Add utility functions for banded-then-dense symmetric matrices.
PiperOrigin-RevId: 528757637 Change-Id: I916d843140322c28e857100a6b305a041f704d53
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Yuval Tassa
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@@ -2748,7 +2748,7 @@ mju_cholSolve
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.. mujoco-include:: mju_cholSolve
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Solve mat * res = vec, where mat is Cholesky-factorized
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Solve (mat*mat') * res = vec, where mat is a Cholesky factor.
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.. _mju_cholUpdate:
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@@ -2759,6 +2759,101 @@ mju_cholUpdate
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Cholesky rank-one update: L*L' +/- x*x'; return rank.
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.. _mju_cholFactorBand:
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mju_cholFactorBand
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~~~~~~~~~~~~~~~~~~
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.. mujoco-include:: mju_cholFactorBand
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Band-dense Cholesky decomposition.
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|br| Add ``diagadd + diagmul*mat_ii`` to diagonal before decomposition.
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|br| Returns the minimum value of the factorized diagonal or 0 if rank-defficient.
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**Symmetric band-dense matrices**
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:ref:`mju_cholFactorBand` and subsequent functions containing the substring "band" operate on matrices which are a
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generalization of symmetric `band matrices <https://en.wikipedia.org/wiki/Band_matrix>`_. *Symmetric band-dense* or
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"arrowhead" matrices have non-zeros along proximal diagonal bands and dense blocks on the bottom rows and right
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columns. These matrices have the property that Cholesky factorization creates no fill-in and can therefore be
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performed efficiently in-place. Matrix structure is defined by three integers:
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- ``ntotal``: the number of rows (columns) of the symmetric matrix.
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- ``nband``: the number of bands under (over) the diagonal, inclusive of the diagonal.
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- ``ndense``: the number of dense rows (columns) at the bottom (right).
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The non-zeros are stored in memory as two contiguous row-major blocks, colored green and blue in the illustration
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below. The first block has size ``nband x (ntotal-ndense)`` and contains the diagonal and the bands below it. The
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second block has size ``ndense x ntotal`` and contains the dense part. Total required memory is the sum of the block
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sizes.
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.. figure:: /images/APIreference/arrowhead.svg
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:width: 750px
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:align: left
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For example, consider an arrowhead matrix with ``nband = 3``, ``ndense = 2`` and ``ntotal = 8``. In this example, the
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total memory required is ``3*(8-2) + 2*8 = 34`` mjtNum's, laid out as follows:
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.. code-block::
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0 1 2
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3 4 5
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6 7 8
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9 10 11
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12 13 14
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15 16 17
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18 19 20 21 22 23 24 25
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26 27 28 29 30 31 32 33
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The diagonal elements are ``2, 5, 8, 11, 14, 17, 24, 33``.
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|br| Elements ``0, 1, 3, 25`` are present in memory but never touched.
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.. _mju_cholSolveBand:
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mju_cholSolveBand
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~~~~~~~~~~~~~~~~~
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.. mujoco-include:: mju_cholSolveBand
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Solve (mat*mat')*res = vec where mat is a band-dense Cholesky factor.
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.. _mju_band2Dense:
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mju_band2Dense
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~~~~~~~~~~~~~~
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.. mujoco-include:: mju_band2Dense
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Convert banded matrix to dense matrix, fill upper triangle if flg_sym>0.
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.. _mju_dense2Band:
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mju_dense2Band
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~~~~~~~~~~~~~~
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.. mujoco-include:: mju_dense2Band
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Convert dense matrix to banded matrix.
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.. _mju_bandMulMatVec:
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mju_bandMulMatVec
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~~~~~~~~~~~~~~~~~
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.. mujoco-include:: mju_bandMulMatVec
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Multiply band-diagonal matrix with nvec vectors, include upper triangle if flg_sym>0.
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.. _mju_bandDiag:
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mju_bandDiag
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~~~~~~~~~~~~
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.. mujoco-include:: mju_bandDiag
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Address of diagonal element i in band-dense matrix representation.
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.. _mju_eig3:
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mju_eig3
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@@ -321,6 +321,51 @@ mju_ceil
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.. _Decompositions:
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.. _mju_cholFactorBand:
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Band-dense Cholesky decomposition.
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|br| Add ``diagadd + diagmul*mat_ii`` to diagonal before decomposition.
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|br| Returns the minimum value of the factorized diagonal or 0 if rank-defficient.
