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Katherine Johnson’s mathematical calculations helped guide some of America’s most important early space missions while she confronted the racial and gender barriers faced by Black women in twentieth-century America.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
Floating-point error mitigation is the minimization of errors caused by the fact that real numbers cannot, in general, be accurately represented in a fixed space. By definition, floating-point error cannot be eliminated, and, at best, can only be managed.
Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator":[1]
Thus under some conditions, the major portion of the significant data digits may lie beyond the capacity of the registers. Therefore, the result obtained may have little meaning if not totally erroneous.
The Z1, developed by Konrad Zuse in 1936, was the first computer with floating-point arithmetic and was thus susceptible to floating-point error. Early computers, however, with operation times measured in milliseconds, could not solve large, complex problems[2] and thus were seldom plagued with floating-point error. Today, however, with supercomputer system performance measured in petaflops, floating-point error is a major concern for computational problem solvers.
The following sections describe the strengths and weaknesses of various means of mitigating floating-point error.
Though not the primary focus of numerical analysis,[3][4]: 5 numerical error analysis exists for the analysis and minimization of floating-point rounding error.
Error analysis by Monte Carlo arithmetic is accomplished by repeatedly injecting small errors into an algorithm's data values and determining the relative effect on the results.
Extension of precision is using of larger representations of real values than the one initially considered. The IEEE 754 standard defines precision as the number of digits available to represent real numbers. A programming language can include single precision (32 bits), double precision (64 bits), and quadruple precision (128 bits). While extension of precision makes the effects of error less likely or less important, the true accuracy of the results is still unknown.
Variable length arithmetic represents numbers as a string of digits of a variable's length limited only by the memory available. Variable-length arithmetic operations are considerably slower than fixed-length format floating-point instructions. When high performance is not a requirement, but high precision is, variable length arithmetic can prove useful, though the actual accuracy of the result may not be known.
The floating-point algorithm known as TwoSum[5] or 2Sum, due to Knuth and Møller, and its simpler, but restricted version FastTwoSum or Fast2Sum (3 operations instead of 6), allow one to get the (exact) error term of a floating-point addition rounded to nearest. One can also obtain the (exact) error term of a floating-point multiplication rounded to nearest in 2 operations with a fused multiply–add (FMA), or 17 operations if the FMA is not available (with an algorithm due to Dekker). These error terms can be used in algorithms in order to improve the accuracy of the final result, e.g. with floating-point expansions or compensated summations.
Operations giving the result of a floating-point addition or multiplication rounded to nearest with its error term (but slightly differing from algorithms mentioned above) have been standardized and recommended in the IEEE 754-2019 standard.
Changing the radix, in particular from binary to decimal, can help to reduce the error and better control the rounding in some applications, such as financial applications.
Interval arithmetic is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Values are intervals, which can be represented in various ways, such as:[6]
"Instead of using a single floating-point number as approximation for the value of a real variable in the mathematical model under investigation, interval arithmetic acknowledges limited precision by associating with the variable a set of reals as possible values. For ease of storage and computation, these sets are restricted to intervals."[7]
The evaluation of interval arithmetic expression may provide a large range of values,[7] and may seriously overestimate the true error boundaries.[8]: 8
Unums ("Universal Numbers") are an extension of variable length arithmetic proposed by John Gustafson.[9] Unums have variable length fields for the exponent and significand lengths and error information is carried in a single bit, the ubit, representing possible error in the least significant bit of the significand (ULP).[9]: 4
The efficacy of unums is questioned by William Kahan.[8]
Bounded floating point is a method proposed and patented by Alan Jorgensen.[10] The data structure includes the standard IEEE 754 data structure and interpretation, as well as information about the error between the true real value represented and the value stored by the floating point representation.[11]
Bounded floating point has been criticized as being derivative of Gustafson's work on unums and interval arithmetic.[10][12]
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Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
Floating-point error mitigation is the minimization of errors caused by the fact that real numbers cannot, in general, be accurately represented in a fixed space. By definition, floating-point error cannot be eliminated, and, at best, can only be managed. Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator": Thus under some conditions, the major portion of the significant data digits may lie beyond the capacity of the registers. Therefore, the result obtained may have little meaning if not totally erroneous. The Z1, developed by Konrad Zuse in 1936, was the first computer with floating-point arithmetic and was thus susceptible to floating-point error. Early computers, however, with operation times measured in milliseconds, could not solve large, complex problems and thus were seldom plagued with floating-point error. Today, however, with supercomputer system performance measured in petaflops, floating-point error is a major concern for computational problem solvers. The following sections describe the strengths and weaknesses of various means of mitigating floating-point error.
RapidCAD is a specially packaged Intel 486DX and a dummy floating-point unit (FPU) designed as pin-compatible replacements for an Intel 80386 processor and 80387 FPU. Because the i486DX has a working on-chip FPU, a dummy FPU package (the "RapidCAD-2") is supplied to go in the Intel 387 FPU socket. The dummy FPU is used to provide the FERR (Floating-point ERRor) signal, necessary for compatibility purposes. Despite being able to execute instructions in the same number of cycles as an i486DX, integer performance on RapidCAD suffers due to the absence of level 1 cache and the bottleneck of the 386 bus. RapidCAD offers minimal improvement in integer performance over a 386DX (typically 10%, at most 35%), but provides substantial improvement in floating-point operations (up to 80% faster) for which it was marketed, primarily for computer-aided design (hence the name). Floating-point performance is mostly improved by moving the FPU onto the CPU core. Other source reports that the performance boost ranged from 20 to 170 percent faster than the regular 386DX based motherboards. It was available at manufacturer's suggested list price of US$499.
As with other spreadsheets, Microsoft Excel works only to limited accuracy because it retains only a certain number of figures to describe numbers (it has limited precision). With some exceptions regarding erroneous values, infinities, and denormalized numbers, Excel calculates in double-precision floating-point format from the IEEE 754 specification (besides numbers, Excel uses a few other data types). Although Excel allows display of up to 30 decimal places, its precision for any specific number is no more than 15 significant figures, and calculations may have an accuracy that is even less due to five issues: round off, truncation, and binary storage, accumulation of the deviations of the operands in calculations, and worst, "catastrophic cancellation" at subtraction of values with similar magnitude. Standard floating-point error mitigation techniques apply.
Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions (common in human-entered data, such as measurements or financial information) and binary (base-2) fractions. The advantage of decimal floating-point representation over decimal fixed-point and integer representation is that it supports a much wider range of values. For example, while a fixed-point representation that allocates 8 decimal digits and 2 decimal places can represent the numbers 123456.78, 8765.43, 123.00, and so on, a floating-point representation with 8 decimal digits could also represent 1.2345678, 1234567.8, 0.000012345678, 12345678000000000, and so on. This wider range can dramatically slow the accumulation of rounding errors during successive calculations; for example, the Kahan summation algorithm can be used in floating point to add many numbers with no asymptotic accumulation of rounding error.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Madam C.J. Walker