A non-linear correction current ICTAT2 (current complementary to the square of absolute temperature) is generated from a current IPTAT (current proportional to absolute temperature) and a current ICTAT (current complementary to absolute temperature), both modified in a circuit having a topology and components which capitalize on the logarithmic relationship between transistor collector current and base-emitter voltage. The resulting ICTAT2 current (current complementary to the square of absolute temperature) is injected into a node of a bandgap reference circuit to compensate for non-linear temperature effects on output voltage. A more general correction circuit generates both IPTAT2 and ICTAT2, and applies each to a respective multiplier which, in a preferred embodiment, is a current DAC configured as a multiplier. Control inputs CTL1 and CTL2 to respective multipliers set the amplitudes of the modified IPTAT2 and ICTAT2 output currents, which are then summed to generate the compensating current Icomp which is injected to the appropriate node in the bandgap reference circuit as described above. By adjusting the relative amplitudes of the IPTAT2 and ICTAT2 currents, a wide range of compensating current versus voltage curves is produced, allowing the optimization of a wide range of bandgap reference circuits. An optimal value for CTL1 is determined by holding CTL2 constant, then measuring curvature at a plurality of CTL1 values. That CTL1 value closest to the interpolated value at which curvature is minimized is then used.