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**Symmetric band-dense matrices**
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:ref:`mju_cholFactorBand` and subsequent functions containing the substring "band" operate on matrices which are a
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generalization of symmetric `band matrices <https://en.wikipedia.org/wiki/Band_matrix>`_. *Symmetric band-dense* or
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"arrowhead" matrices have non-zeros along proximal diagonal bands and dense blocks on the bottom rows and right
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columns. These matrices have the property that Cholesky factorization creates no fill-in and can therefore be
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performed efficiently in-place. Matrix structure is defined by three integers:
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- ``ntotal``: the number of rows (columns) of the symmetric matrix.
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- ``nband``: the number of bands under (over) the diagonal, inclusive of the diagonal.
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- ``ndense``: the number of dense rows (columns) at the bottom (right).
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The non-zeros are stored in memory as two contiguous row-major blocks, colored green and blue in the illustration
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below. The first block has size ``nband x (ntotal-ndense)`` and contains the diagonal and the bands below it. The
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second block has size ``ndense x ntotal`` and contains the dense part. Total required memory is the sum of the block
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sizes.
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.. figure:: /images/APIreference/arrowhead.svg
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:width: 750px
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:align: left
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For example, consider an arrowhead matrix with ``nband = 3``, ``ndense = 2`` and ``ntotal = 8``. In this example, the
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total memory required is ``3*(8-2) + 2*8 = 34`` mjtNum's, laid out as follows:
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.. code-block::
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0 1 2
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3 4 5
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6 7 8
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9 10 11
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12 13 14
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15 16 17
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18 19 20 21 22 23 24 25
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26 27 28 29 30 31 32 33
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The diagonal elements are ``2, 5, 8, 11, 14, 17, 24, 33``.
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|br| Elements ``0, 1, 3, 25`` are present in memory but never touched.
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.. _mju_boxQP:
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Minimize :math:`\tfrac{1}{2} x^T H x + x^T g \quad \text{s.t.} \quad l \le x \le u`, return rank or -1 if failed.
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+13
-5
@@ -5,18 +5,26 @@ Changelog
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Upcoming version (not yet released)
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-----------------------------------
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Plugins
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^^^^^^^
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.. youtube:: hqIMTNGaLF4
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:align: right
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:width: 240px
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Plugins
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^^^^^^^
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- Added touch-grid sensor plugin. See `documentation <https://github.com/deepmind/mujoco/blob/main/plugin/sensor/README.md>`_
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for details, and associated `touch_grid.xml <https://github.com/deepmind/mujoco/blob/main/model/plugin/touch_grid.xml>`_
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example model. The plugin includes `in-scene visualisation <https://youtu.be/0LOJ3WMnqeA>`_.
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- Add ``mj_multiRay`` function for intersecting multiple rays emanating from a single point. This is significantly
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faster than calling ``mj_ray`` multiple times.
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General
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^^^^^^^
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- Added :ref:`mjd_inverseFD` for finite-differenced inverse-dynamics derivatives.
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- Added functions for operations on banded-then-dense "arrowhead" matrices. Such matrices are
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common when doing direct trajectory optimization.
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See :ref:`mju_cholFactorBand` documentation for details.
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- Added :ref:`mj_multiRay` function for intersecting multiple rays emanating from a single point.
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This is significantly faster than calling :ref:`mj_ray` multiple times.
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Version 2.3.5 (April 25, 2023)
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------------------------------
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@@ -2408,6 +2408,16 @@ void mju_trnVecPose(mjtNum res[3], const mjtNum pos[3], const mjtNum quat[4],
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int mju_cholFactor(mjtNum* mat, int n, mjtNum mindiag);
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void mju_cholSolve(mjtNum* res, const mjtNum* mat, const mjtNum* vec, int n);
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int mju_cholUpdate(mjtNum* mat, mjtNum* x, int n, int flg_plus);
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mjtNum mju_cholFactorBand(mjtNum* mat, int ntotal, int nband, int ndense,
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mjtNum diagadd, mjtNum diagmul);
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void mju_cholSolveBand(mjtNum* res, const mjtNum* mat, const mjtNum* vec,
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int ntotal, int nband, int ndense);
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void mju_band2Dense(mjtNum* res, const mjtNum* mat, int ntotal, int nband, int ndense,
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mjtByte flg_sym);
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void mju_dense2Band(mjtNum* res, const mjtNum* mat, int ntotal, int nband, int ndense);
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void mju_bandMulMatVec(mjtNum* res, const mjtNum* mat, const mjtNum* vec,
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int ntotal, int nband, int ndense, int nvec, mjtByte flg_sym);
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int mju_bandDiag(int i, int ntotal, int nband, int ndense);
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int mju_eig3(mjtNum eigval[3], mjtNum eigvec[9], mjtNum quat[4], const mjtNum mat[9]);
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int mju_boxQP(mjtNum* res, mjtNum* R, int* index, const mjtNum* H, const mjtNum* g, int n,
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const mjtNum* lower, const mjtNum* upper);
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