The method converges quadratically near the root. Advantages are fast convergence and ability to refine roots from other methods. Disadvantages include needing the derivative and potential issues with initial guesses or non-convergence. Examples are provided to illustrate the method.
Newton's method is an iterative method for finding roots of functions. It requires calculating the derivative of the function. The method may fail to converge if the derivative does not exist at the root, is discontinuous at the root, or if the root has a multiplicity greater than one. It also may not converge if the starting point is too far from the root or if a stationary point is encountered. When it does converge, the rate is quadratic close to a simple root where the derivative is non-zero, but may be linear or fail to converge in other cases.
The document discusses different methods for finding the roots or zeros of equations, where the root is the value of x that makes the equation f(x) equal to 0. It describes graphical methods, incremental search methods like bisection which divides the interval in half each iteration, and open methods like Newton-Raphson that use calculus to home in on the root. It also addresses challenges in finding multiple roots where the function is tangent to the x-axis at the root.
This document discusses calculating the frequencies of quasinormal modes of black holes using semi-analytical methods. It summarizes that quasinormal modes are oscillations of black holes that emit gravitational waves. The WKB approximation and continued fractions method are used to calculate the frequencies. Results from the WKB method match other semi-analytic results for lower modes. The continued fractions method is also shown to work as another semi-analytic technique.
The document compares the convergence rates of the bisection, Newton-Raphson, and secant methods for finding roots of functions. It finds the root of the function f(x)=x-cos(x) on the interval [0,1] using each method. The bisection method converges at the 52nd iteration, while Newton-Raphson and secant methods converge to the exact root of 0.739085 with an error of 0 at the 8th and 6th iterations respectively. Therefore, the document concludes that the secant method is the most effective of the three for this problem.
1. There are conventional methods for finding multiple roots of equations, such as the Muller method and Bairstow method. The Muller method uses three initial values to obtain the coefficients of a parabola, which are then substituted into the quadratic formula to estimate the root where the parabola intersects the x-axis. The Bairstow method iteratively calculates the factors of a polynomial to determine its roots.
2. Complex roots of equations can be calculated using Euler's formula, which states that every complex number has exactly n complex roots. The formula generates each root by substituting different values for k between 0 and n-1. This allows polar forms of complex numbers to be substituted to find their roots
This document proposes a method for obtaining a sparse polynomial model from time series data. It uses an optimal minimal nonuniform time embedding to construct a time delay kernel from which a polynomial basis is built. A sparse model is then obtained by solving a regularized least squares problem that minimizes error while penalizing model complexity. The method is applied to generate a model of the Mackey-Glass chaotic system from time series data.
The Russian Doll Search algorithm improves upon the Depth First Branch and Bound algorithm for solving constraint optimization problems. It does this by performing n successive searches on nested subproblems, where n is the number of variables in the problem. Each search solves a subproblem involving a subset of the variables and records the optimal solution. This recorded information is then used to improve the lower bound estimate for partial assignments during subsequent searches on larger subproblems, allowing earlier pruning of search branches. On benchmark problems, this approach yields better results than a standard Depth First Branch and Bound.
The Muller method is a root-finding algorithm that estimates the root of a function by fitting a parabola through three points on the graph of the function. It works by first calculating the coefficients of a parabola that passes through three given points (x0, f(x0)), (x1, f(x1)), (x2, f(x2)). Then, the x-value where this parabola intersects the x-axis gives an estimated root. This process is repeated iteratively to converge on an accurate root. The document provides mathematical definitions of the Muller method and an example application with coefficients h = 0.1, x2 = 5, and x1 = 5.
Quasi newton artificial neural network training algorithmsMrinmoy Majumder
Quasi Newton Artificial Neural Network uses the quasi newton network training algorithms. This is another training algorithm which is used for weight update and production of new generation of solutions in a neural network based modeling system.
This document discusses numerical methods for finding roots of equations. It begins by introducing graphical and closed methods, including bisection and false position. It then covers open methods such as fixed point iteration and Newton-Raphson. Multiple roots and polynomial roots are also addressed. Methods for polynomials include Muller's method and Bairstow's method.
