title: Richard Kiel

text: The James Bond film producers spotted Kiel in "Barbary Coast", and thought he was ideal for the role of Jaws in "The Spy Who Loved Me" (1977). He was one of the few Bond villains to appear in two Bond films, later appearing in "Moonraker" (1979). He was often shot with his mouth closed or briefly showing his dangerous smile as he admitted the mouthpiece to simulate the metal teeth was extremely painful to wear and could only be used for a few minutes on every take. Because he suffered from acrophobia (fear of heights), a stunt double was used during the cable car stunt scenes because Kiel refused to be filmed on the top of a cable car more than above the ground. He reprised his role of Jaws in the video game called "", supplying his voice and likeness. This was his second outing as a metal-toothed villain; a year before being cast in "The Spy Who Loved Me", he had played Reace in the comedy-thriller film "Silver Streak" (1976). He used his size for comedic effect, as the "best-dressed giant" Mr. Eddie, in "So Fine" with Ryan O'Neal. Kiel also starred in the film "Pale Rider" (1985). Acting as the main antagonist's henchman, he redeems his character's status by saving the hero from a gunshot to the back.


title: Richard Kiel

text: Although earlier roles had offered him little dialogue, his role in "Happy Gilmore" (1996) was quite the opposite. As Mr. Larson, Happy Gilmore's former employer, Kiel exchanges several one-liners with both Adam Sandler's Happy and Christopher McDonald's Shooter McGavin. Kiel took a quieter profile after "Happy Gilmore"s release, but left semi-retirement to record a role for "Tangled" (2010). In the acclaimed animated Disney film, he portrayed Vlad, a surprisingly softhearted thug who collects ceramic unicorns.


title: Richard Kiel

text: Kiel's first marriage was to Faye Daniels in 1960. They divorced in the early 1970s. He later married Diane Rogers, who was 5 ft 1 in (154 cm) tall. Their marriage lasted for 40 years, until his death. They had four children and nine grandchildren.


title: Richard Kiel

text: Kiel co-authored a biography of the abolitionist Cassius Marcellus Clay titled "Kentucky Lion". In 2002, Kiel published his autobiography, "Making It Big in the Movies". Kiel was also a born-again Christian. His website states his religious conversion helped him to overcome alcoholism.


title: Richard Kiel

text: On September 10, 2014, three days short of his 75th birthday, Kiel died at St. Agnes Medical Center in Fresno, California, of a heart attack. The week before, he had been admitted to the hospital for a broken leg.


title: Fisting

text: Fisting, handballing, fist-fucking, brachiovaginal, or brachioproctic insertion is a sexual activity that involves inserting a hand into the vagina or rectum. Once insertion is complete, the fingers are either clenched into a fist or kept straight. Fisting may be performed without a partner, but it is most often a partnered activity.


title: Fisting

text: Fisting's emergence as a popular sexual practice is commonly attributed to gay male culture and it may not have existed until the twentieth century. Robert Morgan Lawrence, a sex educator, however, believes the practice dates back thousands of years. The most famous fisting club in the world was the Catacombs, located in San Francisco, which operated during the 1970s and 1980s. The Handball Express was another such club. Crisco was commonly used as a lubricant, before more specialized personal lubricants became available.


title: Fisting

text: In the 1980s, it was assumed that unprotected fisting—which often produces small injuries to the anus, permitting microorganisms access to the blood—was an easy route for transmission of HIV. This, combined with sexual squeamishness towards the public fisting culture in gay establishments of San Francisco, led gay writer Randy Shilts to successfully campaign for the closure of venues, such as gay bathhouses and sex clubs, that openly permitted it. Fisting gradually returned as a sexual practice over the next 30 years.


title: Fisting

text: The San Francisco South of Market Leather History Alley consists of four works of art along Ringold Alley honoring leather culture; it opened in 2017. One of the works of art is metal bootprints along the curb which honor 28 people who were an important part of the leather communities of San Francisco; those honored include Steve McEachern, owner of the fisting club the Catacombs, and Bert Herman, leader of the fisting community, author, and publisher.


title: Fisting

text: The "silent duck", also called "duck-billing", is the technique often used in which the person engaging in hand insertion shapes the hand to resemble a duck's beak. Typically, fisting does not involve forcing the clenched fist into the vagina or rectum; this is a practice called "punching". Instead, all five fingers are kept straight and held as close together as possible (forming the beak-like "duck"), then slowly inserted into a well lubricated vagina or rectum.


title: Fisting

text: In more vigorous forms of fisting, such as "punching" or "punchfisting", a fully clenched fist may be inserted and withdrawn slowly.


title: Fisting

text: Fistees who are more experienced may take two fists (double-fisting). In the case of double-fisting, pleasure is derived more from the stretching of the anus or vagina than from the thrusting (in-and-out) movement of hands.


title: Fisting

text: Fisting can cause laceration or perforation of the vagina, perineum, rectum, or colon, resulting in serious injury and even death. In addition, sexual activities that cause air to enter the vagina can lead to a fatal air embolism, and the risk is probably even higher during pregnancy.


title: Fisting

text: Anal fisting carries risks of colorectal perforation; participants are advised to use latex gloves and lubricant, and designate a safeword, the utterance of which will call an immediate halt to the activity. The practice, along with the insertion of hard objects into the anus, has been significantly related to the traumatization of the rectal mucosa in increasing the likelihood of infection, including Hepatitis B.


title: Fisting

text: In the United Kingdom fisting is legal to perform; however, the Crown Prosecution Service considers publication of fisting material to be grounds for prosecution under the Obscene Publications Act 1959. Despite the CPS's guidance, in the case of "R v Peacock" in 2012, Michael Peacock was found not guilty of breaching the Obscene Publications Act for selling DVDs containing anal fisting. There was also an unsuccessful extreme pornography prosecution in 2012 where it was argued by the prosecution that images of anal fisting constitute extreme pornography and thus are illegal to possess because the act is "likely to result in serious injury to a person's anus, breasts or genitals".


title: Fisting

text: In the United States, according to attorney Allan Gelbard there is nothing "per se illegal" about any pornography so long as it is not child pornography, however, it can be found to be obscene. Fisting is on the Cambria List, a list of sex acts which may be prosecutable under U.S. obscenity law, created by lawyer Paul Cambria in 2001. The list is intended to act as guidelines to help producers avoid obscenity lawsuits.


title: Newton's method

text: In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then


title: Newton's method

text: is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as


title: Newton's method

text: until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.


title: Newton's method

text: The idea is to start with an initial guess, then to approximate the function by its tangent line, and finally to compute the -intercept of this tangent line. This -intercept will typically be a better approximation to the original function's root than the first guess, and the method can be iterated.


title: Newton's method

text: We start the process with some arbitrary initial value . (The closer to the zero, the better. But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small interval by appealing to the intermediate value theorem.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that . Furthermore, for a zero of multiplicity 1, the convergence is at least quadratic (see rate of convergence) in a neighbourhood of the zero, which intuitively means that the number of correct digits roughly doubles in every step. More details can be found in the analysis section below.


title: Newton's method

text: Householder's methods are similar but have higher order for even faster convergence. However, the extra computations required for each step can slow down the overall performance relative to Newton's method, particularly if or its derivatives are computationally expensive to evaluate.


title: Newton's method

text: The name "Newton's method" is derived from Isaac Newton's description of a special case of the method in "De analysi per aequationes numero terminorum infinitas" (written in 1669, published in 1711 by William Jones) and in "De metodis fluxionum et serierum infinitarum" (written in 1671, translated and published as "Method of Fluxions" in 1736 by John Colson). However, his method differs substantially from the modern method given above. Newton applied the method only to polynomials, starting with an initial root estimate and extracting a sequence of error corrections. He used each correction to rewrite the polynomial in terms of the remaining error, and then solved for a new correction by neglecting higher-degree terms. He did not explicitly connect the method with derivatives or present a general formula. Newton applied this method to both numerical and algebraic problems, producing Taylor series in the latter case.


title: Newton's method

text: Newton may have derived his method from a similar but less precise method by Vieta. The essence of Vieta's method can be found in the work of the Persian mathematician Sharaf al-Din al-Tusi, while his successor Jamshīd al-Kāshī used a form of Newton's method to solve to find roots of (Ypma 1995). A special case of Newton's method for calculating square roots was known since ancient times and is often called the Babylonian method.


title: Newton's method

text: Newton's method was used by 17th-century Japanese mathematician Seki Kōwa to solve single-variable equations, though the connection with calculus was missing.


title: Newton's method

text: Newton's method was first published in 1685 in "A Treatise of Algebra both Historical and Practical" by John Wallis. In 1690, Joseph Raphson published a simplified description in "Analysis aequationum universalis". Raphson also applied the method only to polynomials, but he avoided Newton's tedious rewriting process by extracting each successive correction from the original polynomial. This allowed him to derive a reusable iterative expression for each problem. Finally, in 1740, Thomas Simpson described Newton's method as an iterative method for solving general nonlinear equations using calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero.


title: Newton's method

text: Arthur Cayley in 1879 in "The Newton–Fourier imaginary problem" was the first to notice the difficulties in generalizing Newton's method to complex roots of polynomials with degree greater than 2 and complex initial values. This opened the way to the study of the theory of iterations of rational functions.


title: Newton's method

text: Newton's method is a powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties with the method.


