Checks a user-supplied gradient of a function.
Required Arguments
FCN
User-supplied subroutine to evaluate the function of which the gradient will be
checked. The usage is
CALL FCN (N, X, F), where
N Length of X. (Input)
X The point at which
the function is evaluated. (Input)
X should not be
changed by FCN.
F The computed function value at the point X. (Output)
FCN must be declared EXTERNAL in the calling program.
GRAD Vector of length N containing the estimated gradient at X. (Input)
X Vector of length N containing the point at which the gradient is to be checked. (Input)
INFO Integer vector of length N. (Output)
INFO(I) = 0 means the user-supplied gradient is a poor estimate of the numerical gradient at the point X(I).
INFO(I) = 1 means the user-supplied gradient is a good estimate of the numerical gradient at the point X(I).
INFO(I) = 2 means the user-supplied gradient disagrees with the numerical gradient at the point X(I), but it might be impossible to calculate the numerical gradient.
INFO(I) = 3 means the user-supplied gradient and the numerical gradient are both zero at X(I), and, therefore, the gradient should be rechecked at a different point.
Optional Arguments
N Dimension of
the problem. (Input)
Default: N = size
(X,1).
FORTRAN 90 Interface
Generic: CALL CHGRD (FCN, GRAD, X, INFO [, ])
Specific: The specific interface names are S_CHGRD and D_CHGRD.
FORTRAN 77 Interface
Single: CALL CHGRD (FCN, GRAD, N, X, INFO)
Double: The double precision name is DCHGRD.
Description
The routine CHGRD uses the following finite-difference formula to estimate the gradient of a function of n variables at x:
where hi = ɛ1/2 max{|xi|, 1/si} sign(xi), ɛ is the machine epsilon, ei is the i-th unit vector, and si is the scaling factor of the i-th variable.
The routine CHGRD checks the user-supplied gradient ∇f(x) by comparing it with the finite-difference gradient g(x). If
where τ = ɛ1/4, then (∇f(x))i, which is the i-th element of ∇f(x), is declared correct; otherwise, CHGRD computes the bounds of calculation error and approximation error. When both bounds are too small to account for the difference, (∇f(x))i is reported as incorrect. In the case of a large error bound, CHGRD uses a nearly optimal stepsize to recompute gi(x) and reports that (∇f(x))i is correct if
Otherwise, (∇f(x))i is considered incorrect unless the error bound for the optimal step is greater than τ |(∇f(x))i|. In this case, the numeric gradient may be impossible to compute correctly. For more details, see Schnabel (1985).
Comments
1. Workspace may be explicitly provided, if desired, by use of C2GRD/DC2GRD. The reference is:
CALL C2GRD (FCN, GRAD, N, X, INFO, FX, XSCALE, EPSFCN, XNEW)
The additional arguments are as follows:
FX The functional value at X.
XSCALE Real vector of length N containing the diagonal scaling matrix.
EPSFCN The relative noise of the function FCN.
XNEW Real work vector of length N.
2. Informational errors
Type Code
4 1 The user-supplied gradient is a poor estimate of the numerical gradient.
Example
The user-supplied gradient of
at (625, 1, 3.125, 0.25) is checked where t = 2.125.
USE
CHGRD_INT
USE WRIRN_INT
IMPLICIT
NONE
!
Declare variables
INTEGER N
PARAMETER (N=4)
!
INTEGER INFO(N)
REAL GRAD(N), X(N)
EXTERNAL DRIV, FCN
!
! Input values for point X
! X = (625.0, 1.0, 3.125, .25)
!
DATA X/625.0E0, 1.0E0, 3.125E0, 0.25E0/
!
CALL DRIV (N, X, GRAD)
!
CALL CHGRD (FCN, GRAD, X, INFO)
CALL WRIRN ('The information vector', INFO, 1, N, 1)
!
END
!
SUBROUTINE FCN (N, X, FX)
INTEGER N
REAL X(N), FX
!
REAL EXP
INTRINSIC EXP
!
FX = X(1) + X(2)*EXP(-1.0E0*(2.125E0-X(3))**2/X(4))
RETURN
END
!
SUBROUTINE DRIV (N, X, GRAD)
INTEGER N
REAL X(N), GRAD(N)
!
REAL EXP
INTRINSIC EXP
!
GRAD(1) = 1.0E0
GRAD(2) = EXP(-1.0E0*(2.125E0-X(3))**2/X(4))
GRAD(3) = X(2)*EXP(-1.0E0*(2.125E0-X(3))**2/X(4))*2.0E0/X(4)* &
(2.125-X(3))
GRAD(4) = X(2)*EXP(-1.0E0*(2.125E0-X(3))**2/X(4))* &
(2.125E0-X(3))**2/(X(4)*X(4))
RETURN
END
Output
The information vector
1
2 3 4
1 1 1
1
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