The Newton Raphson method is an iterative root-finding algorithm used to find the roots or solutions of nonlinear equations. It starts with an initial guess and iteratively refines it using the tangent line to the function to approximate the root, continuing until a desired level of accuracy is achieved. The formula for the Newton Raphson method is: x(n+1) = x(n) - f(x(n))/f'(x(n)). It has fast convergence compared to other root-finding methods and can provide accurate solutions for a wide range of functions. However, it may fail or diverge if the initial guess is far from the root.
This document summarizes a new stochastic optimization method called Complex Simultaneous Perturbation Stochastic Approximation (CSPSA) that can directly optimize real-valued functions of complex variables. CSPSA estimates the complex gradient of the target function within the field of complex numbers and generates a sequence of complex estimates that converges to the optimal solution. The method has advantages over existing approaches that optimize in the real domain, as calculations are simpler using complex variables and the complex structure can improve performance. Numerical tests on quantum state tomography demonstrate CSPSA achieves solutions orders of magnitude closer to the true minimum compared to other methods using the same resources.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
The document discusses von Neumann entropy in quantum computation. It provides definitions of key terms like von Neumann entropy, density matrix, and computational complexity theory. Von Neumann entropy extends concepts of classical entropy to quantum mechanics and characterizes the classical and quantum information capacities of an ensemble. It quantifies the degree of mixing of a quantum state and how much a state departs from a pure state. The von Neumann entropy of a system is computed using the density matrix and eigendecomposition of the system's quantum state.
Seven Requirements of CTC 1 Age 2 Relationship .pdfanubhavnigam2608
Seven Requirements of CTC: 1 Age 2 Relationship 3 Support 4 Dependent
Status 5 Citizenship 6 Length of Residency 7 Family Income 1. Age test
To qualify, a child must have been under age 17 (i.e., 16 years old or younger) at the end of the
tax year for which you claim the credit. 2. Relationship test The child must be your own
child, a stepchild, or a foster child placed with you by a court or authorized agency. An adopted
child is always treated as your own child. (\"An adopted child\" includes a child lawfully placed
with you for legal adoption, even if that adoption is not final by the end of the tax year.)You can
also claim your brother or sister, stepbrother, stepsister. And you can claim descendants of any of
these qualifying people—such as your nieces, nephews and grandchildren—if they meet all the
other tests. 3. Support test To qualify, the child cannot have provided more than half of his
or her own financial support during the tax year. 4. Dependent test You must claim the
child as a dependent on your tax return.Bear in mind that in order for you to claim a child as a
dependent, he or she must: 1) be your child (or adoptive or foster child), sibling, niece, nephew
or grandchild; 2) be under age 19, or under age 24 and a fulltime student for at least five months
of the year; or be permanently disabled, regardless of age; 3) have lived with you for more than
half the year; and 4) have provided no more than half his or her own support for the year. 5.
Citizenship test The child must be a U.S. citizen, a U.S. national or a U.S. resident alien. (For
tax purposes, the term \"U.S. national\" refers to individuals who were born in American Samoa
or in the Commonwealth of the Northern Mariana Islands.) 6. Residence test The child
must have lived with you for more than half of the tax year for which you claim the credit. There
are important exceptions, however: A child who was born (or died) during the tax year is
considered to have lived with you for the entire year.Temporary absences by you or the child for
special circumstances, such as school, vacation, business, medical care, military services or
detention in a juvenile facility, are counted as time the child lived with you. (There are also some
exceptions to the residency test for children of divorced or separated parents. For details, see the
instructions for Form 1040, lines 51 and 6c, or Form 1040A, lines 33 and 6c.) 7. Family
income test The child tax credit is reduced if your modified adjusted gross income (MAGI) is
above certain amounts, which are determined by your tax-filing status. In 2017, the phase out
threshold is $55,000 for married couples filing separately; $75,000 for single, head of household,
and qualifying widow or widower filers; and $110,000 for married couples filing jointly. For
each $1,000 of income above the threshold, your available child tax credit is reduced by $50.
Once all this requirements are met than further this question shall arise Tax Professional
shou.
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This document discusses calculating the frequencies of quasinormal modes of black holes using semi-analytical methods. It summarizes that quasinormal modes are oscillations of black holes that emit gravitational waves. The WKB approximation and continued fractions method are used to calculate the frequencies. Results from the WKB method match other semi-analytic results for lower modes. The continued fractions method is also shown to work as another semi-analytic technique.