title: Newton's method

text: Newton's method requires that the derivative can be calculated directly. An analytical expression for the derivative may not be easily obtainable or could be expensive to evaluate. In these situations, it may be appropriate to approximate the derivative by using the slope of a line through two nearby points on the function. Using this approximation would result in something like the secant method whose convergence is slower than that of Newton's method.


title: Newton's method

text: It is important to review the proof of quadratic convergence of Newton's method before implementing it. Specifically, one should review the assumptions made in the proof. For situations where the method fails to converge, it is because the assumptions made in this proof are not met.


title: Newton's method

text: If the first derivative is not well behaved in the neighborhood of a particular root, the method may overshoot, and diverge from that root. An example of a function with one root, for which the derivative is not well behaved in the neighborhood of the root, is


title: Newton's method

text: for which the root will be overshot and the sequence of will diverge. For , the root will still be overshot, but the sequence will oscillate between two values. For , the root will still be overshot but the sequence will converge, and for the root will not be overshot at all.


title: Newton's method

text: In some cases, Newton's method can be stabilized by using successive over-relaxation, or the speed of convergence can be increased by using the same method.


title: Newton's method

text: If a stationary point of the function is encountered, the derivative is zero and the method will terminate due to division by zero.


title: Newton's method

text: A large error in the initial estimate can contribute to non-convergence of the algorithm. To overcome this problem one can often linearize the function that is being optimized using calculus, logs, differentials, or even using evolutionary algorithms, such as the stochastic tunneling. Good initial estimates lie close to the final globally optimal parameter estimate. In nonlinear regression, the sum of squared errors (SSE) is only "close to" parabolic in the region of the final parameter estimates. Initial estimates found here will allow the Newton–Raphson method to quickly converge. It is only here that the Hessian matrix of the SSE is positive and the first derivative of the SSE is close to zero.


title: Newton's method

text: In a robust implementation of Newton's method, it is common to place limits on the number of iterations, bound the solution to an interval known to contain the root, and combine the method with a more robust root finding method.


title: Newton's method

text: If the root being sought has multiplicity greater than one, the convergence rate is merely linear (errors reduced by a constant factor at each step) unless special steps are taken. When there are two or more roots that are close together then it may take many iterations before the iterates get close enough to one of them for the quadratic convergence to be apparent. However, if the multiplicity of the root is known, the following modified algorithm preserves the quadratic convergence rate:


title: Newton's method

text: This is equivalent to using successive over-relaxation. On the other hand, if the multiplicity of the root is not known, it is possible to estimate after carrying out one or two iterations, and then use that value to increase the rate of convergence.


title: Newton's method

text: If the multiplicity of the root is finite then will have a root at the same location with multiplicity 1. Applying Newton's method to find the root of recovers quadratic convergence in many cases although it generally involves the second derivative of . In a particularly simple case, if then and Newton's method finds the root in a single iteration with


title: Newton's method

text: If is continuously differentiable and its derivative is nonzero at , then there exists a neighborhood of such that for all starting values in that neighborhood, the sequence will converge to .


title: Newton's method

text: If the function is continuously differentiable and its derivative is not 0 at and it has a second derivative at then the convergence is quadratic or faster. If the second derivative is not 0 at then the convergence is merely quadratic. If the third derivative exists and is bounded in a neighborhood of , then:


title: Newton's method

text: If the derivative is 0 at , then the convergence is usually only linear. Specifically, if is twice continuously differentiable, and , then there exists a neighborhood of such that, for all starting values in that neighborhood, the sequence of iterates converges linearly, with rate . Alternatively, if and for ,  in a neighborhood of , being a zero of multiplicity , and if , then there exists a neighborhood of such that, for all starting values in that neighborhood, the sequence of iterates converges linearly.


title: Newton's method

text: In practice, these results are local, and the neighborhood of convergence is not known in advance. But there are also some results on global convergence: for instance, given a right neighborhood of , if is twice differentiable in and if , in , then, for each in the sequence is monotonically decreasing to .


title: Newton's method

text: According to Taylor's theorem, any function which has a continuous second derivative can be represented by an expansion about a point that is close to a root of . Suppose this root is . Then the expansion of about is:


title: Newton's method

text: Equation () shows that the order of convergence is at least quadratic if the following conditions are satisfied:


title: Newton's method

text: The disjoint subsets of the basins of attraction—the regions of the real number line such that within each region iteration from any point leads to one particular root—can be infinite in number and arbitrarily small. For example, for the function , the following initial conditions are in successive basins of attraction:


title: Newton's method

text: Newton's method is only guaranteed to converge if certain conditions are satisfied. If the assumptions made in the proof of quadratic convergence are met, the method will converge. For the following subsections, failure of the method to converge indicates that the assumptions made in the proof were not met.


title: Newton's method

text: In some cases the conditions on the function that are necessary for convergence are satisfied, but the point chosen as the initial point is not in the interval where the method converges. This can happen, for example, if the function whose root is sought approaches zero asymptotically as goes to or . In such cases a different method, such as bisection, should be used to obtain a better estimate for the zero to use as an initial point.


title: Newton's method

text: It has a maximum at and solutions of at . If we start iterating from the stationary point (where the derivative is zero), will be undefined, since the tangent at is parallel to the -axis:


title: Newton's method

text: The same issue occurs if, instead of the starting point, any iteration point is stationary. Even if the derivative is small but not zero, the next iteration will be a far worse approximation.


title: Newton's method

text: and take 0 as the starting point. The first iteration produces 1 and the second iteration returns to 0 so the sequence will alternate between the two without converging to a root. In fact, this 2-cycle is stable: there are neighborhoods around 0 and around 1 from which all points iterate asymptotically to the 2-cycle (and hence not to the root of the function). In general, the behavior of the sequence can be very complex (see Newton fractal). The real solution of this equation is ….


title: Newton's method

text: If the function is not continuously differentiable in a neighborhood of the root then it is possible that Newton's method will always diverge and fail, unless the solution is guessed on the first try.


title: Newton's method

text: A simple example of a function where Newton's method diverges is trying to find the cube root of zero. The cube root is continuous and infinitely differentiable, except for , where its derivative is undefined:


title: Newton's method

text: The algorithm overshoots the solution and lands on the other side of the -axis, farther away than it initially was; applying Newton's method actually doubles the distances from the solution at each iteration.


title: Newton's method

text: In fact, the iterations diverge to infinity for every , where . In the limiting case of (square root), the iterations will alternate indefinitely between points and , so they do not converge in this case either.


title: Newton's method

text: If the derivative is not continuous at the root, then convergence may fail to occur in any neighborhood of the root. Consider the function


title: Newton's method

text: Within any neighborhood of the root, this derivative keeps changing sign as approaches 0 from the right (or from the left) while for .


title: Newton's method

text: So is unbounded near the root, and Newton's method will diverge almost everywhere in any neighborhood of it, even though:


title: Newton's method

text: In some cases the iterates converge but do not converge as quickly as promised. In these cases simpler methods converge just as quickly as Newton's method.


title: Newton's method

text: which has approximately times as many bits of precision as has. This is less than the 2 times as many which would be required for quadratic convergence. So the convergence of Newton's method (in this case) is not quadratic, even though: the function is continuously differentiable everywhere; the derivative is not zero at the root; and is infinitely differentiable except at the desired root.


title: Newton's method

text: When dealing with complex functions, Newton's method can be directly applied to find their zeroes. Each zero has a basin of attraction in the complex plane, the set of all starting values that cause the method to converge to that particular zero. These sets can be mapped as in the image shown. For many complex functions, the boundaries of the basins of attraction are fractals.


title: Newton's method

text: In some cases there are regions in the complex plane which are not in any of these basins of attraction, meaning the iterates do not converge. For example, if one uses a real initial condition to seek a root of , all subsequent iterates will be real numbers and so the iterations cannot converge to either root, since both roots are non-real. In this case almost all real initial conditions lead to chaotic behavior, while some initial conditions iterate either to infinity or to repeating cycles of any finite length.


title: Newton's method

text: Curt McMullen has shown that for any possible purely iterative algorithm similar to Newton's method, the algorithm will diverge on some open regions of the complex plane when applied to some polynomial of degree 4 or higher. However, McMullen gave a generally convergent algorithm for polynomials of degree 3.


title: Newton's method

text: One may also use Newton's method to solve systems of equations, which amounts to finding the (simultaneous) zeroes of continuously differentiable functions formula_28 This is equivalent to finding the zeroes of a single vector-valued function formula_29 In the formulation given above, the scalars are replaced by vectors and instead of dividing the function by its derivative one instead has to left multiply the function by the inverse of its Jacobian matrix . This results in the expression


title: Newton's method

text: Rather than actually computing the inverse of the Jacobian matrix, one may save time and increase numerical stability by solving the system of linear equations


title: Newton's method

text: The -dimensional variant of Newton's method can be used to solve systems of greater than (nonlinear) equations as well if the algorithm uses the generalized inverse of the non-square Jacobian matrix instead of the inverse of . If the nonlinear system has no solution, the method attempts to find a solution in the non-linear least squares sense. See Gauss–Newton algorithm for more information.


title: Newton's method

text: Another generalization is Newton's method to find a root of a functional defined in a Banach space. In this case the formulation is


title: Newton's method

text: where is the Fréchet derivative computed at . One needs the Fréchet derivative to be boundedly invertible at each in order for the method to be applicable. A condition for existence of and convergence to a root is given by the Newton–Kantorovich theorem.


