Newton Scheme Research Articles

A Fully-coupled Newton-Krylov Solution Method for Parallel Unstructured Finite Element Fluid Flow, Heat and Mass Transfer Simulations

This manuscript briefly describes a robust iterative solution methodology used for parallel unstructured finite element simulation of strongly coupled fluid flow, heat and mass transfer. The solution method relies on an inexact Newton scheme and linear system solvers based on preconditioned Krylov subspace methods. The discussion considers computational efficiency and robustness issues related to the proposed schemes. The evaluated preconditioning techniques include simple polynomial expansion and multi-step block iterative methods along with overlapping Schwarz domain decomposition techniques using subdomain solvers based on incomplete factorizations. For this comparison a particular spatial discretization of the governing transport PDEs based on a Galerkin Least Squares (GLS) finite element formulation is used. Results are presented for some standard 2D CFD benchmark problems as well as for a number of 3D problems.

99/00035 ESR study of the macromolecular structure of copper-porphyrin-labeled coal

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

99/00738 On the combustion of heat pump with solar steam turbine

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

Stability analysis for inverse simulation of aircraft

AbstractThe integration inverse method has been extensively investigated in the past decade; however, none of the researches fully addresses the stability analysis of the method that is crucial to successful implementation. This paper presents a stability test to analyse the global stability of the integration inverse method for linear time-invariant systems. A stable solution may be obtained from careful selection of the discretisation interval using the proposed stability test. A discrete model is derived to approximate the Newton's scheme in the inverse method. With this approximate model, the stability of the inverse method can be examined. The stability test is exact for linear systems and can be extended to the inverse method for non-linear aircraft simulations by considering an equivalent linear model for each point of the flight envelopes. Guidelines for selection of appropriate reference points in the inverse simulation are given.

98/03265 Prospects for peat use in metallurgical processes

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

98/03273 Top-charging control of blast-furnace smelting for decreased coke consumption

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

98/03268 Reopening of pulverized coal injection in blast furnace after a pause

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

98/02595 Cost-allocation principles for pipeline capacity and usage : Salant, D. J. and Watkins, G. C. Energy Studies Review, 1996, 8, (2), 91–101

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media

Abstract. The Picard and modified Picard iteration schemes are often used to numerically solve the nonlinear Richards equation governing water flow in variably saturated porous media. While these methods are easy to implement, they are only linearly convergent. Another approach to solve the Richards equation is to use Newton's iterative method. This method, also known as Newton–Raphson iteration, is quadratically convergent and requires the computation of first derivatives. We implemented Newton's scheme into the mixed form of the Richards equation. As compared to the modified Picard scheme, Newton's scheme requires two additional matrices when the mixed form of the Richards equation is used and requires three additional matrices, when the pressure head-based form is used. The modified Picard scheme may actually be viewed as a simplified Newton scheme. Two examples are used to investigate the numerical performance of different forms of the 1D vertical Richards equation and the different iterative solution schemes. In the first example, we simulate infiltration in a homogeneous dry porous medium by solving both, the h based and mixed forms of Richards equation using the modified Picard and Newton schemes. Results shows that, very small time steps are required to obtain an accurate mass balance. These small times steps make the Newton method less attractive. In a second test problem, we simulate variable inflows and outflows in a heterogeneous dry porous medium by solving the mixed form of the Richards equation, using the modified Picard and Newton schemes. Analytical computation of the Jacobian required less CPU time than its computation by perturbation. A combination of the modified Picard and Newton scheme was found to be more efficient than the modified Picard or Newton scheme.

Accelerated Inexact Newton Schemes for Large Systems of Nonlinear Equations

Classical iteration methods for systems, such as Jacobi iteration, can be accelerated considerably by Krylov subspace methods like GMRES @. In this paper, we describe how inexact Newton methods for problems can be accelerated in a similar way and how this leads to a general framework that includes many well-known techniques for solving and systems, as well as new ones. Inexact Newton methods are frequently used in practice to avoid the expensive exact solution of the large system arising in the (possibly also inexact) linearization step of Newton's process. Our framework includes acceleration techniques for the linear steps as well as for the nonlinear steps in Newton's process. The described class of methods, the accelerated inexact Newton (AIN) methods, contains methods like GMRES and GMRESR for systems, Arnoldi and JacDav{} for eigenproblems, and many variants of Newton's method, like damped Newton, for general problems. As numerical experiments suggest, the AIN{} approach may be useful for the construction of efficient schemes for solving problems.