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2. Complex roots of equations can be calculated using Euler's formula, which states that every complex number has exactly n complex roots. The formula generates each root by substituting different values for k between 0 and n-1. This allows polar forms of complex numbers to be substituted to find their roots
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The Muller method is a root-finding algorithm that estimates the root of a function by fitting a parabola through three points on the graph of the function. It works by first calculating the coefficients of a parabola that passes through three given points (x0, f(x0)), (x1, f(x1)), (x2, f(x2)). Then, the x-value where this parabola intersects the x-axis gives an estimated root. This process is repeated iteratively to converge on an accurate root. The document provides mathematical definitions of the Muller method and an example application with coefficients h = 0.1, x2 = 5, and x1 = 5.
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The Newton Raphson method is an iterative root-finding algorithm used to find the roots or solutions of nonlinear equations. It starts with an initial guess and iteratively refines it using the tangent line to the function to approximate the root, continuing until a desired level of accuracy is achieved. The formula for the Newton Raphson method is: x(n+1) = x(n) - f(x(n))/f'(x(n)). It has fast convergence compared to other root-finding methods and can provide accurate solutions for a wide range of functions. However, it may fail or diverge if the initial guess is far from the root.
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Seven Requirements of CTC 1 Age 2 Relationship .pdfanubhavnigam2608
Seven Requirements of CTC: 1 Age 2 Relationship 3 Support 4 Dependent
Status 5 Citizenship 6 Length of Residency 7 Family Income 1. Age test
To qualify, a child must have been under age 17 (i.e., 16 years old or younger) at the end of the
tax year for which you claim the credit. 2. Relationship test The child must be your own
child, a stepchild, or a foster child placed with you by a court or authorized agency. An adopted
child is always treated as your own child. (\"An adopted child\" includes a child lawfully placed
with you for legal adoption, even if that adoption is not final by the end of the tax year.)You can
also claim your brother or sister, stepbrother, stepsister. And you can claim descendants of any of
these qualifying people—such as your nieces, nephews and grandchildren—if they meet all the
other tests. 3. Support test To qualify, the child cannot have provided more than half of his
or her own financial support during the tax year. 4. Dependent test You must claim the
child as a dependent on your tax return.Bear in mind that in order for you to claim a child as a
dependent, he or she must: 1) be your child (or adoptive or foster child), sibling, niece, nephew
or grandchild; 2) be under age 19, or under age 24 and a fulltime student for at least five months
of the year; or be permanently disabled, regardless of age; 3) have lived with you for more than
half the year; and 4) have provided no more than half his or her own support for the year. 5.
Citizenship test The child must be a U.S. citizen, a U.S. national or a U.S. resident alien. (For
tax purposes, the term \"U.S. national\" refers to individuals who were born in American Samoa
or in the Commonwealth of the Northern Mariana Islands.) 6. Residence test The child
must have lived with you for more than half of the tax year for which you claim the credit. There
are important exceptions, however: A child who was born (or died) during the tax year is
considered to have lived with you for the entire year.Temporary absences by you or the child for
special circumstances, such as school, vacation, business, medical care, military services or
detention in a juvenile facility, are counted as time the child lived with you. (There are also some
exceptions to the residency test for children of divorced or separated parents. For details, see the
instructions for Form 1040, lines 51 and 6c, or Form 1040A, lines 33 and 6c.) 7. Family
income test The child tax credit is reduced if your modified adjusted gross income (MAGI) is
above certain amounts, which are determined by your tax-filing status. In 2017, the phase out
threshold is $55,000 for married couples filing separately; $75,000 for single, head of household,
and qualifying widow or widower filers; and $110,000 for married couples filing jointly. For
each $1,000 of income above the threshold, your available child tax credit is reduced by $50.
Once all this requirements are met than further this question shall arise Tax Professional
shou.
the solvents that are used to combine together to.pdfanubhavnigam2608
the solvents that are used to combine together to make the solvent pair should be
miscible. Hexane is nonpolar solvent, while water is a polar solvent. Thus, they are not miscible
and will stay as two layers after they are combined together. Thus, they cannot work together as
a solvent pair in recrystallization
Solution
the solvents that are used to combine together to make the solvent pair should be
miscible. Hexane is nonpolar solvent, while water is a polar solvent. Thus, they are not miscible
and will stay as two layers after they are combined together. Thus, they cannot work together as
a solvent pair in recrystallization.
