(PDF 12) A sampled function (type 0) uses a table of sample values to de ne the

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function Various techniques are used to interpolate values between the sample values

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(PDF 13) An exponential interpolation function (type 2) de nes a set of coef cients for an exponential function

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Syntax

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(PDF 13) A stitching function (type 3) is a combination of other functions, partitioned across a domain

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(PDF 13) A PostScript calculator function (type 4) uses operators from the

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PostScript language to describe an arithmetic expression All function dictionaries share the entries listed in Table 326

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TABLE 326 Entries common to all function dictionaries

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KEY TYPE VALUE

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FunctionType

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integer

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(Required) The function type: 0 2 3 4 Sampled function Exponential interpolation function Stitching function PostScript calculator function

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Domain

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array

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(Required) An array of 2 m numbers, where m is the number of input values For each i from 0 to m 1, Domain2i must be less than or equal to Domain2i+1 , and the ith input value, xi , must lie in the interval Domain2i xi Domain2i+1 Input values outside the declared domain are clipped to the nearest boundary value (Required for type 0 and type 4 functions, optional otherwise; see below) An array of 2 n numbers, where n is the number of output values For each j from 0 to n 1, Range2j must be less than or equal to Range2j+1 , and the jth output value, yj , must lie in the interval Range2j yj Range2j+1 Output values outside the declared range are clipped to the nearest boundary value If this entry is absent, no clipping is done

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Range

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In addition, each type of function dictionary must include entries appropriate to the particular function type The number of output values can usually be inferred from other attributes of the function; if not (as is always the case for type 0 and type 4 functions), the Range entry is required The dimensionality of the function implied by the Domain and Range entries must be consistent with that implied by other attributes of the function

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Functions

391 Type 0 (Sampled) Functions

Type 0 functions use a sequence of sample values (contained in a stream) to provide an approximation for functions whose domains and ranges are bounded The samples are organized as an m-dimensional table in which each entry has n components Sampled functions are highly general and offer reasonably accurate representations of arbitrary analytic functions at low expense For example, a 1-input sinusoidal function can be represented over the range [0 180] with an average error of only 1 percent, using just ten samples and linear interpolation Two-input functions require signi cantly more samples, but usually not a prohibitive number, so long as the function does not have high frequency variations The dimensionality of a sampled function is restricted only by implementation limits However, the number of samples required to represent functions with high dimensionality multiplies rapidly unless the sampling resolution is very low Also, the process of multilinear interpolation becomes computationally intensive if the number of inputs m is greater than 2 The multidimensional spline interpolation is even more computationally intensive In addition to the entries in Table 326, a type 0 function dictionary includes those shown in Table 327 The Domain, Encode, and Size entries determine how the function s input variable values are mapped into the sample table For example, if Size is [21 31], the default Encode array is [0 20 0 30], which maps the entire domain into the full set of sample table entries Other values of Encode may be used To explain the relationship between Domain, Encode, Size, Decode, and Range, we use the following notation: y = Interpolate ( x , x min , x max , y min , y max ) y max y min = y min + ( x x min ) --------------------------- x max x min For a given value of x, Interpolate calculates the y value on the line de ned by the two points (xmin , ymin ) and (xmax , ymax )