97/04852 Coal pulsating combustion model: Xu, W. Ranshao Kexue Yu Jishu., 1996, 2, (3), 260–269. (In Chinese)

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

97/04847 Calculation of the temperature of coal combustion and gasification products in an exit duct: Yanchenko, G. A. Izu. Vyssh. Uchebn. Zaved., Gorn. Zh., 1997, (1–2), 1–4. (In Russian)

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

97/04844 An approach to multivariable combustion control design: Kocijan, J. J. Proc. Cont., 7, (4), 291–301

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

Extension of a Linear Behaviour Model of Shape Memory Alloys for Finite Strain Studies

Within a thermodynamical approach associated to the classical local state postulate, a constitutive macroscopic model, named R L model [1][2], has been built in order to predict the pseudoelastic behaviour of shape memory alloys (SMA). This behaviour occurs with a martensitic transformation. The volume fraction of martensite z is taken as internal variable and the evolution kinetics of z are derived from metallurgical considerations. In its existing form, the R L model is only valid for small strain calculations. In this paper, an extension is proposed for finite tranformations. Using a formulation in a rotating frame, the obtained model is closed form of the original one. For numerical calculations of NiTi thin films, a stress computation algorithm is also presented. It consists on two steps, the first one is an elastic prediction and the second, a possible pseudoelastic correction. The correction follows an implicit Euler-backward method. To ensure the convergence of the Newton scheme for FE simulation, a consistent tangent operator is used. The results obtained with the proposed approach are in good agreement with the experiments.

Newton-Raphson iteration speed-up algorithm for the solution of nonlinear circuit equations in general-purpose CAD programs

The Newton-Raphson method is usually adopted for the solution of nonlinear circuit equations. However, it is well known that convergence of Newton's scheme is highly dependent on the initial guess, and often fails when convenient modifications are lacking. Typical Newton convergence problems in electronic circuits analysis are identified. An algorithm speeding up the convergence of the Newton scheme and making it less dependent on the initial guess is presented. The method is simple and a general-purpose one, applicable to all kinds of circuit equations. Numerical results are presented, and a comparison with previous work is done.

Standard staggered and staggered Newton schemes in thermo-hydro-mechanical problems

The aim of the paper is to compare the rates of iteration convergence of two staggered numerical schemes: one using successive substitutions and the other the Newton method. In the case of very weakly coupled problems the Newton method is one order faster, but there are specific cases when it is slower than the standard staggered algorithm. In both cases a convergence condition must be verified. The behaviour of the investigated staggered procedures in case of thermo-hydro-mechanical problems are shown on examples.

New Version of the Newton Method for Nonsmooth Equations

In this paper, an inexact Newton scheme is presented which produces a sequence of iterates in which the problem functions are differentiable. It is shown that the use of the inexact Newton scheme does not reduce the convergence rate significantly. To improve the algorithm further, we use a classical finite-difference approximation technique in this context. Locally superlinear convergence results are obtained under reasonable assumptions. To globalize the algorithm, we incorporate features designed to improve convergence from an arbitrary starting point. Convergence results are presented under the condition that the generalized Jacobian of the problem function is nonsingular. Finally, implementations are discussed and numerical results are presented.

97/01372 Hydropyrolysis and hydrodenitrogenation—technological aspects of studies on coal

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

97/01373 Low-NOx combustion for pulverized coal a comparison of air staging and reburning

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.Title of program: CANMCatalogue number: ADSNProgram summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations is numerically solved. It can be written in vector form as where F:Rn→Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).Method of solution: A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure [2–4]. In this approach, the solution of a system F(X)=0 is reduced to a solution of evolutionary differential equation ddtF(X(t))=−F(X(t)),X(0)=X0, with respect to an additional continuous parameter t, 0⩽t<∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways [5,6], yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.Restrictions on the complexity of the problem: The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long write-up and program listing for details). The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either to provide a subroutine JAC which calculates the Jacobian matrix or allow the program to calculate it by the forward-difference approximation.Typical Running time: The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 s on the SGI Origin 2000.

Collocation approximation for fourth-order boundary value problems

This paper concerns the numerical methods based on collocation approximation of fourth-order boundary value problems for the linear problem. Canonical polynomials constructed in [1] are modified and used as a new basis for a collocation solution via the perturbed collocation method with and without exponentially fitting. Nonlinear cases are treated by the Newton's linearization scheme of order 4. The Newton's scheme from the Taylor's series expansion of order 4 is given by: is used throughout this paper. Numerical examples are given to illustrate the effectiveness of the methods discussed in this paper.