Solving the following linear equations in the matrix form:
Ax = b
where A is a regular non-symmetric band matrix of dimension M x M.
In this report the band width (number of non-zero elements above the diagonal element or below)
is set to sqrt(M).
This is a very short report to check the performance of sgbsv() and dgbsv() on Apple's devices to solve the problems with non-symmetric band matrices, which appear in some fluid simulations.
-
sgbsv() and dgbsv() run blazingly fast. It can handle matrices of size 4096 x 4096 with band width of 64 in 2 milli seconds.
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The performance does not vary much between 2020 M1 Mac Mini and iPhone 13 Mini.
The following experiments are done with test_nonsymmetric_band_mat.cpp in this directory.
Compiler: Apple clang version 13.0.0 (clang-1300.0.29.3) Target: arm64-apple-darwin20.6.0 Thread model: posix
Devices:
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Mac mini (M1, 2020) Chip Apple M1, Memory 8GB, macOS Big Sur Version 12.4
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iPhone 13 mini, Memory 256GB, iOS 15.5
Please type make all in this directory to reproduce the results on Mac. Please see the section 'Instruction for iOS' for the iOS devices.
The following chart shows the mean running time in log-log scale.
X-axis shows the number of elements in the matrix. For example, 10⁶ indicates the matrix of size (1000x1000).
Y-axis is the time in milliseconds.
- LAPACK 1 1 : sgbsv()
The following chart shows the mean running time in log-log scale.
X-axis shows the number of elements in the matrix. For example, 10⁶ indicates the matrix of size (1000x1000).
Y-axis is the time in milliseconds.
- LAPACK 1 1 : dgbsv()
The following is an excerpt from TestCaseNonsymmetricBandMat_lapack in test_nonsymmetric_band_mat.cpp.
virtual void run() {
int n = this->m_dim;
int kl = this->m_band_width;
int ku = this->m_band_width;
int nrhs = 1;
int ldab = this->m_band_width * 3 + 1;
int* ipiv = new int[this->m_dim];
int ldb = this->m_dim;
int info;
int r = -1;
if constexpr ( std::is_same< float,T >::value ) {
r = sgbsv_( &n, &kl, &ku, &nrhs, m_AB, &ldab, ipiv, m_bx, &ldb, &info );
}
else {
r = dgbsv_( &n, &kl, &ku, &nrhs, m_AB, &ldab, ipiv, m_bx, &ldb, &info );
}
if ( r != 0 ) {
std::cerr << "sposv returned non zeror:" << r << " info:" << info << "\n";
}
for ( int i = 0 ; i < this->m_dim; i++ ) {
const int pivot = ipiv[i] - 1;
this->m_x[pivot] = m_bx[i];
}
delete[] ipiv;
}
The matrix AB passed to the 5th parameter has a special shape, which can be best explained with an example. Assume DIM = 9, KL=KU=3 (band width), then AB must be constructed as follows.
// Format of the Matrix AB \in [ (KL + KL + KU + 1) x DIM ]
//
// !!!PLEASE NOTE THAT AB IS IN COL-MAJOR!!!
//
// +---------------------------------------------+
// 1 | * * * * * * * * * |
// | * * * * * * * * * |
// KL | * * * * * * * * * |
// +=============================================+
// KL+1 | * * * a14 a25 a36 a47 a58 a69 |
// | * * a13 a24 a35 a46 a57 a68 a79 |
// KL+KL | * a12 a23 a34 a45 a56 a67 a78 a89 |
// +---------------------------------------------+
// KL+KL+1 | a11 a22 a33 a44 a55 a66 a77 a88 a99 | <= diagonal entries
// +---------------------------------------------+
// KL+KL+1+1 | a21 a32 a43 a54 a65 a76 a87 a97 * |
// | a31 a42 a53 a64 a75 a86 a97 * * |
// KL+KL+1+KU| a41 52 a63 a74 a85 a96 * * * |
// +---------------------------------------------+
!!!PLEASE NOTE THAT AB IS IN COL-MAJOR!!!
So far this has been tested on iPhone 13 mini 256GB.
-
Open
AppleNumericalComputing/iOSTester_16/iOSTester_16.xcodeprojwith Xcode -
Build a release build
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Run the iOS App in release build
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Press 'Run' on the screen
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Wait until App finished with 'finished!' on the log output.
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Copy and paste the log into
16_nonsymmetric_band_mat/doc_ios/make_log.txt. -
Run the following in the terminal.
$ cd 16_nonsymmetric_banc_mat
$ grep '\(^INT\|^FLOAT\|^DOUBLE\|data element type\)' doc_ios/make_log.txt > doc_ios/make_log_cleaned.txt
$ python ../common/process_log.py -logfile doc_ios/make_log_cleaned.txt -specfile doc_ios/plot_spec.json -show_impl -plot_charts -base_dir doc_ios/
- You will get the PNG files in
16_nonsymmetric_band_mat/doc_ios/.