Concept

Reference node — where it appears

The node of a circuit whose voltage is defined as zero, against which every other node voltage is solved. Any node may be chosen and the physics does not change, but in finite-precision arithmetic the choice decides which differences are stored and how many digits survive.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The best reference is the middle of the chain below 2.37 kHz and ground above it — the node the capacitors return to. computed by solving, not by drawing. The condition number of the nodal matrix of an 8-section RC ladder — 1 kΩ in series, 10 nF and 1 MΩ to ground at each node — solved at s = jω with each of its 10 nodes in turn called zero, against frequency. Drawn: ground, the input, the middle node and the far end; the others lie between. At direct current n4 is best, near the middle as the earlier resistor chain found; from 2.37 kHz ground is — well below a single section's 15.9 kHz corner, where each capacitor's admittance is already 0.15 of a series resistor's conductance, and all 8 together about 1.2 of one.

The reference a sweep should keep

At direct current the best node to call zero in a resistor chain is its middle. Put a capacitor to ground at each node and solve at s = jω, and the best reference moves: the middle of an eight-section RC ladder below 2.37 kHz, ground above it, where every capacitor returns. So a solver that chooses its reference once for a whole sweep is choosing wrongly somewhere, as the earlier essay suspected — but the two wrong choices are not alike. Kept for the whole sweep, ground is never worse than 1.27 times the best condition number, a tenth of a digit. The middle node, best at direct current and kept, is 143 times worse at a megahertz, 2.2 digits. A fifth-order LC filter says the same: ground within 2.18 times throughout.

networks · Nodal analysis

One step of refinement in the same double precision returns the stopband's lost digits: 1.1e-12 at 10 kHz before, under 2.4 units of round-off after. computed by solving, not by drawing. The 5th-order Chebyshev ladder solved at every frequency in double precision (solid), and again after one step of iterative refinement — the residual b − A·x formed in double, the correction solved with the same elimination and added (dashed) — each against the same network eliminated in a pair of doubles. Plain, the response's error tracks the round-off divided by the response (dotted), from 1.1e-16 at 10 Hz to 1.1e-12 at 10 kHz, where the output is 6.78e-7 of the input. One step brings every frequency to within 2.4 units of round-off of the truth. No more digits were used: the loss belonged to the elimination's errors in the small component, and the residual, computed against the original matrix, sees them.

The step that returns the digits

A fifth-order ladder filter solved in double precision loses four decades in its stopband — 1.1 × 10⁻¹² of relative error at ten times the cutoff — on a solution vector that is right to round-off. One step of iterative refinement, in the same double precision, returns all of it: form the residual b − A·x, solve for a correction with the same elimination, add it, and every frequency is within 2.4 units of round-off of a 31-digit solve. The reason is that the plain solve is stable by the norm and not by the row — its componentwise backward error is 5.3 × 10⁻¹³ in the stopband rows — and one step makes it componentwise stable. The bridge walked to balance cannot be helped at all. Its node voltages are already right to the last bit; its digits are lost storing two numbers whose difference is the answer, and again storing its conductances, which moves the network itself by round-off over δ before anything is solved.

networks · Conditioning

Named alongside it

The objects these essays reach for when they reach for this one.

Numerical error Closed form Condition number Conditioning Frequency response Ladder filter Ladder network Model range Modified nodal analysis Verification

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