gold 2/25/2026.
Here are some simple snippets for numerical methods. The goal is to use Tcl's minimalism as a learning tool. Snippets are short procs that let one play with one core concept at a time. All snippets are Playground V9 safe. One approach to the subject of theoretical physics is to consider these Tcl snippets as Toys. Some snippets here are listed as Toys. These Tcl procs are tiny entry points into physics. On the Wiki Playground V9, Change numbers, add loops, or combine them to explore. Tcl's expr and list/dict make it easy to "feel" the "heavy" ideas without heavy machinery.
The TCL Snippets illustrate ideal mathematical behavior only and do not perform full simulation, actual measurements, or state vector evolution. The tool only visualizes ideal math structure, whereas no state vector simulation, probabilities, or actual measurement outcomes are derived. This tool for visualization does not simulate actual measurement outcomes or state vector evolution during operations. These are idealized protocols for tutorial purposes. Primarily, TCL /TK uses its strong points here for book keeping and displays. The example tool is not a full emulator. Meaning, limited scope for tutorial purposes.
In debugging the calculations, some of the printout values reflect roughly 17-digit precision output from a typical double-precision computation. It's not "true exact" beyond 5 significant figures. Extra significant figures are used to check the calculations from other computer set-ups, not necessarily to infer accuracy of data measurements here. Typically, the slight differences in decimal places on far right of decimal point are normal floating-point behavior in Tcl's expr.
The page presents Tcl code snippets. These educational examples aim to make coding accessible through minimalist programming on the Tcl Playground V9 platform. The following analysis examines how the code implements principles, evaluates the floating-point precision observed in outputs, and suggests improvements for clarity and educational value.
This executive summary introduces the Savitzky-Golay (S-G) filter, a digital signal-smoothing tool widely used in data analysis and signal processing. The summary draws on Thomas Konstantinovsky's comprehensive guide published in The Pythoneers on Medium, supplemented by additional technical sources. The goal is to explain how the filter works, why parameter selection matters, and where common drawbacks arise, so that practitioners can apply the tool with confidence.
Abraham Savitzky and Marcel J. E. Golay published the foundational method in 1964. The filter smooths noisy data by sliding a fixed-width window across a dataset and fitting a low-degree polynomial to the data points inside each window position. Linear least squares (LLS), a mathematical method that finds the best-fitting curve through a set of points, performs the fitting at each step. The central value of each fitted polynomial replaces the original data point, producing a smoother output signal. A practical example is smoothing electrocardiogram (ECG) readings, where baseline noise can otherwise obscure a patient's true heart-rate pattern. Unlike a simple moving average, which flattens all variation equally, the S-G filter preserves peaks, valleys, and other meaningful features of the original signal.
Two parameters control the filter's behaviour: window size and polynomial degree. A small window, such as five data points, preserves rapid changes but may leave residual noise. A large window, such as 31 data points, produces a smoother result but risks erasing genuine signal features, such as a sharp spike in a stock-price series. A low polynomial degree, such as degree two or three, captures general trends reliably. A high polynomial degree fits complex curves more closely but can amplify noise rather than reduce it. Selecting the correct combination requires testing on real data, ideally by comparing smoothed output visually against the original.
Three specific drawbacks deserve attention. The first is the edge effect: the filter cannot access enough neighbouring points near the start and end of a dataset, so smoothing quality deteriorates at the boundaries. Financial time series, for instance, may show distorted recent values if the window extends beyond available data. The second drawback is the assumption of evenly spaced data. The filter requires equal spacing between data points; unevenly sampled sensor readings, for example, must be resampled before applying the filter. The third drawback is computational cost: for very large datasets, fitting a polynomial at every window position becomes slow. In Python, the SciPy (Scientific Python) library's savgol_filter function handles this efficiently, but very long datasets may still require optimisation strategies.
The S-G filter offers a reliable balance between noise reduction and signal preservation. Practitioners should test multiple window sizes and polynomial degrees on a small data sample before committing to final settings, and should account for edge distortion by treating boundary values with caution. For unevenly spaced data, preprocessing or an alternative filter is essential. Applied correctly, the S-G filter is a practical and powerful tool across scientific, financial, and engineering domains.
The Savitzky-Golay data filter is a digital filter that is used to smooth out noisy data, particularly in signal processing applications. It works by fitting a polynomial to small subsets of data points and then using that polynomial to smooth out the data.
Some common uses of the Savitzky-Golay data filter include:
Smoothing out noisy signals in biomedical signal processing
Pre-processing data in chemometrics for spectroscopy analysis
Removing noise from seismic data in geophysics
Enhancing images in computer vision applications
Example: Suppose you have a time series dataset of temperature readings that is noisy due to measurement errors. By applying a Savitzky-Golay filter to the data, you can smooth out the fluctuations and get a clearer picture of the underlying trend of the temperature changes over time. This can be useful for identifying patterns or trends in the data that may not be apparent in the raw, noisy dataset.
The Savitzky-Golay filter is a digital signal processing technique used to smooth out noisy data. It was developed by Abraham Savitzky and Marcel Golay in the 1960s. The filter is based on the idea of fitting a polynomial to a set of data points and then using the polynomial to estimate the smoothed values. The Savitzky-Golay filter works by taking a set of data points and fitting a polynomial of a specified degree to a moving window of data points. The polynomial is then used to estimate the smoothed value at the center of the window. The window is then moved to the next set of data points, and the process is repeated. The Savitzky-Golay filter has several advantages over other smoothing techniques. It is able to preserve the shape and features of the original data, while reducing the noise. It is also computationally efficient and can be used for real-time data processing. The filter has been widely used in various fields, including chemistry, biology, and physics. It has been used to smooth out data from instruments such as spectrophotometers and chromatographs.
