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SIAM Journal on Numerical Analysis, 1982
A classical algorithm for solving the system of nonlinear equations $F(x) = 0$ is Newton’s method \[ x_{k + 1} = x_k + s_k ,\quad {\text{where }}F'(x_k )s_k = - F(x_k ),\quad x_0 {\text{ given}}.\]...
Dembo, Ron S. +2 more
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A classical algorithm for solving the system of nonlinear equations $F(x) = 0$ is Newton’s method \[ x_{k + 1} = x_k + s_k ,\quad {\text{where }}F'(x_k )s_k = - F(x_k ),\quad x_0 {\text{ given}}.\]...
Dembo, Ron S. +2 more
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2004
As a basic application of the derivative, we study Newton’s method for computing roots of an equation f (x) = 0. Newton’s method is one of the corner-stones of constructive mathematics. As a preparation we start out using the concept of derivative to analyze the convergence of Fixed Point Iteration.
Kenneth Eriksson +2 more
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As a basic application of the derivative, we study Newton’s method for computing roots of an equation f (x) = 0. Newton’s method is one of the corner-stones of constructive mathematics. As a preparation we start out using the concept of derivative to analyze the convergence of Fixed Point Iteration.
Kenneth Eriksson +2 more
+4 more sources
2019
Newton’s method was invented by Newton to solve the nonlinear one-dimensional problem. Later it was extended to solve multivariable nonlinear optimization problems. It is well known that the method of steepest descent uses only the first-order derivative (gradient).
Shashi Kant Mishra, Bhagwat Ram
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Newton’s method was invented by Newton to solve the nonlinear one-dimensional problem. Later it was extended to solve multivariable nonlinear optimization problems. It is well known that the method of steepest descent uses only the first-order derivative (gradient).
Shashi Kant Mishra, Bhagwat Ram
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1975
Many of the equations arising in practical problems are of a type difficult or impossible to solve by the standard algebraic methods. For example, the equations: $$ 2\sin {\kern 1pt} x - x = 0{\kern 1pt} {\kern 1pt} {e^x} - 2x - 1 = 0,{\kern 1pt} {\kern 1pt} {x^6} - 3x + 1 = 0 $$ have roots which we may estimate, by graphing the functions and ...
Brian Knight, Roger Adams
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Many of the equations arising in practical problems are of a type difficult or impossible to solve by the standard algebraic methods. For example, the equations: $$ 2\sin {\kern 1pt} x - x = 0{\kern 1pt} {\kern 1pt} {e^x} - 2x - 1 = 0,{\kern 1pt} {\kern 1pt} {x^6} - 3x + 1 = 0 $$ have roots which we may estimate, by graphing the functions and ...
Brian Knight, Roger Adams
openaire +1 more source