title: Newton's method

text: In -adic analysis, the standard method to show a polynomial equation in one variable has a -adic root is Hensel's lemma, which uses the recursion from Newton's method on the -adic numbers. Because of the more stable behavior of addition and multiplication in the -adic numbers compared to the real numbers (specifically, the unit ball in the -adics is a ring), convergence in Hensel's lemma can be guaranteed under much simpler hypotheses than in the classical Newton's method on the real line.


title: Newton's method

text: The Newton–Fourier method is Joseph Fourier's extension of Newton's method to provide bounds on the absolute error of the root approximation, while still providing quadratic convergence.


title: Newton's method

text: Assume that is twice continuously differentiable on and that contains a root in this interval. Assume that on this interval (this is the case for instance if , , and , and on this interval). This guarantees that there is a unique root on this interval, call it . If it is concave down instead of concave up then replace by since they have the same roots.


title: Newton's method

text: Let be the right endpoint of the interval and let be the left endpoint of the interval. Given , define


title: Newton's method

text: where the denominator is and not . The iterations will be strictly decreasing to the root while the iterations will be strictly increasing to the root. Also,


title: Newton's method

text: When the Jacobian is unavailable or too expensive to compute at every iteration, a quasi-Newton method can be used.


title: Newton's method

text: A nonlinear equation has multiple solutions in general. But if the initial value is not appropriate, Newton's method may not converge to the desired solution or may converge to the same solution found earlier. When we have already found N solutions of formula_36, then the next root can be found by applying Newton's method to the next equation:


title: Newton's method

text: Hirano's modified Newton method is a modification conserving the convergence of Newton method and avoiding unstableness. It is developed to solve complex polynomials.


title: Newton's method

text: Combining Newton's method with interval arithmetic is very useful in some contexts. This provides a stopping criterion that is more reliable than the usual ones (which are a small value of the function or a small variation of the variable between consecutive iterations). Also, this may detect cases where Newton's method converges theoretically but diverges numerically because of an insufficient floating-point precision (this is typically the case for polynomials of large degree, where a very small change of the variable may change dramatically the value of the function; see Wilkinson's polynomial).


title: Newton's method

text: Consider , where is a real interval, and suppose that we have an interval extension of , meaning that takes as input an interval and outputs an interval such that:


title: Newton's method

text: where . Note that the hypothesis on implies that is well defined and is an interval (see interval arithmetic for further details on interval operations). This naturally leads to the following sequence:


title: Newton's method

text: The mean value theorem ensures that if there is a root of in , then it is also in . Moreover, the hypothesis on ensures that is at most half the size of when is the midpoint of , so this sequence converges towards , where is the root of in .


title: Newton's method

text: If strictly contains 0, the use of extended interval division produces a union of two intervals for ; multiple roots are therefore automatically separated and bounded.


title: Newton's method

text: Newton's method can be used to find a minimum or maximum of a function . The derivative is zero at a minimum or maximum, so local minima and maxima can be found by applying Newton's method to the derivative. The iteration becomes:


title: Newton's method

text: An important application is Newton–Raphson division, which can be used to quickly find the reciprocal of a number , using only multiplication and subtraction, that is to say the number such that . We can rephrase that as finding the zero of . We have .


title: Newton's method

text: The values of that solve the original equation are then the roots of , which may be found via Newton's method.


title: Newton's method

text: A numerical verification for solutions of nonlinear equations has been established by using Newton's method multiple times and forming a set of solution candidates.


title: Newton's method

text: Consider the problem of finding the square root of a number , that is to say the positive number such that . Newton's method is one of many methods of computing square roots. We can rephrase that as finding the zero of . We have .


title: Newton's method

text: For example, for finding the square root of 612 with an initial guess , the sequence given by Newton's method is:


title: Newton's method

text: where the correct digits are underlined. With only a few iterations one can obtain a solution accurate to many decimal places.


title: Newton's method

text: Consider the problem of finding the positive number formula_47 with formula_48. We can rephrase that as finding the zero of formula_49. We have formula_50. Since formula_51 for all formula_47 and formula_53 for formula_54, we know that our solution lies between 0 and 1.


title: Newton's method

text: For example, with an initial guess , the sequence given by Newton's method is (note that a starting value of 0 will lead to an undefined result, showing the importance of using a starting point that is close to the solution):


title: Newton's method

text: The correct digits are underlined in the above example. In particular, is correct to 12 decimal places. We see that the number of correct digits after the decimal point increases from 2 (for ) to 5 and 10, illustrating the quadratic convergence.


title: Newton's method

text: The following is an implementation example of the Newton's method in the Python (version 3.x) programming language for finding a root of a function codice_1 which has derivative codice_2.


title: Newton's method

text: Each new iteration of Newton's method will be denoted by codice_3. We will check during the computation whether the denominator (codice_4) becomes too small (smaller than codice_5), which would be the case if , since otherwise a large amount of error could be introduced.


title: Fast Fourier transform

text: A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from formula_1, which arises if one simply applies the definition of DFT, to formula_2, where formula_3 is the data size. The difference in speed can be enormous, especially for long data sets where "N" may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.


title: Fast Fourier transform

text: Fast Fourier transforms are widely used for applications in engineering, music, science, and mathematics. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805. In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime", and it was included in Top 10 Algorithms of 20th Century by the IEEE magazine "Computing in Science & Engineering".


title: Fast Fourier transform

text: The best-known FFT algorithms depend upon the factorization of "N", but there are FFTs with O("N" log "N") complexity for all "N", even for prime "N". Many FFT algorithms depend only on the fact that formula_4 is an "N"-th primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms. Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/"N" factor, any FFT algorithm can easily be adapted for it.


title: Fast Fourier transform

text: The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished work in 1805 when he needed it to interpolate the orbit of asteroids Pallas and Juno from sample observations. His method was very similar to the one published in 1965 by James Cooley and John Tukey, who are generally credited for the invention of the modern generic FFT algorithm. While Gauss's work predated even Joseph Fourier's results in 1822, he did not analyze the computation time and eventually used other methods to achieve his goal.


title: Fast Fourier transform

text: Between 1805 and 1965, some versions of FFT were published by other authors. Frank Yates in 1932 published his version called "interaction algorithm", which provided efficient computation of Hadamard and Walsh transforms. Yates' algorithm is still used in the field of statistical design and analysis of experiments. In 1942, G. C. Danielson and Cornelius Lanczos published their version to compute DFT for x-ray crystallography, a field where calculation of Fourier transforms presented a formidable bottleneck. While many methods in the past had focused on reducing the constant factor for formula_1 computation by taking advantage of "symmetries", Danielson and Lanczos realized that one could use the "periodicity" and apply a "doubling trick" to "double <nowiki>[N]</nowiki> with only slightly more than double the labor", though like Gauss they did not analyze that this led to formula_2 scaling.


title: Fast Fourier transform

text: James Cooley and John Tukey independently rediscovered these earlier algorithms and published a more general FFT in 1965 that is applicable when "N" is composite and not necessarily a power of 2, as well as analyzing the formula_2 scaling. Tukey came up with the idea during a meeting of President Kennedy's Science Advisory Committee where a discussion topic involved detecting nuclear tests by the Soviet Union by setting up sensors to surround the country from outside. To analyze the output of these sensors, an FFT algorithm would be needed. In discussion with Tukey, Richard Garwin recognized the general applicability of the algorithm not just to national security problems, but also to a wide range of problems including one of immediate interest to him, determining the periodicities of the spin orientations in a 3-D crystal of Helium-3. Garwin gave Tukey's idea to Cooley (both worked at IBM's Watson labs) for implementation. Cooley and Tukey published the paper in a relatively short time of six months. As Tukey did not work at IBM, the patentability of the idea was doubted and the algorithm went into the public domain, which, through the computing revolution of the next decade, made FFT one of the indispensable algorithms in digital signal processing.


title: Fast Fourier transform

text: Evaluating this definition directly requires formula_1 operations: there are "N" outputs "X", and each output requires a sum of "N" terms. An FFT is any method to compute the same results in formula_2 operations. All known FFT algorithms require formula_14 operations, although there is no known proof that lower complexity is impossible.


title: Fast Fourier transform

text: To illustrate the savings of an FFT, consider the count of complex multiplications and additions for formula_15 data points. Evaluating the DFT's sums directly involves formula_16 complex multiplications and formula_17 complex additions, of which formula_18 operations can be saved by eliminating trivial operations such as multiplications by 1, leaving about 30 million operations. In contrast, the radix-2 Cooley–Tukey algorithm, for "N" a power of 2, can compute the same result with only formula_19 complex multiplications (again, ignoring simplifications of multiplications by 1 and similar) and formula_20 complex additions, in total about 30,000 operations — a thousand times less than with direct evaluation. In practice, actual performance on modern computers is usually dominated by factors other than the speed of arithmetic operations and the analysis is a complicated subject (for example, see Frigo & Johnson, 2005), but the overall improvement from formula_1 to formula_2 remains.


title: Fast Fourier transform

text: By far the most commonly used FFT is the Cooley–Tukey algorithm. This is a divide-and-conquer algorithm that recursively breaks down a DFT of any composite size formula_23 into many smaller DFTs of sizes formula_24 and formula_25, along with formula_26 multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966).


title: Fast Fourier transform

text: This method (and the general idea of an FFT) was popularized by a publication of Cooley and Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms).


title: Fast Fourier transform

text: The best known use of the Cooley–Tukey algorithm is to divide the transform into two pieces of size "N"/2 at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey). These are called the "radix-2" and "mixed-radix" cases, respectively (and other variants such as the split-radix FFT have their own names as well). Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below.