The most simple property to calculate this is Ka.pdfanubhavnigam2608
The most simple property to calculate this is Ka*Kb=e-14 -->Ka=(e-14)/Kb Ka for
NH4+: Kb of NH3=(1.8e-5) --> Ka=(e-14)/(1.8e-5)=5.56e-10 Ka for CH3NH3+: Kb for
CH3NH2=4.4e-4 --> Ka=(e-14)/(4.4e-4)=2.27e-11 Kb for F-: Ka for HF=6.8e-4 --> Kb=(e-
14)/(6.8e-4)=1.47e-11
Solution
The most simple property to calculate this is Ka*Kb=e-14 -->Ka=(e-14)/Kb Ka for
NH4+: Kb of NH3=(1.8e-5) --> Ka=(e-14)/(1.8e-5)=5.56e-10 Ka for CH3NH3+: Kb for
CH3NH2=4.4e-4 --> Ka=(e-14)/(4.4e-4)=2.27e-11 Kb for F-: Ka for HF=6.8e-4 --> Kb=(e-
14)/(6.8e-4)=1.47e-11.
The images of the compounds are not visible. Kin.pdfanubhavnigam2608
The images of the compounds are not visible. Kindly post it in other question and i
will let you know.
Solution
The images of the compounds are not visible. Kindly post it in other question and i
will let you know..
Step1 Moles of water produced =18 Step2 The init.pdfanubhavnigam2608
Step1 Moles of water produced =18 Step2 The initial concentrations of octane and
O2 are not known ; If known ,conc of octane can be found out.
Solution
Step1 Moles of water produced =18 Step2 The initial concentrations of octane and
O2 are not known ; If known ,conc of octane can be found out..
This very short document appears to be discussing sulfur dioxide (SO2) as it contains the chemical formula twice with the word "Solution" in between. It seems to be stating that sulfur dioxide is the subject being discussed and that there may be a solution related to it, but provides no other details in the 3 word document.
scientific synonyms for height may be peak or tal.pdfanubhavnigam2608
scientific synonyms for height may be peak or tallness sometimes length(in case of
of a cube)
Solution
scientific synonyms for height may be peak or tallness sometimes length(in case of
of a cube).
This reaction has a rate law of R = k[NOCl2][NO] with a molecularity of 2. The rate (R) of the reaction is directly proportional to the concentrations of NOCl2 and NO, with the proportionality constant being the rate constant (k). This reaction requires collisions between two reactant molecules, NOCl2 and NO, in order for a reaction to occur.
In general, they are alkenes attached to electron.pdfanubhavnigam2608
In general, they are alkenes attached to electron-withdrawing groups (esters,
ketones, nitriles, nitros). so the ans is D.
Solution
In general, they are alkenes attached to electron-withdrawing groups (esters,
ketones, nitriles, nitros). so the ans is D..
This short document appears to be discussing a solution to a problem or issue numbered as I and II. It briefly mentions I and II at the beginning and end but does not provide any other context or details about the topic being discussed. The middle section indicates there is a solution presented but without more information it is difficult to understand what problem is being addressed or what the proposed solution entails.
Group 11 has three elements namely, Cu, Ag and Au.pdfanubhavnigam2608
Group 11 has three elements namely, Cu, Ag and Au out of these elements Cu
exhibits +1 n +2 oxidation states Ag exhibits only +1 n Au exhibits +3
Solution
Group 11 has three elements namely, Cu, Ag and Au out of these elements Cu
exhibits +1 n +2 oxidation states Ag exhibits only +1 n Au exhibits +3.
This very short document does not contain any substantive information to summarize in 3 sentences or less. It consists of only 3 words with no context provided.
This very short document does not contain enough substantive information to generate a multi-sentence summary. It appears to be an incomplete or placeholder document.
When a compound is completely oxidized with oxygen, the products wil.pdfanubhavnigam2608
When a compound is completely oxidized with oxygen, the products will be H2O and CO2
Solution
When a compound is completely oxidized with oxygen, the products will be H2O and CO2.
