Symbolical

Yngel

Features

• Integral equations: u := iesolve(u, x, (a, b), [n_max]) in a solve block, equations of the second kind, u(x) = f(x) + λ·∫ K(x, s)·u(s) ds, linear or not (K(x, s)·g(u(s)), u(x)³…, by Newton's method from the functions of the same names defined before, with an exact Jacobian), and systems of them, (u, v) := iesolve((u, v), x, (a, b)), the bounds constants of the range (Fredholm) or x (Volterra). Two methods chosen automatically: a spectral Nyström method on the nodes of Gauss-Legendre of the pieces of the range cut at the bounds, the integrals cut at x (kernels of Green functions), and, when its convergence is not spectral (kernels singular at s = x, |x − s|^(−α), ln|x − s|, kernels that jump), a product integration (piecewise quadratic interpolation, Gauss-Legendre graded geometrically toward x); the number of nodes or intervals doubled until TOL; the Fredholm alternative and the equations of the first kind are reported; units; the solution u(x) is plotted as the ones of odesolve. Infinite ranges, (0, ∞), (−∞, b), (−∞, ∞), and bounds ∞ in the integrals: an algebraic change of variable whose scale is taken from the kernels, the integrals graded from x; u(∞) is the limit of the solution. Example Integral equations: the capacitor of two discs (Love), the renewal of a fleet of machines (Volterra), a kernel infinite at s = x, the thermal explosion of a slab (nonlinear, two solutions), the light in a semi-infinite atmosphere (an infinite range)
• A function of the worksheet called with arrays in a vectorized expression (→f(v), the points of pdesolve and iesolve) is computed once with the arrays when its body is computed element by element whatever the values, checked at its first and last elements; otherwise for each element as before: about 12 times faster for a kernel of iesolve
• Symbolic evaluation of a matrix equation, A·X = B solve, X: X defined as a vector of symbols (X := [x; y] →) keeps its definition and its elements are the unknowns, instead of being replaced by a scalar; X not defined, when the equation has no scalar solution, is a matrix of the only shape the equation accepts among the sizes of its matrices (A·X = B gives A⁻¹·B, X·A = C a row, a singular system its free elements X₀, X₁)
• Symbolic systems of equations: a matrix whose cells are equalities, [x + 2·y = 5; 3·x + 4·y = 6] solve, x, y →, as find in a solve block; the system can be defined, S := [...] →, and its solutions given to several names, (a, b) := S solve, x, y →; without keyword, or with simplify, substitute…, each equation is transformed. It is no longer a SymPy matrix of relations, which SymPy deprecates
• Lenient units, a preference of the document (settings of the calculation): a mismatch of units is not an error, the calculation goes on with the values in SI base units, in the operations, the functions, the matrices, the ranges, the integrals, the display units and the solvers (find, odesolve, pdesolve, iesolve); a value computed so has unknown units: it is displayed without units, in SI base units, with a warning that gives the first mismatch, and what reads it has unknown units too
• Drawing regions: Insert > Insert Drawing puts at the cursor a region of simple shapes that illustrate the sheet, moved, resized, copied and deleted as an image, with a frame, a background color and its opacity set from its context menu (undoable); it is not computed. It holds segments, polylines, Bézier curves, rectangles and ellipses, with their stroke (color, width, dashes), their fill and its opacity, arrows and dots at the ends of the lines, clipped at its edges and printed as vectors; drawn with the mouse while the region is edited (when it is created, then with a double-click), with the tools of a palette floating above it: segment, rectangle and ellipse by dragging, polyline by clicks (double-click or Enter to end it, a click on its first point to close it); the points fall on a grid of adjustable step, shown while editing; Shift for 45°, squares and circles; each shape is one undoable step. The Select tool selects the shapes with a click, Ctrl+click or a lasso, moves them on the grid within the region, reshapes the shape selected alone by its handles (Shift for 45°, squares and circles) and deletes them with Delete, each an undoable step. The palette sets the color, the width and the pattern of the lines, the fill and its opacity, the arrows and dots at their ends: on the selected shapes (an undoable step) and on the next ones. The Curve tool draws a smooth curve through the points clicked, reshaped