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Horner's Method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials. The algorithm is based on Horner's rule, in which a polynomial is written in ''nested form'': \begin &a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots + a_nx^n \\ = &a_0 + x \bigg(a_1 + x \Big(a_2 + x \big(a_3 + \cdots + x(a_ + x \, a_n) \cdots \big) \Big) \bigg). \end This allows the evaluation of a polynomial of degree with only n multiplications and n additions. This is optimal, since there are polynomials of degree that cannot be evaluated with fewer arithmetic operations. Alternatively, Horner's method and also refers to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Evaluation
In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the polynomial P(x_1, x_2) = 2x_1x_2 + x_1^3 + 4 at x_1=2, x_2=3 consists of computing P(2,3)= 2\cdot 2\cdot 3 + 2^3+4=24. See also For evaluating the univariate polynomial a_nx^n+a_x^+\cdots +a_0, the most naive method would use n multiplications to compute a_n x^n, use n-1 multiplications to compute a_ x^ and so on for a total of \tfrac multiplications and n additions. Using better methods, such as Horner's rule, this can be reduced to n multiplications and n additions. If some preprocessing is allowed, even more savings are possible. Background This problem arises frequently in practice. In computational geometry, polynomials are used to compute function approximations using Taylor polynomials. In cryptography and hash tables, polynomials are used to compute ''k''-independent has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtraction
Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the ''difference'' of 5 and 2 is 3; that is, . While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, Fraction (mathematics), fractions, irrational numbers, Euclidean vector, vectors, decimals, functions, and matrices. In a sense, subtraction is the inverse of addition. That is, if and only if . In words: the difference of two numbers is the number that gives the first one when added to the second one. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as group (mathematics), groups and ring (mathematics), rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an Additive identity, (often denoted as 0) and an identity with respect to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Shift
In computer programming, an arithmetic shift is a shift operator, sometimes termed a signed shift (though it is not restricted to signed operands). The two basic types are the arithmetic left shift and the arithmetic right shift. For binary numbers it is a bitwise operation that shifts all of the bits of its operand; every bit in the operand is simply moved a given number of bit positions, and the vacant bit-positions are filled in. Instead of being filled with all 0s, as in logical shift, when shifting to the right, the leftmost bit (usually the sign bit in signed integer representations) is replicated to fill in all the vacant positions (this is a kind of sign extension). Some authors prefer the terms ''sticky right-shift'' and ''zero-fill right-shift'' for arithmetic and logical shifts respectively. Arithmetic shifts can be useful as efficient ways to perform multiplication or division of signed integers by powers of two. Shifting left by ''n'' bits on a signed or uns ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division By Zero
In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the dividend (numerator). The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplication, multiplied by the divisor. That is, c = \tfrac ab is equivalent to c \cdot b = a. By this definition, the quotient q = \tfrac is nonsensical, as the product q \cdot 0 is always 0 rather than some other number a. Following the ordinary rules of elementary algebra while allowing division by zero can create a mathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic of real numbers and more general numerical structures called field (mathematics), fields leaves division by zero undefined (mathematics), undefined, and situations where division by zero might occur m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Numeral System
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" ( zero) and "1" ( one). A ''binary number'' may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harrio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Multiplier
A binary multiplier is an electronic circuit used in digital electronics, such as a computer, to multiply two binary numbers. A variety of computer arithmetic techniques can be used to implement a digital multiplier. Most techniques involve computing the set of ''partial products,'' which are then summed together using binary adders. This process is similar to long multiplication, except that it uses a base-2 ( binary) numeral system. History Between 1947 and 1949 Arthur Alec Robinson worked for English Electric, as a student apprentice, and then as a development engineer. Crucially during this period he studied for a PhD degree at the University of Manchester, where he worked on the design of the hardware multiplier for the early Mark 1 computer. However, until the late 1970s, most minicomputers did not have a multiply instruction, and so programmers used a "multiply routine" which repeatedly shifts and accumulates partial results, often written using loop unwinding. Main ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microcontroller
A microcontroller (MC, uC, or μC) or microcontroller unit (MCU) is a small computer on a single integrated circuit. A microcontroller contains one or more CPUs (processor cores) along with memory and programmable input/output peripherals. Program memory in the form of NOR flash, OTP ROM, or ferroelectric RAM is also often included on the chip, as well as a small amount of RAM. Microcontrollers are designed for embedded applications, in contrast to the microprocessors used in personal computers or other general-purpose applications consisting of various discrete chips. In modern terminology, a microcontroller is similar to, but less sophisticated than, a system on a chip (SoC). A SoC may include a microcontroller as one of its components but usually integrates it with advanced peripherals like a graphics processing unit (GPU), a Wi-Fi module, or one or more coprocessors. Microcontrollers are used in automatically controlled products and devices, such as automobile engi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instruction-level Parallelism
Instruction-level parallelism (ILP) is the Parallel computing, parallel or simultaneous execution of a sequence of Instruction set, instructions in a computer program. More specifically, ILP refers to the average number of instructions run per step of this parallel execution. Discussion ILP must not be confused with Concurrency (computer science), concurrency. In ILP, there is a single specific Thread (computing), thread of execution of a Process (computing), process. On the other hand, concurrency involves the assignment of multiple threads to a Central processing unit, CPU's core in a strict alternation, or in true parallelism if there are enough CPU cores, ideally one core for each runnable thread. There are two approaches to instruction-level parallelism: Computer hardware, hardware and software. Hardware-level ILP works upon dynamic parallelism, whereas software-level ILP works on static parallelism. Dynamic parallelism means that the processor decides at run time whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating-point
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469/200 = 12.345 = \! \underbrace_\text \! \times \! \underbrace_\text\!\!\!\!\!\!\!\overbrace^ However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use Binary number, base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIMD
Single instruction, multiple data (SIMD) is a type of parallel computer, parallel processing in Flynn's taxonomy. SIMD describes computers with multiple processing elements that perform the same operation on multiple data points simultaneously. SIMD can be internal (part of the hardware design) and it can be directly accessible through an instruction set architecture (ISA), but it should not be confused with an ISA. Such machines exploit Data parallelism, data level parallelism, but not Concurrent computing, concurrency: there are simultaneous (parallel) computations, but each unit performs exactly the same instruction at any given moment (just with different data). A simple example is to add many pairs of numbers together, all of the SIMD units are performing an addition, but each one has different pairs of values to add. SIMD is particularly applicable to common tasks such as adjusting the contrast in a digital image or adjusting the volume of digital audio. Most modern Cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