Calcium metal reacts with liquid water to produce.pdfanubhavnigam2608
Calcium metal reacts with liquid water to produce calcium hydroxidesolution and
hydrogen gas Ca (s) + 2 H2O (l) > Ca(OH)2 (l) + H2 (g) Ca(OH)2 + 2HCl --> CaCl2 + 2H2O
Solution
Calcium metal reacts with liquid water to produce calcium hydroxidesolution and
hydrogen gas Ca (s) + 2 H2O (l) > Ca(OH)2 (l) + H2 (g) Ca(OH)2 + 2HCl --> CaCl2 + 2H2O.
This very short document contains a single word - "Solution" - with no other context provided. It is not possible to provide a meaningful 3 sentence summary as there is insufficient information and content in the original document.
The mineralogical and structural adjustment of solid rocks to physic.pdfanubhavnigam2608
The mineralogical and structural adjustment of solid rocks to physical and chemical conditions
that have been imposed at depths below the near surface zones of weathering and diagenesis and
which differ from conditions under which the rocks in question originated.
The word \"Metamorphism\" comes from the Greek: meta = change, morph = form, so
metamorphism means to change form. In geology this refers to the changes in mineral
assemblage and texture that result from subjecting a rock to conditions such pressures,
temperatures, and chemical environments different from those under which the rock originally
formed.
There are two major kinds of metamorphism: regional and contact.
Regional metamorphism. Most metamorphic rocks are the result of regional metamorphism (also
called dynamothermal metamorphism). These rocks were typically exposed to tectonic forces
and associated high pressures and temperatures. They are usually foliated and deformed and
thought to be remnants of ancient mountain ranges.
Contact metamorphism. Contact metamorphism (also called thermal metamorphism) is the
process by which the country rock that surrounds a hot magma intrusion is metamorphosed by
the high heat flow coming from the intrusion. The zone of metamorphism that surrounds the
intrusion is called the halo (or aureole) and rarely extends more than 100 meters into the country
rock. Geostatic pressure is usually a minor factor, since contact metamorphism generally takes
place less than 10 kilometers from the surface.Metamorphism is defined as follows:
The mineralogical and structural adjustment of solid rocks to physical and chemical conditions
that have been imposed at depths below the near surface zones of weathering and diagenesis
and which differ from conditions under which the rocks in question originated.
The word \"Metamorphism\" comes from the Greek: meta = change, morph = form, so
metamorphism means to change form. In geology this refers to the changes in mineral
assemblage and texture that result from subjecting a rock to conditions such pressures,
temperatures, and chemical environments different from those under which the rock originally
formed.
There are two major kinds of metamorphism: regional and contact.
Regional metamorphism. Most metamorphic rocks are the result of regional metamorphism
(also called dynamothermal metamorphism). These rocks were typically exposed to tectonic
forces and associated high pressures and temperatures. They are usually foliated and deformed
and thought to be remnants of ancient mountain ranges.
Contact metamorphism. Contact metamorphism (also called thermal metamorphism) is the
process by which the country rock that surrounds a hot magma intrusion is metamorphosed by
the high heat flow coming from the intrusion. The zone of metamorphism that surrounds the
intrusion is called the halo (or aureole) and rarely extends more than 100 meters into the
country rock. Geostatic pressure is usually a minor factor, since contact metamorp.
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Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
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Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
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The general idea is to find a root of the polynomial and then apply .pdf
1. The general idea is to find a root of the polynomial and then apply Horner's method to remove
the corresponding factor according to the Ruffini rule.
This iterative scheme is numerically unstable; the approximation errors accumulate during the
successive factorizations, so that the last roots are determined with a polynomial that deviates
widely from a factor of the original polynomial. To reduce this error, it is advisable to find the
roots in increasing order of magnitude.
Wilkinson's polynomial illustrates that high precision may be necessary when computing the
roots of a polynomial given its coefficients: the problem of finding the roots from the
coefficients is in general ill-conditioned.
The most simple method to find a single root fast is Newton's method. One can use Horner's
method twice to efficiently evaluate the value of the polynomial function and its first derivative;
this combination is called Birge–Vieta's method. This method provides quadratic convergence
for simple roots at the cost of two polynomial evaluations per step.