by its anchors and its handles when it is selected alone, staying smooth where it was. The Text tool writes a text at a point, typed in place on one or more lines, with its color, size, bold, italic and alignment; a double-click with the Select tool changes it. The palette only shows the properties of the selected shapes or of the tool. A right click on the shapes brings them to the front or sends them to the back, groups them (Ctrl+G) as a symbol moved and copied as one, ungroups them (Ctrl+Shift+G), cuts, copies, pastes and deletes them; Ctrl+X, Ctrl+C and Ctrl+V copy shapes between drawing regions and between documents. Example Drawings: an RC circuit drawn, then its time constant and its charge computed and plotted
• Colors in the texts: two buttons of the toolbar of the text fields set the color of the text and of its background, on the selected characters or on the characters to type: red, orange, green, blue, violet and gray for the text, translucent yellow, green, blue, red, violet and gray for the background, legible on the white page and on the dark one; no black nor white, but Default, the color of the text of the theme and no background
• Python functions: a sheet calls the functions of a .py file of its folder like the built-in functions, F := smooth(data, 11), the file being declared in the Python tab of Document Settings. The values arrive as numbers and NumPy arrays in SI base units; named arguments, a tuple of arrays received by (t, y) := f(x). The file is reloaded when it changes, runs only once trusted, and its SHA-256 is printed in an annex. Example Python functions
• AI assistant (View > Assistant, a tab of the left dock, off by default: no server is contacted before Turn On the Assistant): a question about the sheet is answered by a model that works on a copy of the sheet through tools (read the sheet, evaluate an expression, add, replace or delete a formula, a text, a block or a plot, rename a quantity, read the help of a function or an example), in a linear syntax of the formulas, H(ω) := 1/(1 - ω^2·L·C + j·ω·R·C), read and written back exactly. Each change of the copy is computed at once and the verdict of the engine is given back to the model: the result, the error, the regions it puts in error or out of it, the names read as units because the sheet does not define them (L, C), a hint of correction; before the answer is shown, the loop sends back to the model the errors it leaves, the formulas written in its answer or in texts and the values it did not compute. The panel shows the work live (calls, verdicts, the thinking on demand), the answer in Markdown and its proposal, applied to the sheet as soon as it arrives, the regions added framed in green, replaced in orange, to delete struck out in red, accepted or rejected one by one or all at once, in one step of Undo; the region ids of the answer are links to the regions. The questions follow one another as a conversation, a question asked while a proposal is pending modifies it; Stop or Escape stops the model, sessions are saved and replayed. Providers: Ollama, local or its :cloud models, OpenRouter and any server compatible with OpenAI (LM Studio, vLLM, llama.cpp), chosen with Provider…, with the models that call tools, their context and their price; the tokens and the cost of each question, stopped beyond a ceiling ($0.50 by default); the key comes from OPENROUTER_API_KEY or OPENAI_API_KEY, otherwise from the settings, never written in the document nor in a session; nothing is sent off the machine without the agreement of the user, and OpenRouter only sends the sheet to providers that neither keep the data nor train on it
• Insert Space (Ctrl+Enter, and the context menu of the page) inserts a line of 10 mm at the height of the cursor: the regions below it move down, a block that holds the cursor grows, a region pushed below the bottom margin goes to the top of the next page, created if needed, with the regions of its line and what follows it. Delete Space (Ctrl+Backspace) removes the space between the regions above and below the cursor, the regions below moving up under the lowest one above, within a block or from the next page when nothing is below the cursor on its page, an emptied last page being removed. Each one is a step of Undo
• Custom document info for easy reusable documents and other integration purposes
• Textfield indenting support.
• Text selection implemented in Textfields
• Super and Sub script implemented for TextField
• Navigator dock for overview and navigation to headers in TextFields