Here is a reference to the original paper by Savitzky and Golay:
Savitzky, A., & Golay, M. J. E. (1964). Smoothing and differentiation of data by simplified least squares procedures. Analytical Chemistry, No. 1627-1639.
The Savitzky-Golay filter, despite its popularity and effectiveness, has some drawbacks:
1. Edge Effects: The Savitzky-Golay filter can produce distorted or inaccurate results at the edges of the data, where there are not enough data points to compute the smoothed value. This can lead to boundary effects, where the filtered data appears distorted or artifacts are introduced.
2. Sensitivity to Noise: The Savitzky-Golay filter is sensitive to noise in the data. If the data contains outliers or noisy regions, the filter may not perform well, and the smoothed data may still contain artifacts.
3. Computational Complexity: The Savitzky-Golay filter can be computationally intensive, especially for large datasets. The filter requires computing the convolution of the data with the coefficient array, which can be slow for large datasets.
4. Choice of Window Size and Coefficients: The performance of the Savitzky-Golay filter heavily depends on the choice of window size and coefficients. If the window size is too small, the filter may not effectively remove noise, while a too-large window size may oversmooth the data. The choice of coefficients also affects the filter's performance, and there is no universal set of coefficients that works well for all datasets.
5. Non-Adaptive: The Savitzky-Golay filter is a non-adaptive filter, meaning it does not adjust to changes in the data. If the data characteristics change over time, the filter may not perform well.
6. Not Suitable for Non-Stationary Data: The Savitzky-Golay filter assumes that the data is stationary, meaning the statistical properties of the data do not change over time. If the data is non-stationary, the filter may not perform well.
7. Lack of Robustness: The Savitzky-Golay filter is not robust to outliers or abnormal data points. If the data contains outliers, the filter may not perform well, and the smoothed data may still contain artifacts.
8. Not Suitable for High-Frequency Noise: The Savitzky-Golay filter is not effective in removing high-frequency noise from the data. If the data contains high-frequency noise, the filter may not perform well.
Gist.
The reorganization follows NASA and Jet Propulsion Laboratory (JPL) guidelines for safety-critical code. Procedures stay under 25 lines, include at least two assertions for validation, use descriptive variable names.
gold 2/25/2026.
Maintain single-purpose TCL procedures under 30 lines each. Test TCL invariants after every transformation. Use descriptive names embedding physics meaning in the TCL code. Convert tabs to spaces uniformly. Print variable states at computation boundaries. These steps transform debugging into verification process reliably. Tool Control Language thrives under disciplined practices in scientific work.
The autotests are "easy" tests. Scaling factor applied to filtered output. Autotests have a forgiveness factor of 20%. If the real data has higher-order curvature, strong high-frequency noise, or outliers, the results might look less "perfect" with some residual wiggles or slight peak broadening. Which is normal. No filter is magic.
Smoothed data points should look excellent on these test cases. Because the filter is now mathematically correct for quadratic smoothing. On constants and pure linears, the output matches input exactly and no distortion. That's the whole point of SG design for preserving low-order polynomials. On noisy/zigzag data, it smooths gently without overshooting much or ringing badly. (unlike some other filters). The "too good" appearance is not gimmicked; it's expected behavior when parameters match the signal's underlying smoothness Here, the quadratic assumptions fit these simple synthetic tests perfectly.
These are pure floating-point rounding artifacts in Tcl's expr (double precision). They are ~1e-16 relative. Way smaller than the 0.20 relative tolerance, so the tests still passe reliably.
The program demonstrates math patterns.
The strict ASCII constraint ensures compatibility with collegiate IT lab environments where students may work across diverse platforms and text editors. The implementation deliberately omits boundary closure bars during active development to simplify debugging, with plans to add them once testing completes.
| Index Number | Field of Use | Python S-G Filter Equivalent | Pseudocode Comment | Quibble Notes |
|---|---|---|---|---|
| 1 | General signal processing / data smoothing | from scipy.signal import savgol_filter; smoothed = savgol_filter(y, window_length=5, polyorder=2) | Apply the filter to convolve the data with precomputed coefficients to reduce noise while preserving signal tendency. | Widely used when simple moving averages distort peaks or trends too much; window_length odd, typically 5–51 depending on noise level. |
| 1.1 | General signal processing / boundary handling | Handle boundaries by mirroring, padding with constants, or using smaller local windows. | Many implementations (including scipy) default to 'interp' or 'nearest'; edge distortion remains a common limitation. | |
| 2 | Analytical chemistry / spectroscopy | smoothed_spectra = savgol_filter(spectral_data, 7, 3) | Fit higher-order polynomial locally for smoothing spectroscopic data. | Original 1964 paper application; preserves peak shapes better than simple filters. |
| 3 | Numerical differentiation / peak detection | first_deriv = savgol_filter(y, 5, 2, deriv=1, delta=h); extrema = np.where(np.diff(np.sign(first_deriv)))0 | Compute smoothed first derivative using the polynomial fit; find zero crossings for extrema. | More stable than finite differences on noisy data; common in chromatography, voltammetry, etc. |
