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GSoC 2025 ‐ Aayush Khanna

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Aayush Khanna edited this page Aug 26, 2025 · 38 revisions

About me

Hey there! I'm Aayush Khanna from Noida, Uttar Pradesh, India. I am a third year undergrad pursuing civil engineering at the Indian institute of Technology (BHU), Varanasi. I am interested in all things related to tech in general! more recently, I've been trying to learn how interpreters work. I am also a huge football enthusiast :)

Project overview

My project aims to advance the state of LAPACK routines in stdlib, by extending conventional LAPACK APIs which ensures easy compatibility with stdlib ndarrays and adding support for both row-major (C-style) and column-major (Fortran-style) storage layouts. The project covers both lower level helper routines and higher level user facing routines. To further optimize these routines, techniques such as loop reordering and loop tiling were be used. A lot of time was spent on benchmarking and testing of these routines against the actual LAPACK implementations as well! The initial goal was to cover all LAPACK routines up to dgeev but we didn't quite make it there unfortunately.

Project recap

I started off by parsing the LAPACK source code into a directed graph where the nodes represent a LAPACK routine and the edges represent dependencies between these routines. To pick out which routines to work on first, I performed a topological sorting and started working on the ones with no dependencies. I've documented the process in this repository!

After that I started implementing these routines one by one, my workflow consisted of writing a base API that took strides and offsets as input parameters, this was to be kept private and not exported. Then I would make the ndarray wrappers over that and another API that was consistent with the LAPACK function signature.

The testing and benchmarking of these routines was a very important part of the project, for testing I would compare the outputs of our implementation and the actual LAPACK routine in various different cases by storing them in a JSON format and using tape to write the tests for it. This process soon became very tedious so I ended up writing a script to auto-generate the ndarray test fixtures for a routine given the standard inputs. This script can be found here. This saved us a lot of time and helped us to move faster.

The existing reference LAPACK implementation is Fortran-based and hence follows a column-major layout by default. However, in JavaScript, we can provide the user with the freedom to choose whether they want to pass the matrix in a row-major or column-major order. This flexibility is important since matrices are represented as arrays in JavaScript, ensuring contiguous memory allocation.

We did this by representing a matrix in linear memory using strides and offsets. For example:

  A   = [ 1, 2, 3 ]
        [ 4, 5, 6 ]
        [ 7, 8, 9 ] (3X3)
A_row = [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] (row-major)
A_col = [ 1, 4, 7, 2, 5, 8, 3, 6, 9 ] (column-major)

here, we would define two strides to iterate over the two dimensions of the matrix, strideA1 and strideA2. For row-major matrices strideA1 would be greater than strideA2 and the opposite otherwise. Note that swapping the strides and the dimensions of a matrix also gives us it's transpose. This allows for various optimizations which I'll talk about later.

To ensure consistency with the LAPACK function signatures, we have had to use a single element typed array to pass elements by value where needed, for example in dlacn2 the LAPACK API is:

SUBROUTINE DLACN2( N, V, X, ISGN, EST, KASE, ISAVE )

where KASE is an integer value that changes repeatedly between multiple function calls to dlacn2. Hence, to pass the variables by reference we used a single-element typed array, that would make out JavaScript API to be:

function dlacn2( N, V, strideV, offsetV, X, strideX, offsetX, ISGN, strideISGN, offsetISGN, EST, offsetEST, KASE, offsetKASE, ISAVE, strideISAVE, offsetISAVE )

and KASE[ offsetKASE ] represents the element that we're passing by reference.

Completed work

LAPACK routines

  • feat: add lapack/base/dlarfb
    Applies a block reflector to a general rectangular matrix from either the left or right side. This routine is crucial for blocked QR factorization algorithms, where multiple Householder reflectors are grouped together to improve cache efficiency and computational performance. The block reflector representation allows for Level 3 BLAS operations, significantly enhancing performance on modern computer architectures. It's an essential building block for high-performance implementations of QR decomposition and related matrix factorizations.