Closely related to Newton's method are Halley's method and Laguerre's method. Using one
additional Horner evaluation, the value of the second derivative is used to obtain methods of
cubic convergence order for simple roots. If one starts from a point x close to a root and uses the
same complexity of 6 function evaluations, these methods perform two steps with a residual of
O(|f(x)|9), compared to three steps of Newton's method with a reduction O(|f(x)|8), giving a
slight advantage to these methods.
When applying these methods to polynomials with real coefficients and real starting points,
Newton's and Halley's method stay inside the real number line. One has to choose complex
starting points to find complex roots. In contrast, the Laguerre method with a square root in its
evaluation will leave the real axis of its own accord.
Another class of methods is based on translating the problem of finding polynomial roots to the
problem of finding eigenvalues of the companion matrix of the polynomial. In principle, one can
use any eigenvalue algorithm to find the roots of the polynomial. However, for efficiency
reasons one prefers methods that employ the structure of the matrix, that is, can be implemented
in matrix-free form. Among these methods are the power method, whose application to the
transpose of the companion matrix is the classical Bernoulli's method to find the root of greatest
modulus. The inverse power method with shifts, which finds some smallest root first, is what
drives the complex (cpoly) variant of the Jenkins–Traub method and gives it its numerical
stability. Additionally, it is insensitive to multiple roots and has fast convergence with order
2. 1
+
2.6
{displaystyle 1+Phi approx 2.6}
even in the presence of clustered roots. This fast convergence comes with a cost of three Horner
evaluations per step, resulting in a residual of O(|f(x)|2+3), which is slower than three steps of
Newton's method.
Finding roots in pairs[edit]
If the given polynomial only has real coefficients, one may wish to avoid computations with
complex numbers. To that effect, one has to find quadratic factors for pairs of conjugate complex
roots. The application of the multi-dimensional Newton's method to this task results in
Bairstow's method. In the framework of inverse power iterations of the companion matrix, the
double shift method of Francis results in the real (rpoly) variant of the Jenkins–Traub method.
Finding all roots at once[edit]
The simple Durand–Kerner and the slightly more complicated Aberth methods simultaneously
find all of the roots using only simple complex number arithmetic. Accelerated algorithms for
multi-point evaluation and interpolation similar to the fast Fourier transform can help speed them
up for large degrees of the polynomial. It is advisable to choose an asymmetric, but evenly
distributed set of initial points.
Another method with this style is the Dandelin–Gräffe method (sometimes also ascribed to
Lobachevsky), which uses polynomial transformations to repeatedly and implicitly square the
roots. This greatly magnifies variances in the roots. Applying Viète's formulas, one obtains easy
approximations for the modulus of the roots, and with some more effort, for the roots
themselves.
Exclusion and enclosure methods[edit]
Several fast tests exist that tell if a segment of the real line or a region of the complex plane
contains no roots. By bounding the modulus of the roots and recursively subdividing the initial
region indicated by these bounds, one can isolate small regions that may contain roots and then
apply other methods to locate them exactly.
All these methods require to find the coefficients of shifted and scaled versions of the
polynomial. For large degrees, FFT-based accelerated methods become viable.
3. For real roots, Sturm's theorem and Descartes' rule of signs with its extension in the
Budan–Fourier theorem provide guides to locating and separating roots. This plus interval
arithmetic combined with Newton's method yields robust and fast algorithms.
The Lehmer–Schur algorithm uses the Schur–Cohn test for circles, Wilf's global bisection
algorithm uses a winding number computation for rectangular regions in the complex plane.
The splitting circle method uses FFT-based polynomial transformations to find large-degree
factors corresponding to clusters of roots. The precision of the factorization is maximized using a
Newton-type iteration. This method is useful for finding the roots of polynomials of high degree
to arbitrary precision; it has almost optimal complexity in this setting.
Method based on the Budan–Fourier theorem or Sturm chains[edit]
The methods in this class give for polynomials with rational coefficients, and when carried out
in rational arithmetic, provably complete enclosures of all real roots by rational intervals. The
test of an interval for real roots using Budan's theorem is computationally simple but may yield
false positive results. However, these will be reliably detected in the following transformation
and refinement of the interval. The test based on Sturm chains is computationally more involved
but gives always the exact number of real roots in the interval.