| 4 | Analytical chemistry / titration analysis | second_deriv = savgol_filter(titration_data, 9, 4, deriv=2); endpoint = np.argmax(np.abs(second_deriv)) | Use second derivative to locate inflection point (endpoint) in potentiometric or photometric titration curves. | Classic example: malonic acid–bromate–bromide Belousov–Zhabotinsky reaction curve shown on Wikipedia. |
| 5 | Spectroscopy / baseline correction | flattened = savgol_filter(absorption_data, 5, 2, deriv=2); baseline = ... # further processing | Second (or higher) derivative removes broad curved baselines while highlighting sharp absorption bands. | Enables more accurate peak height/area measurement in IR, UV-Vis, Raman spectra. |
| 6 | Spectroscopy / resolution enhancement | enhanced = savgol_filter(spectra, 11, 4, deriv=4) | Apply fourth derivative (or second) to sharpen overlapping spectral bands and reduce apparent half-width. | Used in NMR, IR, fluorescence to resolve closely spaced peaks; amplifies high-frequency noise so pre-smoothing critical. |
| 7 | Time series analysis / econometrics | smoothed_ts = savgol_filter(time_series, 11, 1) # degree 1 ≈ robust moving average | Low polynomial degree (0 or 1) acts as a smoothing filter similar to weighted moving average. | Applied to GDP, stock prices, sales data; preserves linear trends better than simple MA. |
| 8 | Image processing / multi-dimensional data | # Use scipy.ndimage or custom 2D kernel convolution with Savitzky-Golay coefficients | Fit local bivariate polynomials to smooth 2D data arrays (images, surfaces). | Used in medical imaging (ultrasound speckle reduction), radar, microscopy; preserves edges/features better than Gaussian blur in some cases. |
| 8.1 | Multi-dimensional signal processing | Compute 2D/3D Savitzky–Golay coefficients via least-squares matrix solution (Gram matrix / Jacobian). | Extension beyond 1D is mathematically straightforward but computationally heavier; implemented in some specialized libraries. |
Note. The Savitzky Golay Filter has been implemented in many computer languages. However, much of the literature and web searhes here deal with Python sipy library. The original paper and its algorithms was on Fortran. For implementation, libraries like SciPy (savgol_filter) or MATLAB (sgolay) precompute the coefficients.
"1","General signal processing / data smoothing","from scipy.signal import savgol_filter; smoothed = savgol_filter(y, window_length=5, polyorder=2)","Apply the filter to convolve the data with precomputed coefficients to reduce noise while preserving signal tendency.","Widely used when simple moving averages distort peaks or trends too much; window_length odd, typically 5–51 depending on noise level." "1.1","General signal processing / boundary handling","","Handle boundaries by mirroring, padding with constants, or using smaller local windows.","Many implementations (including scipy) default to 'interp' or 'nearest'; edge distortion remains a common limitation." "2","Analytical chemistry / spectroscopy","smoothed_spectra = savgol_filter(spectral_data, 7, 3)","Fit higher-order polynomial locally for smoothing spectroscopic data.","Original 1964 paper application; preserves peak shapes better than simple filters." "3","Numerical differentiation / peak detection","first_deriv = savgol_filter(y, 5, 2, deriv=1, delta=h); extrema = np.where(np.diff(np.sign(first_deriv)))[0]","Compute smoothed first derivative using the polynomial fit; find zero crossings for extrema.","More stable than finite differences on noisy data; common in chromatography, voltammetry, etc." "4","Analytical chemistry / titration analysis","second_deriv = savgol_filter(titration_data, 9, 4, deriv=2); endpoint = np.argmax(np.abs(second_deriv))","Use second derivative to locate inflection point (endpoint) in potentiometric or photometric titration curves.","Classic example: malonic acid–bromate–bromide Belousov–Zhabotinsky reaction curve shown on Wikipedia." "5","Spectroscopy / baseline correction","flattened = savgol_filter(absorption_data, 5, 2, deriv=2); baseline = ... # further processing","Second (or higher) derivative removes broad curved baselines while highlighting sharp absorption bands.","Enables more accurate peak height/area measurement in IR, UV-Vis, Raman spectra." "6","Spectroscopy / resolution enhancement","enhanced = savgol_filter(spectra, 11, 4, deriv=4)","Apply fourth derivative (or second) to sharpen overlapping spectral bands and reduce apparent half-width.","Used in NMR, IR, fluorescence to resolve closely spaced peaks; amplifies high-frequency noise so pre-smoothing critical." "7","Time series analysis / econometrics","smoothed_ts = savgol_filter(time_series, 11, 1) # degree 1 ≈ robust moving average","Low polynomial degree (0 or 1) acts as a smoothing filter similar to weighted moving average.","Applied to GDP, stock prices, sales data; preserves linear trends better than simple MA." "8","Image processing / multi-dimensional data","# Use scipy.ndimage or custom 2D kernel convolution with Savitzky-Golay coefficients","Fit local bivariate polynomials to smooth 2D data arrays (images, surfaces).","Used in medical imaging (ultrasound speckle reduction), radar, microscopy; preserves edges/features better than Gaussian blur in some cases." "8.1","Multi-dimensional signal processing","","Compute 2D/3D Savitzky–Golay coefficients via least-squares matrix solution (Gram matrix / Jacobian).","Extension beyond 1D is mathematically straightforward but computationally heavier; implemented in some specialized libraries."
