Morgan - Numerical Methods (523161), страница 49
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It is of the form:f(x) = anxn + an-1xn-1 + . . . + a1x + a0numeratorpositional numbering systemsn2,n1 = n2, n1 > n2 See Chapter 1.numerationlllllThe dividend in a fraction.OctalBase 8.One’s-complementA bit by bit inversion of a number.All ones are made zeros and zerosare made ones.490A numbering system in which the valueof a number is based upon its position,the value of any position is equal to thenumber multiplied by the base of thesystem taken to the power of the position. See Chapter 1.GLOSSARYpositiverational numberPlus.
Those numbers to the right of zeroon the number line. The opposite of anegative number.A number capable of being representedexactly in a particular base.powerA number possessing a fractional extension.Multiplying a value, x, by itself n number of times raises it the the power n.The notation is xn.precisionNumber of digits used to representa value.productThe result of a multiplication.quadword (qword)Four words. On an 8086, this would be64 bits.quotientThe result of a division.radicandThe quantity under the radical.
Three isthe radicand in the expressionwhich represents the square root ofthree.real numberremainderThe difference between the dividendand the product of the divisor and thequotient.resolutionThe constituent parts of a system. Thishas to do with the precision the arithmetic uses to represent values, the greaterthe precision, the more resolution.restoring divisionA form of division in which the divisoris subtracted from the dividend until anunderflow occurs.
At this point, thedivisor is added back into the dividend.The number of times the divisor couldbe subtracted without underflow is returned as the quotient and the last minuend is returned as the remainder.radixrootThe base of a numbering system.The nth root of a number, x, ( written:radix pointis that number when raised tothe nth power is equal to the originalThe division in a number betweenits integer portion and fractional portion. In the decimal system, it is thedecimal point.number (x = an).491NUMERICAL METHODSroundingsineA specified method of reducing the number of digits in a number while adjustingthe remaining digits accordingly.In Figure one, it is the ratio y/r.single-precisionscalingA technique that brings a number withincertain bounds by multiplication or division by a factor.
In a floating point number, the significand is always between1.0 and 2.0 and the exponent is the scaling factor.seedThe initial input to the linear congruentialpsuedo-random number generator.short realThe short real is defined by IEEE 754 asa single precision floating point number.sign-extensionThe sign of the number-one for negative, zero for positive-fills any unusedbits from the MSB of the actual numberto the MSB of the data type. For example, -9H, in two’s complement notation is f7H expressed in eight bits andfff7H in sixteen. Note that the sign bitfills out the data type to the MSB.In accordance with the IEEE format, it isa floating point comprising 32 bits, witha 24 bit significand, eight bit exponent,and sign bit.subtractionThe process opposite to addition.
Deduction or taking away.subtrahendA number you subtract from another.sumThe result of an addition.tangent (tan)In figure one, the ratio y/x denotes thetangent.two’s complementA one’s complement plus one.under flowThis occurs when the result of an operation requires a borrow.whole numberAn integer.significant digitswordThe principal digits in a number.The basic precision offered by a computer. On an 8086, it is 16 bits.significandIn a floating point number, it is the leading bit (implicit or explicit) to the immediate left of the radix point and the fractionto the right.492Previous HomeIndexSymbols32-bit operands 493x256 + 14x16 + 7x1 114-bit quantities 46Aaccuracy 88, 124add64 36addition 21, 33, 136, 164additional system 8arbitrary numbers 281ASCII 164, 179, 182, 187, 192, 200ASCII Adjust 30ASCII Adjust After Addition 164ASCII Adjust After Multiply 164ASCII Adjust After Subtraction 164ASCII Adjust before Division 164ASCII to Single-Precision Float 192associative laws 126atf 195, 193auxiliary carry 25, 40auxiliary carry flag 42, 164Bbase 10, 85, 88bfc_dc 173binary arithmetic 12binary byte 51binary division 63binary multiplication 46binary-to-decimal 187bit pair encoding 56bit-pair 57, 58bn_dnt 166Booth 54, 55branching 26Bresenham 100CC 200carry 24carry flag 34, 92Cartesian coordinate system 239cdiv 67ceil 265Chi-square 288chop 90circle 95circle: 98circular 239, 242circular functions 239close 289cmul 49cmul2 51coefficients 9congruence 16congruent 284, 285conversion 163CORDIC 237core routines 134errorsmultiplication 