Gromacs
2020.4

#include <gromacs/tables/quadraticsplinetable.h>
Quadratic spline interpolation table.
This class interpolates a function specified either as an analytical expression or from userprovided table data.
At initialization, you provide the reference function of vectors as a list of tuples that contain a brief name, the function, and derivative for each function to tabulate. To create a table with two functions this initializer list can for instance look like
{ {"LJ6", lj6Func, lj6Der}, {"LJ12", lj12Func, lj12Der} }
The names are only used so exceptions during initialization can be traced to a specific table.
When interpolating, there are methods to interpolate either 1, 2, or 3 functions in one go. By default these interpolation routines will operate on tables with the same number of functions as specified in the interpolation method (debug builds check that this is consistent with the table). However, it is also possible to use optional template parameters that specify the total number of functions in a table, and what function index to interpolate. For instance, to interpolate the derivative of the second function (i.e., index 1) in a multifunctiontable with three functions in total, you can write
table.evaluateDerivative<3,1>(x,&der);
Here too, debug builds will check that the template parameters are consistent with the table.
The table data is internally adjusted to guarantee that the interpolated derivative is the true derivative of the interpolated potential, which is important to avoid systematic errors for the common case when the derivative is concave/convex in the entire interval. We do this by expressing the difference in the function value at a small offset h relative to a reference value in position 0 with a forward Taylor series expanded around 0, and then doing the opposite of expressing difference in the function at position 0 relative to a reference value in position h when using a backward Taylor expansion:
Summing the equations leads to
To make the second term symmetric too, we can replace it with the average of the Taylor expansion at 0 and h (i.e., using the third derivative). This gives
Thus, if we replace the derivative in the internal quadratic table data with
we will cancel the h^3 term in the error. This will make the integral of the forces match the potential much better (The h^4 term actually disappears, so when summing over 1/h points the remaining error will be O(h^4).
While it is possible to create tables only from function values (i.e., no derivatives), it is recommended to provide derivatives for higher accuracy and to avoid issues with numerical differentiation. Note that the table input should be smooth, i.e. it should not contain noise e.g. from an (iterative) Boltzmann inversion procedure  you have been warned.
Public Member Functions  
QuadraticSplineTable (std::initializer_list< AnalyticalSplineTableInput > analyticalInputList, const std::pair< real, real > &range, real tolerance=defaultTolerance)  
Initialize table data from function. More...  
QuadraticSplineTable (std::initializer_list< NumericalSplineTableInput > numericalInputList, const std::pair< real, real > &range, real tolerance=defaultTolerance)  
Initialize table data from tabulated values and derivatives. More...  
template<int numFuncInTable = 1, int funcIndex = 0, typename T >  
void  evaluateFunctionAndDerivative (T r, T *functionValue, T *derivativeValue) const 
Evaluate both function and derivative, single table function. More...  
template<int numFuncInTable = 1, int funcIndex = 0, typename T >  
void  evaluateFunction (T r, T *functionValue) const 
Evaluate function value only, single table function. More...  
template<int numFuncInTable = 1, int funcIndex = 0, typename T >  
void  evaluateDerivative (T r, T *derivativeValue) const 
Evaluate function derivative only, single table function. More...  
template<int numFuncInTable = 2, int funcIndex0 = 0, int funcIndex1 = 1, typename T >  
void  evaluateFunctionAndDerivative (T r, T *functionValue1, T *derivativeValue1, T *functionValue2, T *derivativeValue2) const 
Evaluate both function and derivative, two table functions. More...  
template<int numFuncInTable = 2, int funcIndex0 = 0, int funcIndex1 = 1, typename T >  
void  evaluateFunction (T r, T *functionValue1, T *functionValue2) const 
Evaluate function value only, two table functions. More...  
template<int numFuncInTable = 2, int funcIndex0 = 0, int funcIndex1 = 1, typename T >  
void  evaluateDerivative (T r, T *derivativeValue1, T *derivativeValue2) const 
Evaluate function derivative only, two table functions. More...  
template<int numFuncInTable = 3, int funcIndex0 = 0, int funcIndex1 = 1, int funcIndex2 = 2, typename T >  
void  evaluateFunctionAndDerivative (T r, T *functionValue1, T *derivativeValue1, T *functionValue2, T *derivativeValue2, T *functionValue3, T *derivativeValue3) const 
Evaluate both function and derivative, three table functions. More...  
template<int numFuncInTable = 3, int funcIndex0 = 0, int funcIndex1 = 1, int funcIndex2 = 2, typename T >  
void  evaluateFunction (T r, T *functionValue1, T *functionValue2, T *functionValue3) const 
Evaluate function value only, three table functions. More...  
template<int numFuncInTable = 3, int funcIndex0 = 0, int funcIndex1 = 1, int funcIndex2 = 2, typename T >  
void  evaluateDerivative (T r, T *derivativeValue1, T *derivativeValue2, T *derivativeValue3) const 
Evaluate function derivative only, three table functions. More...  
real  tableSpacing () const 
Return the table spacing (distance between points) More...  
Static Public Attributes  
static const real  defaultTolerance = 10.0 * 1.19209290e07F 
Default tolerance for tables is 10*GMX_FLOAT_EPS. More...  
gmx::QuadraticSplineTable::QuadraticSplineTable  (  std::initializer_list< AnalyticalSplineTableInput >  analyticalInputList, 
const std::pair< real, real > &  range,  
real  tolerance = defaultTolerance 