title: Fast Fourier transform

text: For "N" = "N""N" with coprime "N" and "N", one can use the prime-factor (Good–Thomas) algorithm (PFA), based on the Chinese remainder theorem, to factorize the DFT similarly to Cooley–Tukey but without the twiddle factors. The Rader–Brenner algorithm (1976) is a Cooley–Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability; it was later superseded by the split-radix variant of Cooley–Tukey (which achieves the same multiplication count but with fewer additions and without sacrificing accuracy). Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms. (The Rader–Brenner and QFT algorithms were proposed for power-of-two sizes, but it is possible that they could be adapted to general composite "N". Bruun's algorithm applies to arbitrary even composite sizes.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial "z" − 1, here into real-coefficient polynomials of the form "z" − 1 and "z" + "az" + 1.


title: Fast Fourier transform

text: Another polynomial viewpoint is exploited by the Winograd FFT algorithm, which factorizes "z" − 1 into cyclotomic polynomials—these often have coefficients of 1, 0, or −1, and therefore require few (if any) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors. Indeed, Winograd showed that the DFT can be computed with only O("N") irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers. In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of "prime" sizes.


title: Fast Fourier transform

text: Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime "N", expresses a DFT of prime size "N" as a cyclic convolution of (composite) size "N" − 1, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods). Another prime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm; it also re-expresses a DFT as a convolution, but this time of the "same" size (which can be zero-padded to a power of two and evaluated by radix-2 Cooley–Tukey FFTs, for example), via the identity


title: Fast Fourier transform

text: Hexagonal fast Fourier transform (HFFT) aims at computing an efficient FFT for the hexagonally-sampled data by using a new addressing scheme for hexagonal grids, called Array Set Addressing (ASA).


title: Fast Fourier transform

text: In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry


title: Fast Fourier transform

text: and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consists of taking an ordinary algorithm (e.g. Cooley–Tukey) and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an "even"-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data), followed by O("N") post-processing operations.


title: Fast Fourier transform

text: It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform (DHT), but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs. Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular.


title: Fast Fourier transform

text: There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of roughly two in time and memory and the DFT becomes the discrete cosine/sine transform(s) (DCT/DST). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via FFTs of real data combined with O("N") pre- and post-processing.


title: Fast Fourier transform

text: A fundamental question of longstanding theoretical interest is to prove lower bounds on the complexity and exact operation counts of fast Fourier transforms, and many open problems remain. It is not rigorously proved whether DFTs truly require formula_29 (i.e., order "formula_30" or greater) operations, even for the simple case of power of two sizes, although no algorithms with lower complexity are known. In particular, the count of arithmetic operations is usually the focus of such questions, although actual performance on modern-day computers is determined by many other factors such as cache or CPU pipeline optimization.


title: Fast Fourier transform

text: Following work by Shmuel Winograd (1978), a tight Θ("N") lower bound is known for the number of real multiplications required by an FFT. It can be shown that only formula_31 irrational real multiplications are required to compute a DFT of power-of-two length formula_32. Moreover, explicit algorithms that achieve this count are known (Heideman & Burrus, 1986; Duhamel, 1990). However, these algorithms require too many additions to be practical, at least on modern computers with hardware multipliers (Duhamel, 1990; Frigo & Johnson, 2005).


title: Fast Fourier transform

text: A tight lower bound is not known on the number of required additions, although lower bounds have been proved under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an formula_33 lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). Pan (1986) proved an formula_33 lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear. For the case of power-of-two "N", Papadimitriou (1979) argued that the number formula_35 of complex-number additions achieved by Cooley–Tukey algorithms is "optimal" under certain assumptions on the graph of the algorithm (his assumptions imply, among other things, that no additive identities in the roots of unity are exploited). (This argument would imply that at least formula_36 real additions are required, although this is not a tight bound because extra additions are required as part of complex-number multiplications.) Thus far, no published FFT algorithm has achieved fewer than formula_35 complex-number additions (or their equivalent) for power-of-two "N".


title: Fast Fourier transform

text: A third problem is to minimize the "total" number of real multiplications and additions, sometimes called the "arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being considered). Again, no tight lower bound has been proven. Since 1968, however, the lowest published count for power-of-two "N" was long achieved by the split-radix FFT algorithm, which requires formula_38 real multiplications and additions for "N" > 1. This was recently reduced to formula_39 (Johnson and Frigo, 2007; Lundy and Van Buskirk, 2007). A slightly larger count (but still better than split radix for "N" ≥ 256) was shown to be provably optimal for "N" ≤ 512 under additional restrictions on the possible algorithms (split-radix-like flowgraphs with unit-modulus multiplicative factors), by reduction to a satisfiability modulo theories problem solvable by brute force (Haynal & Haynal, 2011).


title: Fast Fourier transform

text: Most of the attempts to lower or prove the complexity of FFT algorithms have focused on the ordinary complex-data case, because it is the simplest. However, complex-data FFTs are so closely related to algorithms for related problems such as real-data FFTs, discrete cosine transforms, discrete Hartley transforms, and so on, that any improvement in one of these would immediately lead to improvements in the others (Duhamel & Vetterli, 1990).


title: Fast Fourier transform

text: All of the FFT algorithms discussed above compute the DFT exactly (i.e. neglecting floating-point errors). A few "FFT" algorithms have been proposed, however, that compute the DFT "approximately", with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate FFT algorithm by Edelman et al. (1999) achieves lower communication requirements for parallel computing with the help of a fast multipole method. A wavelet-based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs (time/frequency localization) into account more efficiently than is possible with an exact FFT. Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al. (1995). The Edelman algorithm works equally well for sparse and non-sparse data, since it is based on the compressibility (rank deficiency) of the Fourier matrix itself rather than the compressibility (sparsity) of the data. Conversely, if the data are sparse—that is, if only "K" out of "N" Fourier coefficients are nonzero—then the complexity can be reduced to O("K"log("N")log("N"/"K")), and this has been demonstrated to lead to practical speedups compared to an ordinary FFT for "N"/"K" > 32 in a large-"N" example ("N" = 2) using a probabilistic approximate algorithm (which estimates the largest "K" coefficients to several decimal places).


title: Fast Fourier transform

text: FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small; most FFT algorithms, e.g. Cooley–Tukey, have excellent numerical properties as a consequence of the pairwise summation structure of the algorithms. The upper bound on the relative error for the Cooley–Tukey algorithm is formula_40, compared to formula_41 for the naïve DFT formula, where ε is the machine floating-point relative precision. In fact, the root mean square (rms) errors are much better than these upper bounds, being only formula_42 for Cooley–Tukey and formula_43 for the naïve DFT (Schatzman, 1996). These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT (i.e. the trigonometric function values), and it is not unusual for incautious FFT implementations to have much worse accuracy, e.g. if they use inaccurate trigonometric recurrence formulas. Some FFTs other than Cooley–Tukey, such as the Rader–Brenner algorithm, are intrinsically less stable.


title: Fast Fourier transform

text: In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errors growing as formula_44 for the Cooley–Tukey algorithm (Welch, 1969). Achieving this accuracy requires careful attention to scaling to minimize loss of precision, and fixed-point FFT algorithms involve rescaling at each intermediate stage of decompositions like Cooley–Tukey.


title: Fast Fourier transform

text: To verify the correctness of an FFT implementation, rigorous guarantees can be obtained in formula_2 time by a simple procedure checking the linearity, impulse-response, and time-shift properties of the transform on random inputs (Ergün, 1995).


title: Fast Fourier transform

text: transforms an array "x" with a "d"-dimensional vector of indices formula_47 by a set of "d" nested summations (over formula_48 for each "j"), where the division n/N, defined as formula_49, is performed element-wise. Equivalently, it is the composition of a sequence of "d" sets of one-dimensional DFTs, performed along one dimension at a time (in any order).


title: Fast Fourier transform

text: This compositional viewpoint immediately provides the simplest and most common multidimensional DFT algorithm, known as the row-column algorithm (after the two-dimensional case, below). That is, one simply performs a sequence of "d" one-dimensional FFTs (by any of the above algorithms): first you transform along the "n" dimension, then along the "n" dimension, and so on (or actually, any ordering works). This method is easily shown to have the usual formula_2 complexity, where formula_51 is the total number of data points transformed. In particular, there are "N"/"N" transforms of size "N", etcetera, so the complexity of the sequence of FFTs is:


title: Fast Fourier transform

text: In two dimensions, the "x" can be viewed as an formula_53 matrix, and this algorithm corresponds to first performing the FFT of all the rows (resp. columns), grouping the resulting transformed rows (resp. columns) together as another formula_53 matrix, and then performing the FFT on each of the columns (resp. rows) of this second matrix, and similarly grouping the results into the final result matrix.


title: Fast Fourier transform

text: In more than two dimensions, it is often advantageous for cache locality to group the dimensions recursively. For example, a three-dimensional FFT might first perform two-dimensional FFTs of each planar "slice" for each fixed "n", and then perform the one-dimensional FFTs along the "n" direction. More generally, an asymptotically optimal cache-oblivious algorithm consists of recursively dividing the dimensions into two groups formula_55 and formula_56 that are transformed recursively (rounding if "d" is not even) (see Frigo and Johnson, 2005). Still, this remains a straightforward variation of the row-column algorithm that ultimately requires only a one-dimensional FFT algorithm as the base case, and still has O("N" log "N") complexity. Yet another variation is to perform matrix transpositions in between transforming subsequent dimensions, so that the transforms operate on contiguous data; this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data is extremely time-consuming.