The Savitzky-Golay filter, despite its popularity and effectiveness, has some drawbacks:
| Index Number | Field of Use / Drawback Category | Pseudocode Comment | Quibble Notes |
|---|---|---|---|
| 1 | Boundary / Edge Effects | Handle edges by mirroring data, constant padding, or extrapolation before filtering; otherwise, discard or accept distortion in first/last (window-1)/2 points. | Most implementations suffer from artifacts or bias near boundaries; discontinuities in impulse response cause poor noise suppression at edges; alternatives like Whittaker-Henderson or extrapolation often perform better. |
| 1.1 | Boundary / Edge Effects | For derivatives, edge artifacts are amplified, leading to unreliable values near data ends. | Especially problematic in spectroscopy or peak detection where ends contain important features; common complaint in literature. |
| 2 | Poor High-Frequency Noise Suppression | Apply filter; observe residual high-frequency oscillations or ringing in output. | Stopband attenuation is mediocre (~11-13 dB typical); allows some high frequencies through, unlike better low-pass designs (Gaussian, FIR with optimized windows); reduces SNR unnecessarily in stopband. |
| 2.1 | Poor High-Frequency Noise Suppression | For higher derivatives, residual noise is amplified, making results impractical without additional pre-smoothing. | Key criticism in recent papers: SG should often be replaced for derivative calculations due to this weakness. |
| 3 | Parameter Sensitivity & Selection | Choose window_length (odd, > polyorder) and polyorder; test multiple combinations to balance noise reduction vs. signal distortion. | Performance heavily depends on these; too small window → insufficient smoothing, too large → oversmoothing & lost details; too high polyorder → ill-conditioned fit & artifacts; no universal optimal values. |
| 4 | Computational Complexity | For large N (data length), compute convolution or per-window least-squares fits repeatedly. | O(N * window) time, can be intensive for very long signals or high-order polynomials; less efficient than simple moving average for basic smoothing. |
| 5 | Signal Distortion & Oversmoothing Risk | Fit local polynomial; evaluate at center; higher polyorder preserves features but risks ringing if mismatched. | Can reduce peak heights, broaden peaks, or introduce oscillations if parameters poor; flattens trends or curves inappropriately in some cases. |
| 6 | Lack of Adaptivity / Non-Stationary Data | Apply fixed coefficients across entire signal. | Does not adjust to changing signal statistics (e.g., varying noise levels or trends); performs poorly on non-stationary time series without segmentation. |
| 7 | Sensitivity to Outliers | Directly fit polynomial to window points including outliers. | Not robust; outliers pull fit, creating local artifacts; unlike median-based or robust regression alternatives. |
| 8 | Ill-Conditioning in Extreme Cases | Attempt fit with polyorder close to or exceeding window constraints. | When polyorder high and window large, matrix becomes ill-conditioned → numerical instability or meaningless coefficients. |
(save as savitzky_golay_drawbacks.csv):"index_number","drawback_category","pseudocode_comment","quibble_notes"
"1","Boundary / Edge Effects","Handle edges by mirroring data, constant padding, or extrapolation before filtering; otherwise, discard or accept distortion in first/last (window-1)/2 points.","Most implementations suffer from artifacts or bias near boundaries; discontinuities in impulse response cause poor noise suppression at edges; alternatives like Whittaker-Henderson or extrapolation often perform better." "1.1","Boundary / Edge Effects","For derivatives, edge artifacts are amplified, leading to unreliable values near data ends.","Especially problematic in spectroscopy or peak detection where ends contain important features; common complaint in literature." "2","Poor High-Frequency Noise Suppression","Apply filter; observe residual high-frequency oscillations or ringing in output.","Stopband attenuation is mediocre (~11-13 dB typical); allows some high frequencies through, unlike better low-pass designs (Gaussian, FIR with optimized windows); reduces SNR unnecessarily in stopband." "2.1","Poor High-Frequency Noise Suppression","For higher derivatives, residual noise is amplified, making results impractical without additional pre-smoothing.","Key criticism in recent papers: SG should often be replaced for derivative calculations due to this weakness." "3","Parameter Sensitivity & Selection","Choose window_length (odd, > polyorder) and polyorder; test multiple combinations to balance noise reduction vs. signal distortion.","Performance heavily depends on these; too small window → insufficient smoothing, too large → oversmoothing & lost details; too high polyorder → ill-conditioned fit & artifacts; no universal optimal values." "4","Computational Complexity","For large N (data length), compute convolution or per-window least-squares fits repeatedly.","O(N * window) time, can be intensive for very long signals or high-order polynomials; less efficient than simple moving average for basic smoothing." "5","Signal Distortion & Oversmoothing Risk","Fit local polynomial; evaluate at center; higher polyorder preserves features but risks ringing if mismatched.","Can reduce peak heights, broaden peaks, or introduce oscillations if parameters poor; flattens trends or curves inappropriately in some cases." "6","Lack of Adaptivity / Non-Stationary Data","Apply fixed coefficients across entire signal.","Does not adjust to changing signal statistics (e.g., varying noise levels or trends); performs poorly on non-stationary time series without segmentation." "7","Sensitivity to Outliers","Directly fit polynomial to window points including outliers.","Not robust; outliers pull fit, creating local artifacts; unlike median-based or robust regression alternatives." "8","Ill-Conditioning in Extreme Cases","Attempt fit with polyorder close to or exceeding window constraints.","When polyorder high and window large, matrix becomes ill-conditioned → numerical instability or meaningless coefficients."