135subtraction 135addition 135division 135cosine 16, 89, 96, 125, 224, 241, 274Ddaa 164dcsin 225decimal 164decimal addition and subtraction 40decimal adjust 42decimal and ASCII instructions 30decimal arithmetic 164decimal integers 85denormal arithmetic 124denormals 125dfc_bn 176diminished-radix complement 18div32 74div64 78, 80divide 154division 21, 63, 114, 165, 175, 43, 85, 147493NUMERICAL METHODSdivision by inversion 105division by multiplication 114divisor 108divmul 116, 117divnewt 108, 109dnt_bn 170drawing circles 95Eelementary functions 217error 88, 89, 94, 178error checking 63, 147errors 64exponent 129extended precision 131external routines 132Ffaster shift and add 50finite divided difference approximation 218fixed point 15, 17, 33, 86, 206floating point 8, 15, 17, 86, 206FLADD 136FLADD Routine 140FLADD: The Epilogue 144FLADD: The Prologue 138flceil 265FLDIV 154FLMUL 147floating-point arithmetic 123floating-point conversions 192floating point divide 79floor 262flr 263flsin 274flsqr 270four-bit multiply 47fp_add 132fraction 95fractional arithmetic 15, 33, 87, 88fractional conversions 165fractional multiply 80frxp 259Fta 202fta 200ftf 207494ftfx 212fx_sqr 254GGeneral Purpose Interface Bus 163guard bits 92guard digits 89, 248Hhardware division 69hardware multiply 61hex 179hexasc: 180hidden bit 124, 125Homers rule 248, 259, 274hyperbolic functions 239IIEEE754 17, 19, 87, 123, 127, 129, 131,137, 159, 211IEEE 854 125input 163Instructions 26addition 26add 26add-with-carry 27division 28divide 28modulus 28signed divide 28signed modulus 28multiplication 27multiply 27signed multiply 27negation and signs 28one’s complement 28sign extension 29two’s complement 28shifts, rotates and normalization 29arithmetic shift.
29normalization 29rotate 29rotate-through-carry 29subtraction 27compare 27subtract 27INDEXsubtract-with-carry 27integer conversions 165integers 33ints 206irand 284irandom 287irrational 12Jjamming 90Kk-space 288Llaccum 193Least Significant Bit 12, 26ldxp 261lgl0 219line 101line-Drawing 100linear congruential method 16linear interpolation 77, 217, 224logarithm 21logarithms 219Long real 17long real 86longs 206look-up tables 217loop counter 48Mmantissa 129memory location 51Microprocesors 22Buswidth 22Data type 24flags 24auxiliary carry 25carry 24overflow 24overflow trap 25Parity 25sign 24sticky bit 25zero 24middle-square method 282minimax 274minimax polynomial 259modular 85modular arithmetic 16modularity 125Most Significant Bit (MSB) 18mu132 62, 63mu164a 151multiplication 21, 27, 43, 61, 147, 169, 172multiplication and division 42multiprecision arithmetic 35multiprecision division 71multiprecision subtraction 37multiword division 73Nnatural numbers 7, 8negation and signs 28Newton-Raphson Iteration 105Newton’s Method 253, 270normalization 72, 147, 200normalize 114, 128normalizing 192Not a Number 129number line 7, 9, 18number ray 7numeration 70One’s complement 19, 20, 28original dividend 73original divisor 72output 163overflow 24, 39, 64, 65, 95overflow flag 39overflow trap 25Ppacked binary 40Polyeval 251Polynomial 247polynomial 131, 175, 248polynomial interpretation 50495NUMERICAL METHODSpolynomials 9, 46positional arithmetic 34positional notation 50positional number theory 47positional numbering system 85positional representation 8potency 283power 21power series 247, 274powers 9, 12, 13, 233, 239proportion 108Pseudo-Random Number Generator 281Pwrb 234Qquantities 33quotient 67Rradix complement 18, 19radix conversions 165radix point 12irrational 12random numbers 281range 86real number 85resolution 179restoring division 188rinit 284root 21, 239rotation matrix 239round 160, 172round to nearest 91, 159rounding 25, 89, 90, 159signed 20, 44signed addition and subtraction 38signed arithmetic 28, 38signed magnitude 129signed numbers 43signed-operation 44significant bits 87sine 89, 241, 259sines 16, 96, 125, 224, 273single-precision 206single-precision float to ASCII 200skipping ones and zeros 53software division 65spectral 289spectral.c 282, 288, 289square root 131, 233, 253, 259, 269sticky bit 25sub64 37subtraction 21, 34, 125, 136, 137, 164Sutherland, Ivan 95symbolic fraction 85Ttable-driven conversions 179tables 179, 233tan 239, 240taylorsin 249tb_bndc 188tb_dcbn 182The Radix Point 89the sticky bit 92time-critical code 53truncation 90two’s complement 19, 27, 28SVscaling 93school_sqr 256seed 282shift-and-add algorithm 47shifts, rotates and normalization 29short real 17, 86shuffling 283sign 18, 24sign digit 21sign-magnitude 18, 21, 32Von Neumann, John 282496Wwhole numbers 86Zzero 24NumericalMethods..........................techniques and useful 8086 and pseudo-codeNumerical Methods brings togetherexamples.
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