  • feat: add lapack/base/dorm2r
    Multiplies a general matrix by the orthogonal matrix Q from an LQ factorization determined by dgelqf. This routine is fundamental for solving overdetermined linear least squares problems and computing matrix pseudoinverses. It efficiently applies the orthogonal transformation without explicitly forming the Q matrix, saving both computational time and memory. The routine supports multiplication from either the left or right side and can apply Q or its transpose, making it versatile for various numerical algorithms.

  • feat: add lapack/base/dorg2r
    Generates all or part of the orthogonal matrix Q from a QR factorization computed by dgeqrf. This routine reconstructs the explicit Q matrix from the Householder reflectors stored in the output of QR factorization. It's essential when you need the actual orthogonal matrix for applications like computing orthonormal bases, solving linear systems, or performing eigenvalue computations. The routine efficiently builds the Q matrix column by column, allowing for partial generation when only specific columns are needed.

  • feat: add lapack/base/dlaqr5
    Performs a single small-bulge multi-shift QR sweep on an active submatrix of a Hessenberg matrix. This is an advanced routine used in the implicitly shifted QR algorithm for computing eigenvalues of general matrices. The multi-shift approach allows for better convergence properties and improved numerical stability compared to single-shift methods. It's a core component of modern eigenvalue solvers and represents one of the most sophisticated algorithms in numerical linear algebra for dense matrix eigenproblems.

  • feat: add lapack/base/dlaqr1
    Sets the first row of a multi-shift QR iteration for a Hessenberg matrix. This routine computes the starting vector for the multi-shift QR algorithm by determining appropriate shift values and initializing the bulge-chasing process. It's a specialized utility function that handles the complex initialization phase of the QR eigenvalue algorithm. The routine is critical for ensuring proper convergence and numerical stability in eigenvalue computations for large matrices.

  • feat: add lapack/base/dlartg
    Generates a plane rotation (Givens rotation) that zeros out the second component of a 2-element vector. This fundamental routine computes the cosine and sine values for a rotation matrix that can eliminate specific matrix elements. It's a building block for many matrix algorithms including QR factorization, singular value decomposition, and eigenvalue computation. The routine includes careful scaling to avoid overflow and underflow, making it numerically robust for extreme input values.

  • feat: add lapack/base/dlarfif
    Applies an elementary reflector H to an m-by-n matrix C from either the left or right. The reflector H is represented in the form H = I - tau * v * v^T, where tau is a scalar and v is a vector (note: this appears to be dlarfx based on the description). This routine is fundamental to Householder-based matrix factorizations and transformations. It efficiently applies the reflector without explicitly forming the H matrix, using optimized BLAS operations for better performance.

  • feat: add lapack/base/dlascl
    Multiplies an m-by-n matrix A by the scalar CTO/CFROM, providing robust scaling that avoids overflow and underflow. This routine is essential for numerical stability in many LAPACK algorithms, allowing matrices to be scaled to appropriate ranges before computations. It supports various matrix storage formats including general, upper/lower triangular, and Hessenberg forms. The routine implements careful overflow detection and gradual scaling to maintain numerical accuracy while preventing arithmetic exceptions.

  • feat: add lapack/base/dlanv2
    Computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form. This routine handles the base case for the QR algorithm, determining whether the 2x2 block has real or complex eigenvalues and computing the appropriate factorization. It's crucial for the recursive structure of eigenvalue algorithms and ensures proper handling of complex eigenvalue pairs in real arithmetic. The routine returns rotation matrices and scaling factors that transform the input to Schur form while maintaining numerical stability.

  • feat: add lapack/base/dlacn2
    Estimates the 1-norm of a square matrix A using a reverse communication interface for matrix-vector products. This iterative algorithm efficiently computes matrix norms without requiring explicit access to all matrix elements, making it ideal for large sparse matrices or matrices defined implicitly. The routine uses Higham's algorithm with improved convergence properties and typically requires only a few matrix-vector multiplications. It's essential for condition number estimation and matrix analysis in large-scale computational applications.