Changed

• Divergent integrals are an error that says where they diverge, "The integral does not converge toward ∞" or "near x = 0", instead of a value without meaning (∫₀¹ 1/x² dx gave −1, ∫₁^∞ 1/t dt gave 38.4 with a warning of a relative error of 0.25): when QUADPACK finds the integral probably divergent, or when the calculation stops short of its tolerance and the pieces of the integral do not decrease toward the bound or the singular point; ∫₋₁² 1/x dx, whose principal value is ln 2, is an error too. An integral that converges slowly keeps its value with a warning whose estimated error includes the part not reached, ∫₁^∞ x^(−1.01) dx = 100 instead of 31.9; a vectorized integral that does not converge is computed again by quad. ∫₀^∞ sin(x)/x dx, which QUADPACK cannot tell from a divergent one, is an error rather than a value wrong by 40 % (→ gives π/2)
• iesolve: a solution that oscillates over many wavelengths stays computed by the spectral method: the method is chosen once the solutions are resolved by their nodes (Legendre coefficients), instead of going to the product integration while they are not yet; n_max is 320 nodes by default instead of 160
• Definite integrals whose function is computed element by element are computed by the adaptive rule of Gauss-Kronrod of quad, the function computed at once at the points of all the intervals to refine: many times faster for an oscillating function or a kernel of functions of the worksheet (the field of a solution of iesolve: 2.9 s → 0.17 s); the others as before, by quad
• iesolve stops doubling when its solutions diverge (changes that grow twice in a row) and gives its best solution, the one of the smallest change, with the precision reached; after the product integration, the spectral solution when it is better
• A function of the worksheet computed point by point for more than a second in a region is named in a warning, with what prevents its computation at once (a matrix [·], the transpose, d/dx, max…); when the time limit is reached, the message of the error names them
• Symbolic evaluation: the result of one element of an operation, vᵀ·w, is a number as in the numerical evaluation, not a matrix of one element: adding or removing → no longer changes the shape or the type of a result
• iesolve: the product integration is graded toward x down to 10⁻⁴ of an interval, not to the rounding of the nodes: a kernel computed as a difference near s = x (nᵀ·r on a curved boundary) made it diverge
• The errors of units give the units of the values and name them by their expressions: "Cannot subtract since units don't match: s has no units, s.0 is in m", "An exponent must be dimensionless: -i·β·r is in 1/s (β is in 1/s)", the names of the expression that have units in parentheses, and the probable cause of their units, the first definition where a name without units meets a name with units (from λ := c/f: c has no units, f is in 1/s); a subtraction is no longer called an addition. A curve against a variable without units added to a length is computed with the variable in the units of that length (the profile of a terrain in meters against x); when the curves cannot all be computed so, the message says that the x axis may need a unit, set in the settings of the plot. The same for the variables of a map, and for a variable given to a solution of odesolve or iesolve, whose message says the units of the value given: M.y takes a value in m, not a value without units
• A zero without units is the zero of all units (0 Hz is 0 MHz): in +, −, the comparisons, the cells of a matrix, the bounds of an integral and of a range, it takes the units of the other side: x > 0 with x in meters, ∫ from 0 to x, 0..100·m, f(0) for a length; next to a temperature of a scale (°C, °F), 0 is ambiguous and an error
• The definite integrals of a function of the worksheet called with arrays (the field of a solution of iesolve at the points of a plot, ∫ from 0 to x) are computed for all the elements at once, each one to its own tolerance, instead of one by one; nested integrals too, at the points of the integral around them; the solutions of odesolve, pdesolve and iesolve are called once with the arrays: the field above the ground of the Hviid example is plotted in 1 s instead of 16 s, with the same values
• The calculation lets the window draw itself more often (the switch interval of Python, 0.2 ms instead of 5 ms): scrolling during a symbolic evaluation no longer stops for half a second
• Better navigation of Fractions and Matrices
• Better navigation in and out of plots
• Double-clicking the graph in a plot no longer enters into edit mode. It only opens plot settings.

Fixes

• The curves and the maps of the plots are computed in the background, as the regions, instead of while the page is drawn: a plot each point of which is an integral (the field above the ground of the Hviid example) blocked the window for seconds when it came into view. Meanwhile the plot keeps the curves it had, with an hourglass; Escape interrupts them; printing waits for them
• French translation completed: the toolbar of the text fields, the Functions dock, Insert Function, the labels of the curves, the parameter of a parametric plot, the file dialogs and the dialog of the unsaved changes; the texts of Qt itself (OK, Cancel, Yes, No, the file dialogs, the docks) are translated with the language of the system
• Escape, and Tab without suggestions, typed in a formula or a plot being edited inserted a control character, drawn as an odd character: they type nothing
• Solveblock no longer consumes cursor when navigating into a solveblock with the keyboard
• PlotField labels and expressions better accomodate if field geometry gets larger.