The algorithm for isolating the roots, using Descartes' rule of signs and Vincent's theorem, had
been originally called modified Uspensky's algorithm by its inventors Collins and Akritas.[1]
After going through names like "Collins–Akritas method" and "Descartes' method" (too
confusing if ones considers Fourier's article[2]), it was finally François Boulier, of Lille
University, who gave it the name Vincent–Collins–Akritas (VCA) method,[3] p. 24, based on
"Uspensky's method" not existing[4] and neither does "Descartes' method".[5] This
algorithm has been improved by Rouillier and Zimmerman,[6] and the resulting implementation
is, to date,[when?] the fastest bisection method. It has the same worst case complexity as the
Sturm algorithm, but is almost always much faster. It is the default algorithm of Maple root-
finding function fsolve. Another method based on Vincent's theorem is the
Vincent–Akritas–Strzeboski (VAS) method;[7] it has been shown[8] that the VAS (continued
fractions) method is faster than the fastest implementation of the VCA (bisection) method,[6]
which was independently confirmed elsewhere;[9] more precisely, for the Mignotte polynomials
of high degree, VAS is about 50,000 times faster than the fastest implementation of VCA. VAS
is the default algorithm for root isolation in Mathematica, Sage, SymPy, Xcas. See Budan's
theorem for a description of the historical background of these methods. For a comparison
between Sturm's method and VAS, use the functions realroot(poly) and time(realroot(poly)) of
Xcas. By default, to isolate the real roots of poly realroot uses the VAS method; to use Sturm's
method, write realroot(sturm, poly). See also the External links for a pointer to an
iPhone/iPod/iPad application that does the same thing.
4. Solution
The general idea is to find a root of the polynomial and then apply Horner's method to remove
the corresponding factor according to the Ruffini rule.
This iterative scheme is numerically unstable; the approximation errors accumulate during the
successive factorizations, so that the last roots are determined with a polynomial that deviates
widely from a factor of the original polynomial. To reduce this error, it is advisable to find the
roots in increasing order of magnitude.
Wilkinson's polynomial illustrates that high precision may be necessary when computing the
roots of a polynomial given its coefficients: the problem of finding the roots from the
coefficients is in general ill-conditioned.
The most simple method to find a single root fast is Newton's method. One can use Horner's
method twice to efficiently evaluate the value of the polynomial function and its first derivative;
this combination is called Birge–Vieta's method. This method provides quadratic convergence
for simple roots at the cost of two polynomial evaluations per step.
Closely related to Newton's method are Halley's method and Laguerre's method. Using one
additional Horner evaluation, the value of the second derivative is used to obtain methods of
cubic convergence order for simple roots. If one starts from a point x close to a root and uses the
same complexity of 6 function evaluations, these methods perform two steps with a residual of
O(|f(x)|9), compared to three steps of Newton's method with a reduction O(|f(x)|8), giving a
slight advantage to these methods.
When applying these methods to polynomials with real coefficients and real starting points,
Newton's and Halley's method stay inside the real number line. One has to choose complex
starting points to find complex roots. In contrast, the Laguerre method with a square root in its
evaluation will leave the real axis of its own accord.
Another class of methods is based on translating the problem of finding polynomial roots to the
problem of finding eigenvalues of the companion matrix of the polynomial. In principle, one can
use any eigenvalue algorithm to find the roots of the polynomial. However, for efficiency
reasons one prefers methods that employ the structure of the matrix, that is, can be implemented
in matrix-free form. Among these methods are the power method, whose application to the
transpose of the companion matrix is the classical Bernoulli's method to find the root of greatest
modulus. The inverse power method with shifts, which finds some smallest root first, is what
drives the complex (cpoly) variant of the Jenkins–Traub method and gives it its numerical
stability. Additionally, it is insensitive to multiple roots and has fast convergence with order
5. 1
+
2.6
{displaystyle 1+Phi approx 2.6}
even in the presence of clustered roots. This fast convergence comes with a cost of three Horner
evaluations per step, resulting in a residual of O(|f(x)|2+3), which is slower than three steps of
Newton's method.