| Index Number | Test Category / Purpose | Pseudocode Comment | Quibble Notes |
|---|---|---|---|
| 1 | Ramp preservation (linear trend exact) | Input linear sequence {1..10}; expect interior points reproduced exactly (3.0 to 8.0) with scale=35. | Core proof that quadratic SG preserves polynomials of degree ≤1 without distortion; exact match expected and achieved. |
| 1.1 | Same ramp with non-standard scale | Same input; scale=6.0; expect amplified linear values (e.g. 17.5, 23.333…). | Verifies scaling math is correct even when deliberately wrong (not preserving unity gain); useful for debugging division logic. |
| 2 | Constant signal preservation (unity gain) | Input all 5.0; expect all outputs exactly 5.0 with scale=35. | Primary correctness check: coefficients sum to 35 → constant K maps to K; fails dramatically with old /3.0 scaling (gives 1.667). |
| 2.1 | Repeated constant test (redundancy) | Identical constant test run again as "confirm". | Ensures result is not accidental or state-dependent; good practice in defensive test suites. |
| 3 | Noisy / zigzag smoothing behavior | Alternating pattern {1 3 2 4 3 5 4 6 5 7}; expect smoothed values close to underlying trend (3.0–5.0 range). | Tests actual smoothing on non-polynomial data; small deviations (~0.03) show gentle low-pass effect without ringing. |
| 3.1 | Same zigzag with non-standard scale | Same zigzag input; scale=12.0; expect scaled-up smoothed values. | Confirms weighted sum / scale math holds under arbitrary scaling; helps isolate coefficient application from normalization. |
| 4 | General floating-point robustness | All tests use ~20% relative + 0.0001 absolute tolerance. | Loose tolerance accommodates Tcl expr double-precision rounding (e.g. 2.9714285714285716 vs 2.9714285714285715). *** Very forgiving margin; catches gross errors but not subtle precision bugs. |
| 5 | Boundary / window size enforcement | Implicit: tests require ≥5 points; only interior points 2…(N-3) smoothed. | No explicit test for short input (<5), but assertions catch it; no test for very long inputs or edge cases. *** Could add short-list failure test and single-point/empty input rejection for completeness. |
| 6 | Coefficient list validation | Asserts exactly 5 coefficients. | Prevents misuse with wrong window size; good defensive style. *** No test deliberately passing invalid coeff count (would crash via assert). |
Save as savitzky_golay_autotests_summary.csv. LIst of Header is "index_number","test_category_purpose","pseudocode_comment","quibble_notes"
"1","Ramp preservation (linear trend exact)","Input linear sequence {1..10}; expect interior points reproduced exactly (3.0 to 8.0) with scale=35.","Core proof that quadratic SG preserves polynomials of degree ≤1 without distortion; exact match expected and achieved."
"1.1","Same ramp with non-standard scale","Same input; scale=6.0; expect amplified linear values (e.g. 17.5, 23.333…).","Verifies scaling math is correct even when deliberately wrong (not preserving unity gain); useful for debugging division logic."
"2","Constant signal preservation (unity gain)","Input all 5.0; expect all outputs exactly 5.0 with scale=35.","Primary correctness check: coefficients sum to 35 → constant K maps to K; fails dramatically with old /3.0 scaling (gives 1.667)."
"2.1","Repeated constant test (redundancy)","Identical constant test run again as ""confirm"".","Ensures result is not accidental or state-dependent; good practice in defensive test suites."
"3","Noisy / zigzag smoothing behavior","Alternating pattern {1 3 2 4 3 5 4 6 5 7}; expect smoothed values close to underlying trend (3.0–5.0 range).","Tests actual smoothing on non-polynomial data; small deviations (~0.03) show gentle low-pass effect without ringing."
"3.1","Same zigzag with non-standard scale","Same zigzag input; scale=12.0; expect scaled-up smoothed values.","Confirms weighted sum / scale math holds under arbitrary scaling; helps isolate coefficient application from normalization."
"4","General floating-point robustness","All tests use ~20% relative + 0.0001 absolute tolerance.","Loose tolerance accommodates Tcl expr double-precision rounding (e.g. 2.9714285714285716 vs 2.9714285714285715)."
"5","Boundary / window size enforcement","Implicit: tests require ≥5 points; only interior points 2…(N-3) smoothed.","No explicit test for short input (<5), but assertions catch it; no test for very long inputs or edge cases."
"6","Coefficient list validation","Asserts exactly 5 coefficients.","Prevents misuse with wrong window size; good defensive style."Note. These Snippets on Theoretical Physics are a set, not stand alones. Recommend read all of the set.
This is a draft, still debugging on Playground V9. convert to strict 7-bit ASCII for Playground V9.
# Savitzky-Golay Five-Point Smoothing Filter - VERSION V7
# Tcl/Tk 8.6+ 7-bit ASCII safe. NASA/JPL defensive programming style.
# Filter math: weightedSum / normalizationScaleFactor where coefficients sum
# to 3.0 for standard 5-point smoothing behavior.
# Savitzky-Golay Five-Point Smoothing Filter - VERSION V7
# NASA/JPL defensive programming style.
# Compatible with Tcl/Tk (Tool Command Language / Toolkit) 8.6+
# Written for Windows 11 on ActiveState Tcl.
# Working under strict 7-bit ASCII encoding.
# Optimized for collegiate information technology lab environments.
# TCL club, 02/26/2026
#
# KEY CHANGE FROM V6:
# Coefficients updated from non-standard {-2 -1 1 1 2} (sum=1, scale=3)
# to standard quadratic Savitzky-Golay {-3 12 17 12 -3} (sum=35, scale=35).
# This preserves constant and linear signals without amplitude distortion.
# ===========================================================================
console show
# ---------------------------------------------------------------------------
# assertConditionIsTrue
# Purpose: Defensive assertion with clear failure message for maintainers.
# ---------------------------------------------------------------------------
proc assertConditionIsTrue {conditionBoolean failureMessage} {
if {!$conditionBoolean} {
error "ASSERTION FAILED: $failureMessage"
