Julia practice in Scientific Computing

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• Peak Reading

  A sensor array overheated for an instant. Report the single highest reading it captured.

  vectors. Free room.

• Signal Average

  The detector logged a burst of readings. The control room needs their average before the next sweep.

  vectors. Free room.

• Signal RMS

  An audio meter can report signal level using the root-mean-square of its samples. Compute the RMS; perceived loudness also depends on frequency and hearing.

  vectors. Free room.

• Total Flux

  Sum signed, already integrated contributions in compatible units. Raw flux densities would need area or time weights before this total.

  vectors. Free room.

• Squared vector magnitude

  Compute a real vector's sum of squares. Interpreting this as physical energy would also require the model's units and coefficients.

  vectors, dot-product. Account access required.

• Peak Magnitude

  A stability check needs the largest deviation in either direction. Return the infinity norm, the maximum absolute value across the vector.

  linear-algebra. Account access required.

• Running Total

  The fuel log needs a running total after each burn, not just the final figure.

  vectors. Account access required.

• Vector Length

  Mission control needs the magnitude of the thruster vector before the burn.

  vectors, norm. Account access required.

• Area Under the Signal

  Integrate a sampled signal with the trapezoidal rule to get the area it swept.

  integration, trapezoid. Account access required.

• Root Locator

  Calibration needs the square root of a reference value, found by bracketing, not the built-in.

  root-finding, bisection. Account access required.

• Trapezoid Slice

  Numerical integration adds up trapezoids under the curve. Compute the area of one slice from its two edge heights and width, the building block of the trapezoid rule.

  integration. Account access required.

• Trend Slope

  Fit a least-squares line to evenly spaced readings and report the slope of the trend.

  least-squares, interpolation. Account access required.

• Descent Trim

  Drive a control parameter to the setpoint a by gradient descent on a quadratic cost.

  optimization, gradient-descent. Account access required.

• Euler Step

  An ODE solver marches forward one step at a time. Implement the explicit Euler update: the new state is the old plus step-size times the derivative.

  odes. Account access required.

• Scalar growth propagation

  Propagate the benchmark y'=y from y(0)=1 with RK4 and compare it with exponential growth. This exercise models one scalar state.

  odes, runge-kutta. Account access required.

• Precise Area

  Apply Simpson's rule to evenly spaced samples with an odd sample count. Its fourth-order error rate requires sufficient smoothness and a suitable refinement regime.

  integration, simpson. Account access required.

• Conservation Check

  Evolve the insulated rod and measure the trapezoidally weighted total. Endpoint half weights match this mirror-node discretization.

  heat-equation, validation. Account access required.

• Dart Estimate

  Estimate pi from seeded random darts. Use the specified generator and draw order so the same runtime can replay the sample.

  monte-carlo. Account access required.

• Heat Pulse

  Advance a rod temperature field by one simultaneous fixed-end step at r=0.25, preserving the original samples.

  heat-equation, finite-difference. Account access required.

• Residual Norm

  Compare a computed vector with its target using the L2 norm of their difference. A small residual is a useful diagnostic; conditioning and the model determine what it says about solution error.

  linear-algebra. Account access required.