title: Fast Fourier transform

text: There are other multidimensional FFT algorithms that are distinct from the row-column algorithm, although all of them have formula_2 complexity. Perhaps the simplest non-row-column FFT is the vector-radix FFT algorithm, which is a generalization of the ordinary Cooley–Tukey algorithm where one divides the transform dimensions by a vector formula_58 of radices at each step. (This may also have cache benefits.) The simplest case of vector-radix is where all of the radices are equal (e.g. vector-radix-2 divides "all" of the dimensions by two), but this is not necessary. Vector radix with only a single non-unit radix at a time, i.e. formula_59, is essentially a row-column algorithm. Other, more complicated, methods include polynomial transform algorithms due to Nussbaumer (1977), which view the transform in terms of convolutions and polynomial products. See Duhamel and Vetterli (1990) for more information and references.


title: Fast Fourier transform

text: An formula_60 generalization to spherical harmonics on the sphere "S" with "N" nodes was described by Mohlenkamp, along with an algorithm conjectured (but not proven) to have formula_61 complexity; Mohlenkamp also provides an implementation in the libftsh library. A spherical-harmonic algorithm with formula_62complexity is described by Rokhlin and Tygert.


title: Fast Fourier transform

text: The fast folding algorithm is analogous to the FFT, except that it operates on a series of binned waveforms rather than a series of real or complex scalar values. Rotation (which in the FFT is multiplication by a complex phasor) is a circular shift of the component waveform.


title: Fast Fourier transform

text: Various groups have also published "FFT" algorithms for non-equispaced data, as reviewed in Potts "et al." (2001). Such algorithms do not strictly compute the DFT (which is only defined for equispaced data), but rather some approximation thereof (a non-uniform discrete Fourier transform, or NDFT, which itself is often computed only approximately). More generally there are various other methods of spectral estimation.


title: Fast Fourier transform

text: The FFT's importance derives from the fact that it has made working in the frequency domain equally computationally feasible as working in the temporal or spatial domain. Some of the important applications of the FFT include:


title: Jalen Green

text: Jalen Romande Green (born February 9, 2002) is an American professional basketball player for the Houston Rockets of the National Basketball Association (NBA). He is the third player in the NBA of Filipino descent, following Jordan Clarkson and Raymond Townsend. He was a consensus five-star recruit and the best shooting guard in the 2020 class, with ESPN ranking him number one overall. He finished his high school career at Prolific Prep in Napa, California, and he chose to forgo college basketball to join the NBA G League Ignite team in its inaugural season. Green has won three gold medals with the United States at the junior level and was named Most Valuable Player (MVP) of the 2018 FIBA Under-17 World Cup. Green was selected by the Houston Rockets with the second overall pick in the 2021 NBA draft.


title: Jalen Green

text: Green was born in Merced, California. He grew up in Livingston, California. When Green was in third grade, he moved with his family back to Fresno, California. By sixth grade, he was playing Amateur Athletic Union basketball and practicing for five hours each day.


title: Jalen Green

text: For his first three years of high school, Green played basketball for San Joaquin Memorial High School in Fresno. As a freshman, he was a full-time starter and averaged 18.1 points and nine rebounds per game. He led his team to a CIF Central Section Division II runner-up finish and the CIF Division II quarterfinals. He earned MaxPreps Freshman All-American second team and CIF Central Section rookie of the year honors. In his sophomore season, Green averaged 27.9 points and 7.7 rebounds per game, leading San Joaquin Memorial to a Central Section Division II title and the CIF Open Division playoffs. He was named MaxPreps National Sophomore of the Year and made the "USA Today" All-USA California second team.


title: Jalen Green

text: As a junior, Green averaged 30.1 points, 7.8 rebounds, and 3.6 assists per game for San Joaquin Memorial. He won his second consecutive Central Division II championship. In the title game, Green surpassed the school career scoring record of 2,288 held by Roscoe Pondexter since 1971. He also helped his team reach the CIF Northern California Division I quarterfinals. Green was named "USA Today" All-USA California player of the year and appeared on the All-USA second team and MaxPreps All-American second team.


title: Jalen Green

text: For his senior season, he transferred to Prolific Prep in Napa, California. He helped his team win the Grind Session World championship. In March 2020, he shared Grind Session most valuable player honors with Daishen Nix. Green averaged 31.5 points, 7.5 rebounds, and five assists per game, leading his team to a 31–3 record. He was named "Sports Illustrated" All-American player of the year and to the MaxPreps All-American first team. Green was selected to play in the McDonald's All-American Game, the Jordan Brand Classic, and the Nike Hoop Summit, but all three games were canceled due to the COVID-19 pandemic.


title: Jalen Green

text: Green was a consensus five-star recruit and the number one shooting guard in the 2020 recruiting class, according to major recruiting services 247Sports, ESPN and Rivals. He was ranked as the top recruit in his class by ESPN. He received offers from many NCAA Division I basketball programs, including Arizona, Florida State, and USC before turning 15 years old. On April 16, 2020, Green announced that he would join the NBA G League, forgoing college basketball. He chose the G League over offers from Auburn, Oregon, and Memphis.


title: Jalen Green

text: On April 16, 2020, Green signed a one-year, $600,000 contract with the NBA G League Ignite, a developmental team affiliated with the NBA G League. He became the first player to join the team. On March 8, 2021, Green recorded a season-high 30 points, seven assists, and five rebounds in a 127–102 loss to the Raptors 905 in the first round of the playoffs. He averaged 17.9 points, 4.1 rebounds, and 2.8 assists per game.


title: Jalen Green

text: On July 29, 2021, Green was selected by the Houston Rockets second overall in the 2021 NBA draft, Making him the first ever player being drafted out from the G-League, being behind from former Ignite teammate Jonathan Kuminga and Isaiah Todd. On August 8, he made his summer league debut in a 84–76 win against the Cleveland Cavaliers where he posted 23 points, five rebounds, and two assists in 30 minutes. He made the All-Summer League Second Team after missing the last three of five games due to soreness on his right hamstring. He made his preseason debut on October 5 in a 125–119 win against the Washington Wizards with 12 points, six rebounds, and two assists. On October 20, Green made his NBA debut, putting up nine points, four rebounds, and four assists in a 124–106 loss to the Minnesota Timberwolves. On October 24, Green put up 30 points, with eight three-pointers made, in a 107–97 loss to the Boston Celtics, becoming the first rookie to put up at least 30 points and eight threes in a game in Rockets history. Unfortunately for Green, he started out his rookie season suffering with a hamstring injury on his lower left leg after a home game win against the Chicago Bulls and been sidelined out of the rotation for a while. After missing 14-games in a month from his hamstring injury, Green returned to the starting line up on December 24, scoring 20 points by going on a 6-for-9 from the three point line in a loss road trip game to the Indiana Pacers.


title: Jalen Green

text: On February 19, 2022, Green participated in the Slam Dunk Contest, finishing in third place. On March 9, Green scored 32 points along with three rebounds, three assists, and one blocked shot in 38 minutes which marked his second 30-point game in a 130–139 overtime win against the Los Angeles Lakers. On March 28, Green scored his third 30-points game along with four rebounds and assists in a 123–120 loss against the San Antonio Spurs. He joined with Allen Iverson as the only NBA rookie to score a 30-plus point in five straight games in a row since 1997. In his final game as a rookie, Green scored a season career-high with 41 points in a 130–114 loss to the Atlanta Hawks, which was the first 40-point game by a Rockets rookie since Hakeem Olajuwon. At the end of the regular season, he was named Rookie of the Month for March and April. He was selected to the NBA All-Rookie First Team with averages of 17.3 points, 3.4 rebounds, and 2.6 assists.


title: Jalen Green

text: Prior to the 2022–23 season, Green changed his jersey number from 0 to 4. In his second game of the season, Green played around a total of 35 minutes with a record of 33 points with four threes, five rebounds, two assists and a steal in a 129–122 loss to the Grizzlies. On November 7, 2022, He helped the Rockets to beat the Orlando Magic with 34 points, shooting 66.7% from the field and playing around a total of 38 minutes. He became the sixth guard in NBA history to score at least 30 points ten or more times before turning 21, joining with other elite guards in Luka Dončić, Devin Booker, Anthony Edwards, LaMelo Ball, and Kyrie Irving. On November 26, he secured a career-high nine assists along with 28 points and three rebounds in a 118–105 win against the Oklahoma City Thunder. In December 2, Green logged 20 points in the third quarter leading with 30 points in a 122-121 comeback game win against the Phoenix Suns.


title: Jalen Green

text: Green represents the United States internationally but has also shown interest in playing for the Philippines in the future due to his partial Filipino background. He made his national team debut for the United States at the 2017 FIBA Under-16 Americas Championship in Formosa, Argentina. In five games, he averaged 9.8 points, two rebounds, and one steal per game, helping his team win the gold medal. He was named MVP of the 2018 FIBA Under-17 Basketball World Cup in Argentina after averaging a team-high 15.7 points, 2.3 rebounds, and 1.7 assists per game and winning the gold medal. Green won another gold medal with the United States at the 2019 FIBA Under-19 Basketball World Cup in Heraklion, Greece. As the youngest member of his team, he averaged 10.1 points, 2.1 rebounds, and 1.7 steals per game.


