Table-driven Modeling Subcommittee October 11, 2010 – 8am PDT – 5pm CEST (updated slides) Joachim Haase

Agenda Call to order Volunteer for minutes Approval of agenda Review and approve minutes from the last meeting Information regarding the voting process of the requirements document Preparation of phase 2 - VHDL-AMS implementation aspects Next meeting AOB Adjourn

Approve minutes from September 13, 2010 meeting Minutes available in the Table-driven modeling member-only area Thanks to Alain for preparing the minutes Review of IEEE Patent policy

Action item from September 13, 2010 meeting Alain: Update requirement document from the meeting discussion Requirements for Table Look-Up Modeling with VHDL-AMS v Updates TDM-R2: rephrased. TDM-R5: requirement on complexe dependent variable added. TDM-R7: rationale updated. TDM-R14: change from 'shall' to 'should'.

Review and approval of the requirements document Mail of Ernst Christen Return vote to by Sunday, October 17, Options Affirmative (no comments) Affirmative (with comments) Do Not Approve (Negative with comments) Do Not Approve (Negative without comments) Abstain

Further activities Collection and evaluation of vote results Continuation of activities Phase 2: Discussion of VHDL(-AMS) implementation aspects Starting point for phase 2 Representation of data point information Description of applied methods Interpolation Extrapolation Handling of data point information saved in files Some details in the slides of the September 13, 2010 meeting

1-dimensional case – Data representation Given data points Representation of independent given values Given by Description of dependent given values Array with real values (case 1) Array with complex values (case 2) Independent values at knots Dependent values at knots Case 1: y i are real Case 2: y i are complex

1-dimensional case - Interpolation methods Description of interpolation method to describe f with MethodDescription between knots General ConditionsAdditional conditions Closest data pointconstantNot nessecary Piecewise constantconstantNot nessecary Piecewise linear Meet all (m+1) knotsNot nessecary Quadratic splinePolynomial of degree 2 Meet all (m+1) knots 1st order derivatives at (m-1) inner knots are continuous ? (one condition is missed to determine all parameters of polynomials) Complete cubic spline Polynomial of degree 3 Meet all (m+1) knots 1st and 2nd order derivatives at (m-1) inner knots are continuous First order derivatives at endpoints are given k 0 = f(x 0 +0) and k m =f(x m -0) Natural cubic spline Second order derivatives at endpoints are 0 Not-a-Knot cubic spline Third order derivatives at x 1 and x m-1 are continuous

1-dimensional case – Extrapolation methods (on the left) Extrapolation methodInterpolation methodConditions for x < x 0 (left side) Constant with given fixed value a 0 allf(x) = a 0 Constant with fixed value at border allf(x) = y 0 Linear continuationall (?) Continuation given by interpolation method allContinuation of description for intervall [x 0, x 1 ] PeriodicClosest data point, piecewiese constant, piecewise linear y 0 = y m Quadratic spliney 0 = y m, f‘(x 0 +0)=f‘(x m -0) (all) cubic spline methods y 0 = y m, f‘(x 0 +0)=f‘(x m -0), f‘‘(x 0 +0)=f‘‘(x m -0), ErrorallERROR

1-dimensional case – Extrapolation methods (on the right) Extrapolation methodInterpolation methodConditions for x ≥x m (right side) Constant with given fixed value a m allf(x) = a m Constant with fixed value at border allf(x) = y m Linear continuationall (?) Continuation given by interpolation method allContinuation of description for intervall [x m-1, x m ] PeriodicClosest data point, piecewiese constant, piecewise linear y 0 = y m Quadratic spliney 0 = y m, f‘(x 0 +0)=f‘(x m -0) (all) cubic spline methods y 0 = y m, f‘(x 0 +0)=f‘(x m -0), f‘‘(x 0 +0)=f‘‘(x m -0), ErrorallERROR

1-dimensional case – pp-form Spline interpolants are described piecewise (i=1, …, m) If the polynomials have to be evaluated at many points, then it is efficient to first convert them to piecwise polynomial form (pp-form) There are special data structure for pp-forms (see de Boor, 1978; Dahlquist/Björck, 2008; Octave; …)

Multidimensional case Given data points Representation of Arrays that characterize independent variables Characterization for interpolation and extrapolation methods (left and right side) for each dimension Mulidimensional array with values Characterization of a place that is not filled (NaN) – array is not fully populated Discussion: It should be tried to define values by (n+1) tupels?

Transformation of data sources Functionality to extract data representations from Verilog-AMS files IBIS files Touchstone files Excel CSV has to be defined Not used in the meeting

Some VHDL-AMS implementation aspects Type declaration Complex array Description of interpolation and extrapolation methods Assignement of method to arrays with independent variables (x) Description of function interfacing Overloading of functions Same interface for multidimensional interpolation with different dimenations Not used in the meeting

Action items Discussion of useful combinations of interpolation and extrapolation methods for the 1-dimensional case Summarize the results in a text document

Next meeting Monday, November 8, 2010 – 8am PST (UTC-8) = 5pm CET (UTC+1)