}
}
# ---------------------------------------------------------------------------
# smoothDataWithSavitzkyGolay
# Purpose: Apply 5-point Savitzky-Golay smoothing filter to numeric data list.
# Window: indices (center-2), (center-1), center, (center+1), (center+2)
# Returns: Smoothed values for centers at input indices 2...(N-3)
# ---------------------------------------------------------------------------
proc smoothDataWithSavitzkyGolay {inputDataList normalizationScaleFactor filterCoefficientList} {
set inputDataLength [llength $inputDataList]
set coefficientCount [llength $filterCoefficientList]
assertConditionIsTrue [expr {$inputDataLength >= 5}] \
"input must have 5 or more samples for 5-point window"
assertConditionIsTrue [expr {$coefficientCount == 5}] \
"filter must have exactly 5 coefficients"
assertConditionIsTrue [expr {$normalizationScaleFactor != 0.0}] \
"normalization scale factor cannot be zero"
set smoothedDataList {}
for {set centerIndex 2} {$centerIndex < [expr {$inputDataLength - 2}]} {incr centerIndex} {
set weightedSum 0.0
for {set coeffIndex 0} {$coeffIndex < 5} {incr coeffIndex} {
set offset [expr {$coeffIndex - 2}]
set dataIndex [expr {$centerIndex + $offset}]
assertConditionIsTrue \
[expr {$dataIndex >= 0 && $dataIndex < $inputDataLength}] \
"data index out of bounds"
set coeffValue [lindex $filterCoefficientList $coeffIndex]
set dataValue [lindex $inputDataList $dataIndex]
set weightedSum [expr {$weightedSum + $coeffValue * $dataValue}]
}
set smoothedValue [expr {$weightedSum / $normalizationScaleFactor}]
lappend smoothedDataList $smoothedValue
}
assertConditionIsTrue [expr {[llength $smoothedDataList] > 0}] \
"filter must produce at least one smoothed value"
return $smoothedDataList
}
# ---------------------------------------------------------------------------
# Standard Savitzky-Golay quadratic 5-point coefficients.
# Coefficients: {-3 12 17 12 -3}, normalization scale = 35.
# Coefficient sum equals 35, so dividing by 35 gives unity gain on constants.
# A constant input of value K produces a smoothed output of K, not K/35.
# A linear ramp passes through unchanged: center value is reproduced exactly.
# ---------------------------------------------------------------------------
set defaultFilterCoefficients {-3 12 17 12 -3}
set defaultScaleFactor 35.0
# ---------------------------------------------------------------------------
# numericAbsoluteDifference
# Purpose: Safe absolute value computation for tolerance comparisons.
# ---------------------------------------------------------------------------
proc numericAbsoluteDifference {valueA valueB} {
set difference [expr {$valueA - $valueB}]
expr {$difference < 0.0 ? 0.0 - $difference : $difference}
}
# ---------------------------------------------------------------------------
# numericApproximatelyEqualWithMargin
# Purpose: 20% relative and absolute tolerance for floating point autotests.
# Tolerance = absMargin + relMargin * max(1.0, |expected|)
# ---------------------------------------------------------------------------
proc numericApproximatelyEqualWithMargin {expectedValue actualValue relativeMargin absoluteMargin} {
set absDiff [numericAbsoluteDifference $expectedValue $actualValue]
set refMag [expr {abs($expectedValue) < 1.0 ? 1.0 : abs($expectedValue)}]
set tolerance [expr {$absoluteMargin + $relativeMargin * $refMag}]
expr {$absDiff <= $tolerance ? 1 : 0}
}
# ---------------------------------------------------------------------------
# listApproximatelyEqualWithMargin
# Purpose: Element-wise numeric tolerance test for output lists.
# ---------------------------------------------------------------------------
proc listApproximatelyEqualWithMargin {expectedList actualList relativeMargin absoluteMargin} {
if {[llength $expectedList] != [llength $actualList]} {
return 0
}
foreach expected $expectedList actual $actualList {
if {![numericApproximatelyEqualWithMargin $expected $actual $relativeMargin $absoluteMargin]} {
return 0
}
}
return 1
}
# ---------------------------------------------------------------------------
# runSingleSavitzkyGolayAutotest
# Purpose: Execute and report one complete autotest case.
# ---------------------------------------------------------------------------
proc runSingleSavitzkyGolayAutotest {testName inputDataList scaleFactor coeffList expectedList relativeMargin absoluteMargin} {
puts "---- AUTOTEST: $testName ----"
puts "input: $inputDataList"
puts "scale: $scaleFactor"
puts "coeffs: $coeffList"
puts "expect: $expectedList"
set actual [smoothDataWithSavitzkyGolay $inputDataList $scaleFactor $coeffList]
puts "actual: $actual"
set pass [listApproximatelyEqualWithMargin $expectedList $actual $relativeMargin $absoluteMargin]
puts "RESULT: [expr {$pass ? "PASS" : "FAIL"}] $pass"
puts "-----------------------------"
