This is part of the HicEst documentation

SOLVE: Linear and Nonlinear Equations and Data Fit


Find roots or minima of nonlinear algebraic equations or perform least-square-fit to data of an overdetermined system (Levenberg-Marquardt).

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Bookmarks:
⇾callback_function ⇾cubic_equations ⇾function_minimum ⇾inline_expression ⇾inversed_matrix ⇾linear_equations ⇾quadratic_equations ⇾quintic_equation ⇾root_of_function ⇾tolerance_weights ⇾transpose_matrix

Optional keywords:
(Syntax of optional keywords)
Const DATA DataIdx ERror Iters Limit MaXIters NUL NumDiff STDdev Speed1 SpeedHigh SpeedLow SpeedStop Unknown UnknownIdx
keywords type mini sample keyword sequence is insignificant
NUL xpr NUL=chi() required to return a value to SOLVE:
  • chi() is an
    inline_expression
    or a
    callback_function
    that SOLVE tries to minimize. This can be used to determine
    • the root(s) of an expression, e.g.:
      • NUL=x^2 -1
    • the minimum of a function, e.g.:
      • NUL=x^2 + 1
    • the parameters of a function to least-square fit some data like
      • NUL = ( theory(nr) - data(nr) ) / error(nr)
    The function can include
    tolerance_weights
    for each data point (equation) to account for relative weights or errors, like error(nr) in the example.
Unknown NUM u=x
  • scalar argument required for cases with a single root or a single least-square fit parameter
  • x is the initial guess (default 0) on input, and is the root on output
  • the Callback function in NUL=... can be entered directly in this case (e.g. NUL=x^2-1)
  • AXIS(MiN=..., MaX=...) and LINE(..., Points=...) come handy to visualize the equation
  • Example: find a root and a function minimum of a
    quintic_equation
    (1 root, 1 equation) SOLVE-quintic.png
    • chi2 = SOLVE( Unknown=x, NUL=x^5 - x + 1 )
    results usually depend on initial x, e.g.:
Unknown vec u=vec
  • a vector argument is required for cases with m>1 roots. The solution vector must be dimensioned with m elements:
  • example of a least-square fit to the Gaussian y=h* EXP(-((x-x0)/w)^2). Measured data are in the vectors x and y
    • parms = ALIAS(h, x0, w) ! for better legibility form a vector from the 3 parameters
    • parms = (8, 5, 2) ! guess some initial parameters for h, x0, w
    • chi2 = SOLVE(Unknown=parms, DATA=m, DataIdx=nr, NUL=1-h* EXP( -((x(nr)-x0)/w)^2)/y(nr) )
  • note the normalization of the NUL expression to y(nr) to force equal weights for all data points in the example. SOLVE-lsf.png
  • common practice is also the normalization to the measuring errors.
DataIdx NUM di=nr required for cases with m>1 roots:
  • DataIdx is needed to evaluate the correct NUL=... equation in the callback function
  • Example:
    • XY = ALIAS(x, y) ! same as REAL XY(2) with named elements: x == XY(1) and y == XY(2)
    • XY = (2, 7) ! define initial values for x and y
    • chi2 = SOLVE( U=XY, NUL=null(), DataIdx=nr ) ! callback function is null()
    • ! return values are x=7.497601303, y=2.768650594, chi2=3.7395492E-7
    • END ! of "main"
    • FUNCTION null() ! called by the solver of the SOLVE function. All variables are global
    • IF(nr == 1) null = x - SIN(x)* COSH(y) ! equation 1
    • IF(nr == 2) null = y - COS(x)* SINH(y) ! equation 2
    • END ! of function null()
DATA num data=d
  • required for a least-square fit with less roots (parameters p) than equations (data d)
  • for each parameter 1...p the function evaluation NUL=... is called d times
  • p is the dimension of vec in Unknown=vec
UnknownIdx NUM ui=np
  • this root index (or parameter index) can help to debug and organize the callback function
  • allows the script to perform index-invariant but timely part evaluations by testing np, e.g.:
    • IF(np == 0) THEN
    • ! sub-expressions valid for >1 parameters could go here
    • ENDIF
  • np = 0 : return the base-chi needed to approximate the derivatives ∂chi/∂root.
Const num c=2+8
  • optional to keep selected parameters constant while iterationg unstable models
  • c=2^(j-1) for unknown j to be kept constant, e.g.:
    • SOLVE(..., Const=10, ...) ! constant parameters 2 and 4
Limit num lim=10
  • to limit the next root iteration to root/Limit .... root*Limit
  • Limit must be > 1
  • default is no Limit (denoted by Limit=0)
MaXIters num mxi=5
  • maximum number of iterations
  • default is MaXIters = 1000
NumDiff num nc=delta
  • to approximate the differential ∂chi/∂p ≈ (chi(p+delta) - chi(p)) / delta. This can be useful if the callback function involves approximation techniques like numeric solutions to differential equations
  • default NumDiff = 1E-5
Speed1 num s1=1e5
  • starting value of the iteration acceleration factor on success
  • default Speed1 = 1000
SpeedStop num ss=1e-3
  • to stop iterations after multiple failures
  • default SpeedStop = 1e-10
ERror LBL Err=999 on error jump to an error label, e.g.:
  • SOLVE(....., ERROR=999)
Iters NUM i=n
  • returns in n the actual number of iterations performed
SpeedHigh NUM sh=high
  • returns the maximum successful acceleration factor
SpeedLow NUM sl=low
  • returns the minimum successful acceleration factor
STDdev vec std=dp
  • returns a RMS error estimate of Unknown(s)
  • dp should have the same dimension as vec in Unknown=vec
  • meaningful values only for normally distributed errors



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