) 
Initialize table data from function.
analyticalInputList  Initializer list with one or more functions to tabulate, specified as pairs containing analytical functions and their derivatives. The function will also be called for values smaller than the lower limit of the range, but we avoid calling it for 0.0 if that value is not included in the range. 
range  Range over which the function will be tabulated. Constructor will throw gmx::APIError for negative values. Due to the way the numerical derivative evaluation depends on machine precision internally, this range must be at least 0.001, or the constructor throws gmx::APIError. 
tolerance  Requested accuracy of the table. This will be used to calculate the required internal spacing. If this cannot be achieved (for instance because the table would require too much memory) the constructor will throw gmx::ToleranceError. 
gmx::ToleranceError  if the requested tolerance cannot be achieved, and gmx::APIError for other incorrect input. 
gmx::QuadraticSplineTable::QuadraticSplineTable  (  std::initializer_list< NumericalSplineTableInput >  numericalInputList, 
const std::pair< real, real > &  range,  
real  tolerance = defaultTolerance 

) 
Initialize table data from tabulated values and derivatives.
numericalInputList  Initializer list with one or more functions to tabulate, specified as pairs containing containing vectors for the function values and their derivatives. Data points are separated by the spacing parameter, starting from 0. Values below the lower limit of the range will be used to attempt defining the table, but we avoid using index 0 unless 0.0 is included in the range. Some extra points beyond range.second are required to reinterpolate values, so add some margin. The constructor will throw gmx::APIError if the input vectors are too short to cover the requested range (and they must always be at least five points). 
range  Range over which the function will be tabulated. Constructor will throw gmx::APIError for negative values, or if the value/derivative vector does not cover the range. 
tolerance  Requested accuracy of the table in the range. This will be used to calculate the required internal spacing and possibly reinterpolate. The constructor will throw gmx::ToleranceError if the input spacing is too coarse to achieve this accuracy. 

inline 
Evaluate function derivative only, single table function.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 1 
funcIndex  Index of function to evaluate in table, default is 0 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function derivative  
[out]  derivativeValue  Function derivative 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate function derivative only, two table functions.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 2 
funcIndex0  Index of 1st function to evaluate in table, default is 0 
funcIndex1  Index of 2nd function to evaluate in table, default is 1 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function derivative  
[out]  derivativeValue1  Interpolated derivative for first function 
[out]  derivativeValue2  Interpolated derivative for second function 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate function derivative only, three table functions.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 3 
funcIndex0  Index of 1st function to evaluate in table, default is 0 
funcIndex1  Index of 2nd function to evaluate in table, default is 1 
funcIndex2  Index of 3rd function to evaluate in table, default is 2 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function derivative  
[out]  derivativeValue1  Interpolated derivative for first function 
[out]  derivativeValue2  Interpolated derivative for second function 
[out]  derivativeValue3  Interpolated derivative for third function 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate function value only, single table function.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 1 
funcIndex  Index of function to evaluate in table, default is 0 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function value  
[out]  functionValue  Function value 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate function value only, two table functions.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 2 
funcIndex0  Index of 1st function to evaluate in table, default is 0 
funcIndex1  Index of 2nd function to evaluate in table, default is 1 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function value  
[out]  functionValue1  Interpolated value for first function 
[out]  functionValue2  Interpolated value for second function 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate function value only, three table functions.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 3 
funcIndex0  Index of 1st function to evaluate in table, default is 0 
funcIndex1  Index of 2nd function to evaluate in table, default is 1 
funcIndex2  Index of 3rd function to evaluate in table, default is 2 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function value  
[out]  functionValue1  Interpolated value for first function 
[out]  functionValue2  Interpolated value for second function 
[out]  functionValue3  Interpolated value for third function 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate both function and derivative, single table function.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 1 
funcIndex  Index of function to evaluate in table, default is 0 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function and derivative  
[out]  functionValue  Function value 
[out]  derivativeValue  Function derivative 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate both function and derivative, two table functions.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 2 
funcIndex0  Index of 1st function to evaluate in table, default is 0 
funcIndex1  Index of 2nd function to evaluate in table, default is 1 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function and derivative  
[out]  functionValue1  Interpolated value for first function 
[out]  derivativeValue1  Interpolated derivative for first function 
[out]  functionValue2  Interpolated value for second function 
[out]  derivativeValue2  Interpolated derivative for second function 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Evaluate both function and derivative, three table functions.
This is a templated method where the template can be either real or SimdReal.
numFuncInTable  Number of separate functions in table, default is 3 
funcIndex0  Index of 1st function to evaluate in table, default is 0 
funcIndex1  Index of 2nd function to evaluate in table, default is 1 
funcIndex2  Index of 3rd function to evaluate in table, default is 2 
T  Type (SimdReal or real) of lookup and result 
r  Points for which to evaluate function and derivative  
[out]  functionValue1  Interpolated value for first function 
[out]  derivativeValue1  Interpolated derivative for first function 
[out]  functionValue2  Interpolated value for second function 
[out]  derivativeValue2  Interpolated derivative for second function 
[out]  functionValue3  Interpolated value for third function 
[out]  derivativeValue3  Interpolated derivative for third function 
For debug builds we assert that the input values fall in the range specified when constructing the table.

inline 
Return the table spacing (distance between points)
You should never have to use this for normal code, but due to the way tables are constructed internally we need this in the unit tests to check relative tolerances over each interval.

static 
Default tolerance for tables is 10*GMX_FLOAT_EPS.