  • feat: add lapack/base/dlatrs
    Solves triangular systems of equations with scaling to prevent overflow, handling both upper and lower triangular matrices. This robust solver automatically detects potential overflow conditions and applies appropriate scaling to maintain numerical stability. It supports various matrix orientations (normal or transpose) and can handle both unit and non-unit diagonal triangular matrices. The routine is essential for solving ill-conditioned triangular systems that arise in many numerical algorithms, providing reliable solutions even when standard back-substitution would fail.

  • feat: add lapack/base/dlange
    Computes various matrix norms including the 1-norm, infinity-norm, Frobenius norm, and maximum absolute element value. This versatile routine is fundamental for matrix analysis, providing essential measures of matrix magnitude and conditioning. It efficiently handles different matrix storage formats and implements optimized algorithms for each norm type. The routine is critical for error analysis, convergence testing, and numerical stability assessment in matrix computations, serving as a foundation for many higher-level algorithms.

  • feat: add lapack/base/dlarfg
    Generates an elementary reflector H = I - tau * v * v^T that zeros out all but the first element of a vector. This Householder transformation is fundamental to many matrix factorization algorithms including QR, LQ, and bidiagonal decompositions. The routine carefully handles special cases and scaling to ensure numerical stability and prevent overflow/underflow. It's one of the most frequently used LAPACK routines, forming the foundation for orthogonal matrix factorizations used throughout numerical linear algebra.

  • feat: add lapack/base/dgebak
    Transforms eigenvectors of a balanced matrix back to eigenvectors of the original unbalanced matrix. This routine is the inverse operation to matrix balancing (dgebal) and is essential for completing eigenvalue computations. Matrix balancing improves numerical stability by scaling rows and columns to have similar norms, but the computed eigenvectors must be transformed back to correspond to the original matrix. The routine handles both permutation and scaling transformations applied during the balancing process.

  • feat: add lapack/base/dppequ
    Computes row and column scaling factors for a symmetric positive definite matrix stored in packed format. These scaling factors equilibrate the matrix so that the scaled matrix has unit row and column norms in the infinity norm. Matrix equilibration significantly improves the numerical stability and convergence properties of iterative solvers and direct factorization methods. The routine is particularly important for ill-conditioned matrices where proper scaling can mean the difference between convergent and divergent numerical algorithms.

  • feat: add lapack/base/dgtts2
    Solves systems of linear equations with tridiagonal coefficient matrices using LU factorization with partial pivoting. This specialized solver is optimized for the banded structure of tridiagonal matrices, achieving O(n) complexity compared to O(n³) for general matrices. It handles multiple right-hand sides efficiently and supports various matrix orientations. Tridiagonal systems frequently arise in numerical methods for differential equations, spline interpolation, and many scientific computing applications where efficient solution is critical.

  • feat: add lapack/base/dladiv
    Performs complex division (a + bi) / (c + di) with careful scaling to avoid overflow and underflow. This routine implements Smith's algorithm for robust complex arithmetic, ensuring accurate results even when the operands have vastly different magnitudes. It's essential for numerical stability in complex matrix computations and eigenvalue algorithms. The routine handles special cases like division by zero and provides consistently reliable complex division throughout LAPACK's complex arithmetic operations.

  • feat: add lapack/base/dlaset
    Initializes an m-by-n matrix A with specified values for the diagonal and off-diagonal elements. This utility routine efficiently sets up matrices with specific patterns, supporting various matrix types including general, upper triangular, lower triangular, and trapezoidal forms. It's frequently used for initializing matrices in algorithms, setting up test cases, and preparing matrices for factorization. The routine provides a clean interface for common matrix initialization patterns while maintaining high performance through optimized memory access.