title: Jalen Green

text: Standing at 6 feet and 4 inches (1.98 m) with a wingspan of 6 feet and 7.5 inches (2.02 m) and standing reach of 8 feet and 5 inches (2.57 m), Green primarily plays as a shooting guard. He is known for his elite athleticism, handling skills, and versatile scoring abilities that makes him an on-and off-the-ball threat in a half-court setting and in transition. He has drawn comparisons to Zach LaVine, Ray Allen, Kelly Oubre Jr., Bradley Beal, Clyde Drexler, and Kobe Bryant. Green utilizes an "explosive" first step that allows him to attack the rim and speed past on-ball defenders.


title: Jalen Green

text: Scouts have noted that though his efficiency was "solid" in the NBA G League, he can be a consistent mercurial shooter. His defense, passing abilities, and small frame at 178 pounds have also been noted by scouts.


title: Jalen Green

text: Green's mother, Bree Puruganan, is of partial Filipino descent through her grandfather. His step-father, Marcus Green, was a basketball teammate of NBA player DeShawn Stevenson at Washington Union High School in Fresno. He has a younger sister.


title: Alan Greenspan

text: Alan Greenspan (born March 6, 1926) is an American economist who served as the 13th chairman of the Federal Reserve from 1987 to 2006. He works as a private adviser and provides consulting for firms through his company, Greenspan Associates LLC.


title: Alan Greenspan

text: First nominated to the Federal Reserve by President Ronald Reagan in August 1987, he was reappointed at successive four-year intervals until retiring on January 31, 2006, after the second-longest tenure in the position, behind only William McChesney Martin. President George W. Bush appointed Ben Bernanke as his successor.


title: Alan Greenspan

text: Greenspan came to the Federal Reserve Board from a consulting career. Although he was subdued in his public appearances, favorable media coverage raised his profile to a point that several observers likened him to a "rock star". Democratic leaders of Congress criticized him for politicizing his office because of his support for Social Security privatization and tax cuts.


title: Alan Greenspan

text: Many have argued that the "easy-money" policies of the Fed during Greenspan's tenure, including the practice known as the "Greenspan put", were a leading cause of the dot-com bubble and subprime mortgage crisis (the latter occurring within a year of his leaving the Fed), which, said "The Wall Street Journal", "tarnished his reputation". Yale economist Robert Shiller argues that "once stocks fell, real estate became the primary outlet for the speculative frenzy that the stock market had unleashed". Greenspan argues that the housing bubble was not a result of low-interest short-term rates but rather a worldwide phenomenon caused by the progressive decline in long-term interest rates - a direct consequence of the relationship between high savings rates in the developing world and its inverse in the developed world.


title: Alan Greenspan

text: Greenspan was born in the Washington Heights area of New York City. His father, Herbert Greenspan, was of Romanian Jewish descent, and his mother, Rose Goldsmith, was of Hungarian Jewish descent. After his parents divorced, Greenspan grew up with his mother in the household of his maternal grandparents who were born in Russia. His father worked as a stockbroker and consultant in New York City.


title: Alan Greenspan

text: Greenspan attended George Washington High School from 1940 until he graduated in June 1943, where one of his classmates was John Kemeny. He played clarinet and saxophone along with Stan Getz. He further studied clarinet at the Juilliard School from 1943 to 1944. Among his bandmates in the Woody Herman band was Leonard Garment, Richard Nixon's special counsel. In 1945, Greenspan attended New York University's Stern School of Business, where he earned a B.A. degree in economics "summa cum laude" in 1948 and an M.A. degree in economics in 1950. At Columbia University, he pursued advanced economic studies under Arthur Burns but withdrew because of his increasing work demand at Townsend-Greenspan & Company.


title: Alan Greenspan

text: In 1977, Greenspan obtained a Ph.D. in economics from New York University. His dissertation is not available from the university since it was removed at Greenspan's request in 1987, when he became chairman of the Federal Reserve Board. In April 2008, however, "Barron's" obtained a copy and notes that it includes "a discussion of soaring housing prices and their effect on consumer spending; it even anticipates a bursting housing bubble".


title: Alan Greenspan

text: During his economics studies at New York University, Greenspan worked under Eugene Banks, a managing director at the Wall Street investment bank Brown Brothers Harriman, in the firm's equity research department. From 1948 to 1953, Greenspan worked as an analyst at the National Industrial Conference Board (currently known as the Conference Board), a business- and industry-oriented think tank in New York City. Before he was appointed chairman of the Federal Reserve, from 1955 to 1987, Greenspan was chairman and president of Townsend-Greenspan & Co., Inc., an economics consulting firm in New York City. His 32-year stint there was interrupted only from 1974 to 1977, when he served as chairman of the Council of Economic Advisers, under President Gerald Ford.


title: Alan Greenspan

text: In mid-1968, Greenspan agreed to serve as Richard Nixon's coordinator on domestic policy in the nomination campaign. Greenspan has also served as a corporate director for Aluminum Company of America (Alcoa); Automatic Data Processing; Capital Cities/ABC, Inc.; General Foods; J.P. Morgan & Co.; Morgan Guaranty Trust Company; Mobil Corporation; and the Pittston Company. He was a director of the Council on Foreign Relations foreign policy organization between 1982 and 1988. He also served as a member of the influential Washington-based financial advisory body, the Group of Thirty in 1984.


title: Alan Greenspan

text: On June 2, 1987, President Ronald Reagan nominated Greenspan as a successor to Paul Volcker, as chairman of the Board of Governors of the Federal Reserve, and the Senate confirmed him on August 11, 1987. Investor, author and commentator Jim Rogers has said that Greenspan lobbied to get this chairmanship.


title: Alan Greenspan

text: Two months after his confirmation, Greenspan said immediately following the 1987 stock market crash that the Fed "affirmed today its readiness to serve as a source of liquidity to support the economic and financial system". Although the Federal Reserve followed its announcement with monetary policy actions, which became known as the Greenspan put, George H. W. Bush attributed his re-election loss to a sluggish response. Democratic president Bill Clinton reappointed Greenspan, and consulted him on economic matters. Greenspan lent support to Clinton's 1993 deficit reduction program. Greenspan was fundamentally a monetarist and Austrian Economist in orientation on the economy, and his monetary policy decisions largely followed standard Taylor rule prescriptions (see Taylor 1993 and 1999). Greenspan also played a key role in organizing the U.S. bailout of Mexico during the 1994–1995 Mexican peso crisis.


title: Alan Greenspan

text: In 2000, Greenspan raised interest rates several times; these actions were believed by many to have caused the bursting of the dot-com bubble. According to Nobel laureate Paul Krugman, however, "he didn't raise interest rates to curb the market's enthusiasm; he didn't even seek to impose margin requirements on stock market investors. Instead, he waited until the bubble burst, as it did in 2000, then tried to clean up the mess afterward". E. Ray Canterbery agrees with Krugman's criticism.


title: Alan Greenspan

text: In January 2001, Greenspan, in support of President Bush's proposed tax decrease, stated that the federal surplus could accommodate a significant tax cut while paying down the national debt.


title: Alan Greenspan

text: In autumn 2001, as a decisive reaction to the September 11 attacks and various corporate scandals which undermined the economy, the Greenspan-led Federal Reserve initiated a series of interest cuts that brought down the federal funds rate to 1% in 2004. While presenting the Federal Reserve's Monetary Policy Report in July 2002, he said that "It is not that humans have become any more greedy than in generations past. It is that the avenues to express greed had grown so enormously", and suggested that financial markets need to be more regulated. His critics, led by Steve Forbes, attributed the rapid rise in commodity prices and gold to Greenspan's loose monetary policy, which Forbes believed had caused excessive asset inflation and a weak dollar. By late 2004, the price of gold was higher than its 12-year moving average.


title: Alan Greenspan

text: Greenspan advised senior members of the George W. Bush administration to depose Saddam Hussein for the sake of the oil markets. He believed that even a moderate disruption to the flow of oil could translate into high oil prices, which could lead to "chaos" in the global economy and bring the industrial world "to its knees". He feared that Saddam could seize control of the Straits of Hormuz and restrict the transport of oil through them. In a 2007 interview, he said, "people do not realize in this country, for example, how tenuous our ties to international energy are. That is, we on a daily basis require continuous flow. If that flow is shut off, it causes catastrophic effects in the industrial world. And it's that which made him [Saddam] far more important to get out than bin Laden."


title: Alan Greenspan

text: On May 18, 2004, Greenspan was nominated by President George W. Bush to serve for an unprecedented fifth term as chairman of the Federal Reserve. He was previously appointed to the post by Presidents Reagan, George H. W. Bush, and Bill Clinton.


title: Alan Greenspan

text: In a May 2005 speech, Greenspan stated: "Two years ago at this conference I argued that the growing array of derivatives and the related application of more-sophisticated methods for measuring and managing risks had been key factors underlying the remarkable resilience of the banking system, which had recently shrugged off severe shocks to the economy and the financial system. At the same time, I indicated some concerns about the risks associated with derivatives, including the risks posed by concentration in certain derivatives markets, notably the over-the-counter (OTC) markets for U.S. dollar interest rate options."


title: Alan Greenspan

text: Greenspan opposed tariffs against the People's Republic of China for its refusal to let the yuan rise, suggesting instead that any American workers displaced by Chinese trade could be compensated through unemployment insurance and retraining programs.