Finding roots in pairs[edit]
If the given polynomial only has real coefficients, one may wish to avoid computations with
complex numbers. To that effect, one has to find quadratic factors for pairs of conjugate complex
roots. The application of the multi-dimensional Newton's method to this task results in
Bairstow's method. In the framework of inverse power iterations of the companion matrix, the
double shift method of Francis results in the real (rpoly) variant of the Jenkins–Traub method.
Finding all roots at once[edit]
The simple Durand–Kerner and the slightly more complicated Aberth methods simultaneously
find all of the roots using only simple complex number arithmetic. Accelerated algorithms for
multi-point evaluation and interpolation similar to the fast Fourier transform can help speed them
up for large degrees of the polynomial. It is advisable to choose an asymmetric, but evenly
distributed set of initial points.
Another method with this style is the Dandelin–Gräffe method (sometimes also ascribed to
Lobachevsky), which uses polynomial transformations to repeatedly and implicitly square the
roots. This greatly magnifies variances in the roots. Applying Viète's formulas, one obtains easy
approximations for the modulus of the roots, and with some more effort, for the roots
themselves.
Exclusion and enclosure methods[edit]
Several fast tests exist that tell if a segment of the real line or a region of the complex plane
contains no roots. By bounding the modulus of the roots and recursively subdividing the initial
region indicated by these bounds, one can isolate small regions that may contain roots and then
apply other methods to locate them exactly.
6. All these methods require to find the coefficients of shifted and scaled versions of the
polynomial. For large degrees, FFT-based accelerated methods become viable.
For real roots, Sturm's theorem and Descartes' rule of signs with its extension in the
Budan–Fourier theorem provide guides to locating and separating roots. This plus interval
arithmetic combined with Newton's method yields robust and fast algorithms.
The Lehmer–Schur algorithm uses the Schur–Cohn test for circles, Wilf's global bisection
algorithm uses a winding number computation for rectangular regions in the complex plane.
The splitting circle method uses FFT-based polynomial transformations to find large-degree
factors corresponding to clusters of roots. The precision of the factorization is maximized using a
Newton-type iteration. This method is useful for finding the roots of polynomials of high degree
to arbitrary precision; it has almost optimal complexity in this setting.
Method based on the Budan–Fourier theorem or Sturm chains[edit]
The methods in this class give for polynomials with rational coefficients, and when carried out
in rational arithmetic, provably complete enclosures of all real roots by rational intervals. The
test of an interval for real roots using Budan's theorem is computationally simple but may yield
false positive results. However, these will be reliably detected in the following transformation
and refinement of the interval. The test based on Sturm chains is computationally more involved
but gives always the exact number of real roots in the interval.
The algorithm for isolating the roots, using Descartes' rule of signs and Vincent's theorem, had
been originally called modified Uspensky's algorithm by its inventors Collins and Akritas.[1]
After going through names like "Collins–Akritas method" and "Descartes' method" (too
confusing if ones considers Fourier's article[2]), it was finally François Boulier, of Lille
University, who gave it the name Vincent–Collins–Akritas (VCA) method,[3] p. 24, based on
"Uspensky's method" not existing[4] and neither does "Descartes' method".[5] This
algorithm has been improved by Rouillier and Zimmerman,[6] and the resulting implementation
is, to date,[when?] the fastest bisection method. It has the same worst case complexity as the
Sturm algorithm, but is almost always much faster. It is the default algorithm of Maple root-
finding function fsolve. Another method based on Vincent's theorem is the
Vincent–Akritas–Strzeboski (VAS) method;[7] it has been shown[8] that the VAS (continued
fractions) method is faster than the fastest implementation of the VCA (bisection) method,[6]
which was independently confirmed elsewhere;[9] more precisely, for the Mignotte polynomials
of high degree, VAS is about 50,000 times faster than the fastest implementation of VCA. VAS
is the default algorithm for root isolation in Mathematica, Sage, SymPy, Xcas. See Budan's
theorem for a description of the historical background of these methods. For a comparison
between Sturm's method and VAS, use the functions realroot(poly) and time(realroot(poly)) of
Xcas. By default, to isolate the real roots of poly realroot uses the VAS method; to use Sturm's
7. method, write realroot(sturm, poly). See also the External links for a pointer to an
iPhone/iPod/iPad application that does the same thing.