  • feat: add lapack/base/dlapy3
    Computes sqrt(x² + y² + z²) in a numerically stable manner that avoids overflow and underflow. This routine implements careful scaling to handle extreme input values that would cause standard arithmetic to fail. It's essential for computing 3D vector norms, matrix condition numbers, and various geometric calculations in numerical algorithms. The routine ensures accurate results across the full range of floating-point numbers, making it suitable for robust scientific computing applications where numerical stability is paramount.

  • feat: add lapack/base/dlapy2
    Computes sqrt(x² + y²) using a numerically stable algorithm that prevents overflow and underflow. This fundamental routine is used throughout LAPACK for computing 2D vector norms, complex number magnitudes, and Euclidean distances. It implements careful scaling and handles edge cases to ensure reliable results for all input ranges. The routine is critical for numerical stability in many matrix algorithms, particularly those involving Givens rotations, eigenvalue computations, and singular value decompositions.

  • feat: add lapack/base/iladlc
    Scans a matrix from right to left to find the last column containing a non-zero element. This utility routine is essential for determining the effective rank and dimensions of matrices in various algorithms. It's particularly useful in sparse matrix computations and rank-revealing factorizations where trailing zeros can be ignored for efficiency. The routine provides a clean interface for matrix dimension analysis and helps optimize algorithms by avoiding unnecessary computations on zero columns.

  • feat: add lapack/base/iladlr
    Scans a matrix from bottom to top to find the last row containing a non-zero element. This companion to iladlc helps determine the effective dimensions of matrices by identifying trailing zero rows. It's crucial for rank determination, matrix compression, and optimizing algorithms that can skip computations involving zero rows. The routine is widely used in sparse matrix algorithms and rank-revealing decompositions where identifying the true working dimensions of a matrix significantly improves computational efficiency.

Cleanup and maintainance

Current state

I'm still working on dlaqr5 and dorm2r but other than that, the JavaScript implementations of all routines that are listed above are complete.

What remains

There are around 1700 LAPACK routines in all, while we obviously cannot cover them all in a 12 week project, my work provides a proof of concept for various types of LAPACK routines.

A lot of LAPACK routines are yet to be implemented, the correct order to implement them can be seen by the topological sort that I performed.

Apart from that I would love to work on adding the C and Fortran implementations of these routines and high level ndarray wrappers to make these routines production ready!

Challenges and lessons learned

The biggest challenge that I faced was trying to keep up with the weekly targets that I had set in the proposal. I massively underestimated the time it takes to implement a LAPACK routine as per stdlib standards and while I communicated this to my mentor early on, I will continue chipping away at these routines one by one after the GSoC period as well!

Dealing with the intricacies of row-major and column-major storage layout is also something that I had to get used to. It took me a while to build the intuition to spot the loops which can be optimised further or where a matrix can be transposed and passed to another routine etc.

The biggest lesson that I've learnt by this project is to be patient while solving a problem. A lot of times while working on complicated routines like dlarfb or dlatrs I felt the urge to give up but just then something would click. Working through these routines has really helped me step out of my comfort zone and helped me go deeper into the technical weeds of numerical computing! Which is why I chose this project in the first place.

I could have done a better job in estimating the timeline for the project as well but I was not as familiar with implementing LAPACK routines as I am now which is why I ended up with a very unsustainable weekly plan.

Conclusion

This was a very fruitful project overall which has taught me a lot about various optimizations and was a very nice introduction to high performance numerical computing. Other than that, it's also helped me elevate my problem solving ability and helped me step outside my comfort zone. I would like to thank the Org admins Athan Reines and Phillip Buckhardt for the opportunity, guidance and support throughout this journey. I would also like to thank Karan Anand, Gunj Joshi, Gururaj Gurram and Shabareesh Shetty for their support as well. It's been almost a year since I opened my first PR to stdlib and this is a wonderful reminder of how far I've come and yet I've only scratched the surface. stdlib is a wonderful project that I hold very close to my heart, the people that I've met while contributing here are wonderful and I would definitely be active in the community after the GSoC period as well.

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