}
# ---------------------------------------------------------------------------
# AUTOTEST SUITE - 20% TOLERANCE
# Standard SG quadratic 5-point coefficients {-3 12 17 12 -3}, scale=35.
#
# Expected values are computed as: sum(coeff_i * data_i) / scaleFactor
#
# Test 1 example (ramp, center=2):
# (-3*1 + 12*2 + 17*3 + 12*4 + -3*5) / 35 = 105/35 = 3.0
#
# Test 3 example (constant=5, any center):
# (-3+12+17+12-3)*5 / 35 = 35*5/35 = 5.0 -- constant is preserved exactly
# ---------------------------------------------------------------------------
set toleranceRelative 0.20
set toleranceAbsolute 0.0001
# Test 1: ramp 1 to 10, standard SG coefficients, scale=35
# Expected: filter reproduces the interior ramp values exactly (3 through 8).
runSingleSavitzkyGolayAutotest "ramp_standard_scale35" \
{1 2 3 4 5 6 7 8 9 10} 35.0 {-3 12 17 12 -3} \
{3.0 4.0 5.0 6.0 7.0 8.0} \
$toleranceRelative $toleranceAbsolute
# Test 2: alternating zigzag pattern, standard SG coefficients, scale=35
# Expected: filter smooths the zigzag toward the underlying linear trend.
# center=2: 106/35=3.0286, center=3: 104/35=2.9714, center=4: 141/35=4.0286
# center=5: 139/35=3.9714, center=6: 176/35=5.0286, center=7: 174/35=4.9714
runSingleSavitzkyGolayAutotest "alternating_standard_scale35" \
{1 3 2 4 3 5 4 6 5 7} 35.0 {-3 12 17 12 -3} \
{3.0285714285714285 2.9714285714285716 4.028571428571429 3.9714285714285716 5.028571428571429 4.971428571428572} \
$toleranceRelative $toleranceAbsolute
# Test 3: constant input of 5.0, standard SG coefficients, scale=35
# Expected: every output equals 5.0 -- the primary correctness proof.
# This test FAILS with the old coefficients {-2 -1 1 1 2} / 3.0 (gives 1.667).
runSingleSavitzkyGolayAutotest "constant_standard_scale35" \
{5 5 5 5 5 5 5 5 5 5} 35.0 {-3 12 17 12 -3} \
{5.0 5.0 5.0 5.0 5.0 5.0} \
$toleranceRelative $toleranceAbsolute
# Test 4: ramp 1 to 10, standard SG coefficients, non-standard scale=6
# Demonstrates deliberate amplitude scaling: output = ramp_value * (35/6).
# center=2: 105/6=17.5, center=3: 140/6=23.333, center=7: 280/6=46.667
runSingleSavitzkyGolayAutotest "ramp_standard_scale6" \
{1 2 3 4 5 6 7 8 9 10} 6.0 {-3 12 17 12 -3} \
{17.5 23.333333333333332 29.166666666666668 35.0 40.833333333333336 46.666666666666664} \
$toleranceRelative $toleranceAbsolute
# Test 5: alternating zigzag, standard SG coefficients, non-standard scale=12
# center=2: 106/12=8.833, center=3: 104/12=8.667, center=4: 141/12=11.75
# center=5: 139/12=11.583, center=6: 176/12=14.667, center=7: 174/12=14.5
runSingleSavitzkyGolayAutotest "alternating_standard_scale12" \
{1 3 2 4 3 5 4 6 5 7} 12.0 {-3 12 17 12 -3} \
{8.833333333333334 8.666666666666666 11.75 11.583333333333334 14.666666666666666 14.5} \
$toleranceRelative $toleranceAbsolute
# Test 6: constant input of 5.0, standard SG coefficients, scale=35 repeated
# Confirms constant preservation is not data-dependent: any constant passes.
runSingleSavitzkyGolayAutotest "constant_standard_scale35_confirm" \
{5 5 5 5 5 5 5 5 5 5} 35.0 {-3 12 17 12 -3} \
{5.0 5.0 5.0 5.0 5.0 5.0} \
$toleranceRelative $toleranceAbsolute
# End of deck
---- AUTOTEST: ramp_standard_scale35 ---- input: 1 2 3 4 5 6 7 8 9 10 scale: 35.0 coeffs: -3 12 17 12 -3 expect: 3.0 4.0 5.0 6.0 7.0 8.0 actual: 3.0 4.0 5.0 6.0 7.0 8.0 RESULT: PASS 1 ----------------------------- ---- AUTOTEST: alternating_standard_scale35 ---- input: 1 3 2 4 3 5 4 6 5 7 scale: 35.0 coeffs: -3 12 17 12 -3 expect: 3.0285714285714285 2.9714285714285716 4.028571428571429 3.9714285714285716 5.028571428571429 4.971428571428572 actual: 3.0285714285714285 2.9714285714285715 4.0285714285714285 3.9714285714285715 5.0285714285714285 4.9714285714285715 RESULT: PASS 1 ----------------------------- ---- AUTOTEST: constant_standard_scale35 ---- input: 5 5 5 5 5 5 5 5 5 5 scale: 35.0 coeffs: -3 12 17 12 -3 expect: 5.0 5.0 5.0 5.0 5.0 5.0 actual: 5.0 5.0 5.0 5.0 5.0 5.0 RESULT: PASS 1 ----------------------------- ---- AUTOTEST: ramp_standard_scale6 ---- input: 1 2 3 4 5 6 7 8 9 10 scale: 6.0 coeffs: -3 12 17 12 -3 expect: 17.5 23.333333333333332 29.166666666666668 35.0 40.833333333333336 46.666666666666664 actual: 17.5 23.333333333333332 29.166666666666668 35.0 40.833333333333336 46.666666666666664 RESULT: PASS 1 ----------------------------- ---- AUTOTEST: alternating_standard_scale12 ---- input: 1 3 2 4 3 5 4 6 5 7 scale: 12.0 coeffs: -3 12 17 12 -3 expect: 8.833333333333334 8.666666666666666 11.75 11.583333333333334 14.666666666666666 14.5 actual: 8.833333333333334 8.666666666666666 11.75 11.583333333333334 14.666666666666666 14.5 RESULT: PASS 1 ----------------------------- ---- AUTOTEST: constant_standard_scale35_confirm ---- input: 5 5 5 5 5 5 5 5 5 5 scale: 35.0 coeffs: -3 12 17 12 -3 expect: 5.0 5.0 5.0 5.0 5.0 5.0 actual: 5.0 5.0 5.0 5.0 5.0 5.0 RESULT: PASS 1 ----------------------------- (bin) 1 %
Note. Program is working on Playground V9. But troubles in grabbing the extra long output file from Playground here. This is a reduced output form here.
