Polynomial Array
An algebraic matrix where each row or column contains the sequential terms of a geometric progression facilitates the fitting of polynomial curves to empirical data. A Vandermonde matrix helps solve for the linearization coefficients of a transducer.
Numerical Condition
The structural layout of the array causes it to become increasingly ill-conditioned as the degree of the polynomial rises. When utilizing a Vandermonde matrix, small measurement errors in the calibration points can lead to large variations in the computed coefficients. Numerical solvers often use orthogonal polynomials or scaling techniques to mitigate this sensitivity.
Calculation Protocol
Calibration systems generate the array by evaluating the input stimulations at each reference point. The Vandermonde matrix is then multiplied by the column vector of the measured outputs to solve the system of linear equations. This step is typically executed during the factory test phase using floating-point processing, with the resulting coefficients stored in the sensor’s non-volatile memory to be used during real-time operation.
Accuracy Verification
Traceability of the linearization process is verified by comparing the computed curve against independent reference measurements. If the Vandermonde matrix calculation yields coefficients that produce high residual errors, the calibration is rejected. The test software logs the condition number of the matrix to ensure numerical stability.