title: Alan Greenspan

text: Greenspan's term as a member of the board ended on January 31, 2006, and Ben Bernanke was confirmed as his successor.


title: Alan Greenspan

text: Immediately after leaving the Fed, Greenspan formed an economic consulting firm, Greenspan Associates LLC. He also accepted an honorary (unpaid) position at HM Treasury in the United Kingdom.


title: Alan Greenspan

text: On February 26, 2007, Greenspan forecast a possible recession in the United States before or in early 2008. Stabilizing corporate profits are said to have influenced his comments. The following day, the Dow Jones Industrial Average decreased by 416 points, losing 3.3% of its value.


title: Alan Greenspan

text: In May 2007, Greenspan was hired as a special consultant by Pacific Investment Management Company (PIMCO) to participate in their quarterly economic forums and speak privately with the bond managers about Fed interest rate policy.


title: Alan Greenspan

text: In August 2007, Deutsche Bank announced that it would be retaining Greenspan as a senior advisor to its investment banking team and clients.


title: Alan Greenspan

text: In mid-January 2008, hedge fund Paulson & Co. hired Greenspan as an adviser. According to the terms of their agreement, he was not to advise any other hedge fund while working for Paulson. In 2007, Paulson had foreseen the collapse of the sub-prime housing market and hired Goldman Sachs to package their sub-prime holdings into derivatives and sell them. Some economic commentators blamed this collapse on Greenspan's policies while at the Fed.


title: Alan Greenspan

text: On April 30, 2009, Greenspan offered a defense of the H-1B visa program, telling a U.S. Senate subcommittee that the visa quota is "far too small to meet the need" and saying that it protects U.S. workers from global competition, creating a "privileged elite". Testifying on immigration reform before the Subcommittee on Immigration, Border Security and Citizenship, he said more skilled immigration was needed "as the economy copes with the forthcoming retirement wave of skilled baby boomers".


title: Alan Greenspan

text: Greenspan wrote a memoir titled "The Age of Turbulence: Adventures in a New World", published September 17, 2007. Greenspan says that he wrote the book in longhand mostly while soaking in the bathtub, a habit he regularly employs since injuring his back in 1971. Greenspan discusses in his book, among other things, his history in government and economics, capitalism and other economic systems, current issues in the global economy, and future issues that face the global economy. In the book, Greenspan criticizes President George W. Bush, Vice President Dick Cheney, and the Republican-controlled Congress for abandoning the Republican Party's principles on spending and deficits. Greenspan's criticisms of President Bush include his refusal to veto spending bills, sending the country into increasingly deep deficits, and for "putting political imperatives ahead of sound economic policies". Greenspan writes, "They swapped principle for power. They ended up with neither. They deserved to lose [the 2006 election]". He praised Bill Clinton above all the other presidents for whom he'd worked for his "consistent, disciplined focus on long-term economic growth". Although he respected what he saw as Richard Nixon's immense intelligence, Greenspan found him to be "sadly paranoid, misanthropic and cynical". He said of Gerald Ford that he "was as close to normal as you get in a president, but he was never elected". Regarding future U.S. economic policy, Greenspan recommends improving the U.S. primary and secondary education systems. He asserts this would narrow the inequality between the minority of high-income earners and most workers whose wages have not grown in proportion with globalization and the nation's GDP growth.


title: Alan Greenspan

text: In the early 1950s, Greenspan began an association with novelist and philosopher Ayn Rand. Greenspan was introduced to Rand by his first wife, Joan Mitchell. Rand nicknamed Greenspan "the undertaker" because of his penchant for dark clothing and reserved demeanor. Although Greenspan was initially a logical positivist, he was converted to Rand's philosophy of Objectivism by her associate Nathaniel Branden. He became one of the members of Rand's inner circle, the Ayn Rand Collective, who read "Atlas Shrugged" while it was being written. During the 1950s and 1960s Greenspan was a proponent of Objectivism, writing articles for Objectivist newsletters and contributing several essays for Rand's 1966 book "" including an essay supporting the gold standard. During the 1960s Greenspan offered a ten-lecture course, "The Economics of a Free Society", under the auspices of the Nathaniel Branden Institute. The course highlighted the causes of prosperity and depression, the consequences of government intervention, and the fallacies of collectivist economics. Rand stood beside him at his 1974 swearing-in as chairman of the Council of Economic Advisers. Greenspan and Rand remained friends until her death in 1982.


title: Alan Greenspan

text: Greenspan has come under criticism from Harry Binswanger, who believes his actions while at work for the Federal Reserve and his publicly expressed opinions on other issues show abandonment of Objectivist and free market principles. When questioned in relation to this, however, he has said that in a democratic society individuals have to make compromises with each other over conflicting ideas of how money should be handled. He said he himself had to make such compromises, because he believes that "we did extremely well" without a central bank and with a gold standard. In a congressional hearing on October 23, 2008, Greenspan admitted that his free-market ideology shunning certain regulations was flawed. When asked about free markets and Rand's ideas, however, Greenspan clarified his stance on "laissez faire" capitalism and asserted that in a democratic society there could be no better alternative. He stated that the errors that were made stemmed not from the principle, but from the application of competitive markets in "assuming what the nature of risks would be".


title: Alan Greenspan

text: E. Ray Canterbery has chronicled Greenspan's relationship with Rand, and has concluded that the influence has had pernicious effects on Greenspan's monetary policy.


title: Alan Greenspan

text: In the wake of the subprime mortgage and credit crisis in 2007, Greenspan stated that there was a bubble in the U.S. housing market, warning in 2007 of "large double digit declines" in home values "larger than most people expect". Greenspan also noted, however, "I really didn't get it until very late in 2005 and 2006."


title: Alan Greenspan

text: Greenspan stated that the housing bubble was "fundamentally engendered by the decline in real long-term interest rates", though he also claims that long-term interest rates are beyond the control of central banks because "the market value of global long-term securities is approaching $100 trillion" and thus these and other asset markets are large enough that they "now swamp the resources of central banks".


title: Alan Greenspan

text: After the September 11, 2001 attacks, the Federal Open Market Committee voted to reduce the federal funds rate from 3.5% to 3.0%. Then, after the accounting scandals of 2002, the Fed dropped the federal funds rate from then current 1.25% to 1.00%. Greenspan stated that this drop in rates would have the effect of leading to a surge in home sales and refinancing, adding that "Besides sustaining the demand for new construction, mortgage markets have also been a powerful stabilizing force over the past two years of economic distress by facilitating the extraction of some of the equity that homeowners have built up over the years".


title: Alan Greenspan

text: According to some, however, Greenspan's policies of adjusting interest rates to historic lows contributed to a housing bubble in the United States. The Federal Reserve acknowledged the connection between lower interest rates, higher home values, and the increased liquidity the higher home values bring to the overall economy: "Like other asset prices, house prices are influenced by interest rates, and in some countries, the housing market is a key channel of monetary policy transmission".


title: Alan Greenspan

text: In a February 23, 2004, speech, Greenspan suggested that more homeowners should consider taking out adjustable-rate mortgages (ARMs) where the interest rate adjusts itself to the current interest in the market. The Fed's own funds rate was at a then all-time-low of 1%. A few months after his recommendation, Greenspan began raising interest rates, in a series of rate hikes that would bring the funds rate to 5.25% about two years later. A triggering factor in the 2007 subprime mortgage financial crisis is believed to be the many subprime ARMs that reset at much higher interest rates than what the borrower paid during the first few years of the mortgage.


title: Alan Greenspan

text: In 2008, Greenspan expressed great frustration that the February 23 speech was used to criticize him on ARMs and the subprime mortgage crisis, and stated that he had made countervailing comments eight days after it that praised traditional fixed-rate mortgages. In that speech, Greenspan had suggested that lenders should offer to home purchasers a greater variety of "mortgage product alternatives" other than traditional fixed-rate mortgages. Greenspan also praised the rise of the subprime mortgage industry and its tools for assessing credit-worthiness:


title: Alan Greenspan

text: The subprime mortgage industry collapsed in March 2007, with many of the largest lenders filing for bankruptcy protection in the face of spiraling foreclosure rates. For these reasons, Greenspan has been criticized for his role in the rise of the housing bubble and the subsequent problems in the mortgage industry, as well as "engineering" the housing bubble itself.


title: Alan Greenspan

text: In 2004 "Businessweek" magazine analysts argued: "It was the Federal Reserve-engineered decline in rates that inflated the housing bubble ... the most troublesome aspect of the price runup is that many recent buyers are squeezing into houses that they can barely afford by taking advantage of the lower rates available from adjustable-rate mortgages. That leaves them fully exposed to rising rates.


title: Alan Greenspan

text: In September 2008 Joseph Stiglitz stated that Greenspan "didn't really believe in regulation; when the excesses of the financial system were noted, (he and others) called for self-regulation—an oxymoron". Greenspan, according to "The New York Times", says he himself is blameless. On April 6, 2005, Greenspan called for a substantial increase in the regulation of Fannie Mae and Freddie Mac: "Appearing before the Senate Banking Committee, the Fed chairman, Alan Greenspan, said the enormous portfolios of the companies—nearly a quarter of the home-mortgage market—posed significant risks to the nation's financial system should either company face significant problems." Despite this, Greenspan still claims to be a firm believer in free markets, although in his 2007 biography he wrote, "History has not dealt kindly with the aftermath of protracted periods of low risk premiums" as seen before the credit crisis of 2008.