(tcl) 10 % set toleranceRelative 0.20
0.20
(tcl) 11 % set toleranceAbsolute 0.0001
0.0001
(tcl) 12 % runSingleSavitzkyGolayAutotest "ramp_standard_scale35" {1 2 3 4 5 6 7 8 9 10} 35.0 {-3 12 17 12 -3} {3.0 4.0 5.0 6.0 7.0 8.0} $toleranceRelative $toleranceAbsolute
---- AUTOTEST: ramp_standard_scale35 ----
input: 1 2 3 4 5 6 7 8 9 10
scale: 35.0
coeffs: -3 12 17 12 -3
expect: 3.0 4.0 5.0 6.0 7.0 8.0
actual: 3.0 4.0 5.0 6.0 7.0 8.0
RESULT: PASS 1
-----------------------------
(tcl) 13 % runSingleSavitzkyGolayAutotest "alternating_standard_scale35" {1 3 2 4 3 5 4 6 5 7} 35.0 {-3 12 17 12 -3} {3.0285714285714285 2.9714285714285716 4.028571428571429 3.9714285714285716 5.028571428571429 4.971428571428572} $toleranceRelative $toleranceAbsolute
---- AUTOTEST: alternating_standard_scale35 ----
input: 1 3 2 4 3 5 4 6 5 7
scale: 35.0
coeffs: -3 12 17 12 -3
expect: 3.0285714285714285 2.9714285714285716 4.028571428571429 3.9714285714285716 5.028571428571429 4.971428571428572
actual: 3.0285714285714285 2.9714285714285715 4.0285714285714285 3.9714285714285715 5.0285714285714285 4.9714285714285715
RESULT: PASS 1
-----------------------------
(tcl) 14 % runSingleSavitzkyGolayAutotest "constant_standard_scale35" {5 5 5 5 5 5 5 5 5 5} 35.0 {-3 12 17 12 -3} {5.0 5.0 5.0 5.0 5.0 5.0} $toleranceRelative $toleranceAbsolute
---- AUTOTEST: constant_standard_scale35 ----
input: 5 5 5 5 5 5 5 5 5 5
scale: 35.0
coeffs: -3 12 17 12 -3
expect: 5.0 5.0 5.0 5.0 5.0 5.0
actual: 5.0 5.0 5.0 5.0 5.0 5.0
RESULT: PASS 1
-----------------------------
(tcl) 15 % runSingleSavitzkyGolayAutotest "ramp_standard_scale6" {1 2 3 4 5 6 7 8 9 10} 6.0 {-3 12 17 12 -3} {17.5 23.333333333333332 29.166666666666668 35.0 40.833333333333336 46.666666666666664} $toleranceRelative $toleranceAbsolute
---- AUTOTEST: ramp_standard_scale6 ----
input: 1 2 3 4 5 6 7 8 9 10
scale: 6.0
coeffs: -3 12 17 12 -3
expect: 17.5 23.333333333333332 29.166666666666668 35.0 40.833333333333336 46.666666666666664
actual: 17.5 23.333333333333332 29.166666666666668 35.0 40.833333333333336 46.666666666666664
RESULT: PASS 1
-----------------------------
(tcl) 16 % runSingleSavitzkyGolayAutotest "alternating_standard_scale12" {1 3 2 4 3 5 4 6 5 7} 12.0 {-3 12 17 12 -3} {8.833333333333334 8.666666666666666 11.75 11.583333333333334 14.666666666666666 14.5} $toleranceRelative $toleranceAbsolute
---- AUTOTEST: alternating_standard_scale12 ----
input: 1 3 2 4 3 5 4 6 5 7
scale: 12.0
coeffs: -3 12 17 12 -3
expect: 8.833333333333334 8.666666666666666 11.75 11.583333333333334 14.666666666666666 14.5
actual: 8.833333333333334 8.666666666666666 11.75 11.583333333333334 14.666666666666666 14.5
RESULT: PASS 1
-----------------------------
(tcl) 17 % runSingleSavitzkyGolayAutotest "constant_standard_scale35_confirm" {5 5 5 5 5 5 5 5 5 5} 35.0 {-3 12 17 12 -3} {5.0 5.0 5.0 5.0 5.0 5.0} $toleranceRelative $toleranceAbsolute
---- AUTOTEST: constant_standard_scale35_confirm ----
input: 5 5 5 5 5 5 5 5 5 5
scale: 35.0
coeffs: -3 12 17 12 -3
expect: 5.0 5.0 5.0 5.0 5.0 5.0
actual: 5.0 5.0 5.0 5.0 5.0 5.0
RESULT: PASS 1
-----------------------------
(tcl) 18 % This is a draft, still debugging on Playground V9. convert to strict 7-bit ASCII for Playground V9.
This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, gold 5Jan2026
gold 2/25/2026. Added categories, so can find message in Wiki.
gold 2/3/2025. Testing, encountered initial difficulty in saving work? Long code blocks with or unmatched wiki markup can sometimes confuse the Tcl Wiki formatting engine, especially if fences are not balanced or a line begins with markup it treats specially.
gold 2/14/2026. Added Automatic Dump of Examples, Using ActiveState.
gold 2/14/2026. convert to strict 7-bit ASCII for Playground V9. reporting error at bottom. program should run to completion with automatic test suite.
gold 2/14/2026. convert to strict 7-bit ASCII for Playground V9. variables need to be human readable and very explanatory. avoid variables with single letter names. Assume a future maintainer either AI or human would have to maintain code with info content in program. the program is working the numbers correctly . so minimal changes.
Please place any comments here with your wiki MONIKER and date, Thanks.gold 1/30/2026
Note. Testing computer methods and computer programs, maybe wrong numbers.
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