hompack_nf Module

Array used during interpolation and Newton-step calculations.

Matrix for QR factorizations used in Newton-step calculations.

Array used in QR-factorization and Newton-step computations.

Array used during interpolation and Newton-step calculations.

Array used during interpolation and Newton-step calculations.

Array used for estimating the optimal next step size.

Array used for estimating the optimal next step size.

Pivot indices used by the QR factorization.

Absolute error tolerance.

Relative error tolerance.

Curvature-based error tolerance.

Optimal step size for the next step to be attempted by stepnf.

||yp - ypold|| at the previous step.

Total arc length of the solution path followed by the algorithm.

Problem type and status flag.

Limit on the number of steps allowed in the main loop of fixpnf.

Problem dimension.

Number of homotopy/Jacobian evaluations performed. This counter is incremented once for each call to tangnf, where the homotopy map and its Jacobian are assembled and used to compute a tangent vector and Newton correction.

Flag to indicate the first call to stepnf.

Flag to indicate that the return state of stepnf is a crash.

Parameter vector $a$ for curve-tracking problems, or initial point for the fixed-point and zero-finding problems.

Current point on the zero curve.

Previous point found on the zero curve.

Unit tangent vector to the zero curve at y.

Unit tangent vector to the zero curve at yold.

Linear-algebra workspace.

Relative error tolerance for the normal-flow iteration used while tracking the zero curve. If nonpositive, the default value arcre=0.5*sqrt(ansre) is used.

Absolute error tolerance for the normal-flow iteration used while tracking the zero curve. If nonpositive, the default value arcae=0.5*sqrt(ansae) is used.

Relative error tolerance required of the final solution at lambda=1.

Absolute error tolerance required of the final solution at lambda=1.

Curvature switch. If the curvature of any component of y exceeds this value, the tolerances for the normal-flow iteration are switched from arcre and arcae to the finer tolerances ansre and ansae, respectively.

Step-size control parameters: (lideal, rideal, dideal, hmin, hmax, bmin, bmax, p). Elements that are nonpositive on input are replaced by default values.

Maximum number of steps allowed in the main loop.

Logical I/O unit for intermediate output. * 0 : No output is printed (default). * 6 : Standard output (mapped internally to output_unit) * otherwise : Output to the specified I/O unit.

Optional flag used only by the polynomial-system driver POLSYS1H.

Arc length of the zero curve until the solution was found.

Final value of the homotopy parameter $\lambda$ at the solution.

Final value of the independent variable $x$ at the solution.

Number of homotopy/Jacobian evaluations.

Status.

Solver object.

Problem type. Must be one of the following: * fix_point : solve $F(x) = x$. * zero_find : solve $F(x) = 0$. * curve_track : track a zero curve of $\rho(a,\lambda,x) = 0$.

User-supplied function and Jacobian evaluation subroutines. Required callbacks depend on problem_type as follows: * For problem_type = fix_point or zero_find, callbacks f and fjac must be provided. * For problem_type = curve_track, callbacks rho and rhojac must be provided.

Problem dimension, i.e., the dimension of the independent variable $x$.

Dimension of the parameter vector $a$ for curve-tracking problems. Required if problem_type is curve_track.

Configuration parameters. If not provided, default values are used.

Solver object.

Initial point for the solver. For fixed-point and zero-finding problems, this is the initial guess $a$. For curve-tracking problems, this is the initial solution $x_0$ at $\lambda=0$.

Parameter vector $a$ for curve-tracking problems. Required if the solver was initialized with problem_type = curve_track.

Solver object.

Preallocated state for fixpnf. Set to start conditions on the first call, and updated on subsequent calls when iflag=2 or iflag=3.

User-supplied function and Jacobian evaluation subroutines.

Problem dimension.

Initial point for the solver. For fixed-point and zero-finding problems, this is identical to the initial guess $a$. For curve-tracking problems, this is the initial solution at $\lambda=0$.

Problem type and status flag.

Configuration parameters for the solver. Optional settings will be set to default values. Tolerances may be updated (increased) on output if iflag=2.

Parameter vector a for curve-tracking problems. Ignored for fixed-point and zero-finding problems.

State variables for fixpnf.

User-supplied function and Jacobian evaluation subroutines.

Relative convergence tolerance. Iteration is considered converged when |y(1) - 1| <= relerr + abserr and the Newton correction satisfies ||z|| <= relerr*||x|| + abserr.

Absolute convergence tolerance. Used together with relerr in the convergence criteria.

State variables for fixpnf.

User-supplied function and Jacobian evaluation subroutines.

Step-size control parameters: (lideal, rideal, dideal, hmin, hmax, bmin, bmax, p).

User-supplied function and Jacobian evaluation subroutines.

Controls computation of the homotopy residual norm.

Problem dimension.

Current point on the homotopy zero curve. Shape: (n+1). Contains (lambda, x).

Unit tangent vector to the zero curve at y. Shape: (n+1). Represents dy/ds, where s is arc length along the zero curve.

Unit tangent vector at the previous point on the zero curve. Shape: (n+1). Used to determine a consistent orientation for the newly computed tangent vector.

Parameter vector used in the homotopy map.

Number of homotopy/Jacobian evaluations performed. Incremented by one on every successful call.

Problem type and return status flag.

Workspace containing the Jacobian matrix and its QR factorization. Shape: (n, n+2).

Workspace array used during QR factorization and related linear algebra operations. Shape: (3*n+3).

Newton correction vector. Shape: (n+1). Equal to the negative pseudoinverse of the homotopy Jacobian applied to the homotopy residual.

Pivot indices produced by the QR factorization. Shape: (n+1).

Workspace object.

Problem dimension.

Error status of the allocation.

State object.

Problem dimension.

Dimension of the parameter vector a.

Error status of the allocation.