title: Alan Greenspan

text: In 2009 Robert Reich wrote that "Greenspan's worst move was to contribute to the giant housing bubble and the worst worldwide crash since the Great Depression. In 2004 he lowered interest rates to 1%, enabling banks to borrow money for free, adjusted for inflation. Naturally, the banks wanted to borrow as much as they possibly could, then lend it out, earning nice profits. The situation screamed for government oversight of lending institutions, lest the banks lend to unfit borrowers. He refused, trusting the market to weed out bad credit risks. It did not."


title: Alan Greenspan

text: In congressional testimony on October 23, 2008, Greenspan finally conceded error on regulation. "The New York Times" wrote, "a humbled Mr. Greenspan admitted that he had put too much faith in the self-correcting power of free markets and had failed to anticipate the self-destructive power of wanton mortgage lending ... Mr. Greenspan refused to accept blame for the crisis but acknowledged that his belief in deregulation had been shaken". Although many Republican lawmakers tried to blame the housing bubble on Fannie Mae and Freddie Mac, Greenspan placed far more blame on Wall Street for bundling subprime mortgages into securities.


title: Alan Greenspan

text: In March 2008, Greenspan wrote an article for the "Financial Times" Economists' Forum in which he said that the 2008-financial crisis in the United States is likely to be judged as the most wrenching since the end of World War II. In it he argued: "We will never be able to anticipate all discontinuities in financial markets." He concluded: "It is important, indeed crucial, that any reforms in, and adjustments to, the structure of markets and regulation not inhibit our most reliable and effective safeguards against cumulative economic failure: market flexibility and open competition." The article attracted a number of critical responses from forum contributors, who, finding causation between Greenspan's policies and the discontinuities in financial markets that followed, criticized Greenspan mainly for what many believed to be his unbalanced and immovable ideological suppositions about global capitalism and free competitive markets. Notable critics included J. Bradford DeLong, Paul Krugman, Alice Rivlin, Michael Hudson, and Willem Buiter.


title: Alan Greenspan

text: Greenspan responded to his critics in a follow-up article in which he defended his ideology as applied to his conceptual and policy framework, which, among other things, prohibited him from exerting real pressure against the burgeoning housing bubble or, in his words, "leaning against the wind". Greenspan argued, "My view of the range of dispersion of outcomes has been shaken, but not my judgment that free competitive markets are by far the unrivaled way to organize economies". He concluded: "We have tried regulation ranging from heavy to central planning. None meaningfully worked. Do we wish to retest the evidence?" "Financial Times" associate editor and chief economics commentator Martin Wolf defended Greenspan primarily as a scapegoat for the market turmoil. Several notable contributors in defense of Greenspan included Stephen S. Roach, Allan Meltzer, and Robert Brusca.


title: Alan Greenspan

text: However, an October 15, 2008, article in "The Washington Post" analyzing the origins of the economic crisis claims that Greenspan vehemently opposed any regulation of derivatives, and actively sought to undermine the office of the Commodity Futures Trading Commission when the commission sought to initiate regulation of derivatives. Meanwhile, Greenspan recommended improving mark-to-market regulations to avoid having derivatives or other complex assets marked to a distressed or illiquid market during times of material adverse conditions seen during the late 2000s credit crisis.


title: Alan Greenspan

text: Greenspan was not alone in his opposition to derivatives regulation. In a 1999 government report that was a key driver in the passage of the Commodity Futures Modernization Act of 2000—legislation that clarified that most over-the-counter derivatives were outside the regulatory authority of any government agency—Greenspan was joined by Treasury Secretary Lawrence Summers, Securities and Exchange Commission Chairman Arthur Levitt, and Commodity Futures Trading Commission Chairman William Ranier in concluding that "under many circumstances, the trading of financial derivatives by eligible swap participants should be excluded from the CEA" (Commodity Exchange Act). Other government agencies also supported that view.


title: Alan Greenspan

text: In Congressional testimony on October 23, 2008, Greenspan acknowledged that he was "partially" wrong in opposing regulation and stated "Those of us who have looked to the self-interest of lending institutions to protect shareholder's equity—myself especially—are in a state of shocked disbelief." Referring to his free-market ideology, Greenspan said: "I have found a flaw. I don't know how significant or permanent it is. But I have been very distressed by that fact." When Representative Henry Waxman (D-CA) pressed him to clarify his words. "In other words, you found that your view of the world, your ideology, was not right, it was not working," Waxman said. "Absolutely, precisely", Greenspan replied. "You know, that's precisely the reason I was shocked, because I have been going for 40 years or more with very considerable evidence that it was working exceptionally well." Greenspan admitted fault in opposing regulation of derivatives and acknowledged that financial institutions didn't protect shareholders and investments as well as he expected.


title: Alan Greenspan

text: Matt Taibbi described the Greenspan put and its bad consequences saying: "every time the banks blew up a speculative bubble, they could go back to the Fed and borrow money at zero or one or two percent, and then start the game all over", thereby making it "almost impossible" for the banks to lose money. He also called Greenspan a "classic con man" who, through political savvy, "flattered and bullshitted his way up the Matterhorn of American power and ... jacked himself off to the attention of Wall Street for 20 consecutive years".


title: Alan Greenspan

text: In the documentary film "Inside Job", Greenspan is cited as one of the persons responsible for the financial crisis of 2007–2008. He is also named in "Time" magazine as one of the "25 People to Blame for the Financial Crisis".


title: Alan Greenspan

text: In March 2005, in reaction to Greenspan's support of President Bush's plan to partially privatize Social Security, then-Democratic Senate Minority Leader Harry Reid attacked Greenspan as "one of the biggest political hacks we have in Washington" and criticized him for supporting Bush's 2001 tax cut plan. Then-Democratic House Minority Leader Nancy Pelosi added that there were serious questions about the Fed's independence as a result of Greenspan's public statements. Greenspan also received criticism from Democratic Congressman Barney Frank and others for supporting Bush's Social Security plans favoring private accounts. Greenspan had said Bush's model has "the seeds of developing full funding by its very nature. As I've said before, I've always supported moves to full funding in the context of a private account".


title: Alan Greenspan

text: Others, like Republican Senator Mitch McConnell, disagreed that Greenspan was too deferential to Bush, stating that Greenspan "has been an independent player at the Fed for a long time under both parties and made an enormous positive contribution".


title: Alan Greenspan

text: Economist Paul Krugman wrote that Greenspan was a "three-card maestro" with a "lack of sincerity" who, "by repeatedly shilling for whatever the Bush administration wants, has betrayed the trust placed in the Fed chairman".


title: Alan Greenspan

text: Republican Senator Jim Bunning, who opposed Greenspan's fifth reconfirmation, charged that Greenspan should comment only on monetary policy, not fiscal policy. Greenspan had used his position as Fed chairman to comment upon fiscal policy as early as 1993, however, when he supported President Clinton's deficit reduction plan, which included tax increases and budget cuts.


title: Alan Greenspan

text: In an October 2011 lecture addressing the Occupy movement, Noam Chomsky characterized portions of Greenspan's February 1997 testimony to the U.S. Senate as an example of the self-serving attitudes of the so-called 1%. In that testimony, Greenspan had stated that growing worker insecurity is a significant factor keeping inflation and inflation expectation low, thereby promoting long-term investment.


title: Alan Greenspan

text: Greenspan has married twice. His first marriage was to Canadian artist Joan Mitchell in October 1952; the marriage ended in annulment 10 months later. He dated newswoman Barbara Walters in the late 1970s. In December 1984, Greenspan began dating journalist Andrea Mitchell. Greenspan at the time was 58 and Mitchell was 38. In April 1997, they were married by Supreme Court Justice Ruth Bader Ginsburg.


title: Alan Greenspan

text: In 1976, Greenspan received the U.S. Senator John Heinz Award for Greatest Public Service by an Elected or Appointed Official, an award given out annually by Jefferson Awards.


title: Alan Greenspan

text: In 2004, Greenspan received the Dwight D. Eisenhower Medal for Leadership and Service, from Eisenhower Fellowships. In 2005, he became the first recipient of the Harry S. Truman Medal for Economic Policy, presented by the Harry S. Truman Library Institute. In 2007, Greenspan was the recipient of the inaugural Thomas Jefferson Foundation Medal in Citizen Leadership, presented by the University of Virginia.


title: Alan Greenspan

text: On April 19, 2012, Greenspan received the Eugene J. Keogh Award for Distinguished Public Service from NYU.


title: Unified Payments Interface

text: Unified Payments Interface (UPI) is an instant real-time payment system developed by National Payments Corporation of India (NPCI). The interface facilitates inter-bank peer-to-peer (P2P) and person-to-merchant (P2M) transactions. It is used on mobile devices to instantly transfer funds between two bank accounts. The mobile number on the device is required to be registered with the bank. The UPI ID of the recipient can be used to transfer money. It runs as an open source application programming interface (API) on top of Immediate Payment Service (IMPS), and is regulated by the Reserve Bank of India (RBI).


title: Unified Payments Interface

text: As of September 2022, there were 358 banks available on the platform with a monthly volume of 6.7 billion transactions amounting to .


title: Unified Payments Interface

text: As of May 2021, the platform has over 10 crore (100 million) monthly active users in India. The proportion of UPI transactions in total volume of digital transactions grew from 23% in 2018–19 to 55% in 2020–21 with an average value of ₹1,849 per transaction. Digital transactions worth ₹8.31 lakh crore were made via the platform in January 2022. In FY 2021–22, the value of transactions crossed $1 trillion.