├── .codecov.yml ├── .gitignore ├── tox.ini ├── .pre-commit-config.yaml ├── pyproject.toml ├── setup.py ├── accupy ├── __init__.py ├── dot.py ├── sums.py └── ill_cond.py ├── justfile ├── .github └── workflows │ └── ci.yml ├── setup.cfg ├── tests ├── test_dot.py └── test_sums.py ├── src └── pybind11.cpp ├── README.md └── LICENSE /.codecov.yml: -------------------------------------------------------------------------------- 1 | comment: no 2 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | *.pyc 2 | *.swp 3 | *.prof 4 | MANIFEST 5 | dist/ 6 | build/ 7 | .coverage 8 | .cache/ 9 | *.egg-info/ 10 | .pytest_cache/ 11 | *.png 12 | *.svg 13 | .tox/ 14 | -------------------------------------------------------------------------------- /tox.ini: -------------------------------------------------------------------------------- 1 | [tox] 2 | envlist = py3 3 | isolated_build = True 4 | 5 | [testenv] 6 | deps = 7 | dufte 8 | perfplot 9 | pytest 10 | pytest-cov 11 | commands = 12 | pytest {posargs} 13 | -------------------------------------------------------------------------------- /.pre-commit-config.yaml: -------------------------------------------------------------------------------- 1 | repos: 2 | - repo: https://github.com/PyCQA/isort 3 | rev: 5.9.1 4 | hooks: 5 | - id: isort 6 | 7 | - repo: https://github.com/python/black 8 | rev: 21.6b0 9 | hooks: 10 | - id: black 11 | language_version: python3 12 | 13 | - repo: https://github.com/PyCQA/pylint 14 | rev: v2.9.5 15 | hooks: 16 | - id: pylint 17 | -------------------------------------------------------------------------------- /pyproject.toml: -------------------------------------------------------------------------------- 1 | [build-system] 2 | requires = ["setuptools>=42", "wheel", "pybind11>=2.6.0"] 3 | build-backend = "setuptools.build_meta" 4 | 5 | [tool.isort] 6 | profile = "black" 7 | 8 | [tool.pylint.messages_control] 9 | max-line-length = 88 10 | disable = [ 11 | "invalid-name", 12 | "missing-function-docstring", 13 | "missing-module-docstring", 14 | "import-error", 15 | "duplicate-code", 16 | ] 17 | -------------------------------------------------------------------------------- /setup.py: -------------------------------------------------------------------------------- 1 | # https://github.com/pybind/python_example 2 | from pybind11.setup_helpers import Pybind11Extension, build_ext 3 | from setuptools import setup 4 | 5 | ext_modules = [ 6 | Pybind11Extension( 7 | "_accupy", 8 | ["src/pybind11.cpp"], 9 | include_dirs=["/usr/include/eigen3/"], 10 | ) 11 | ] 12 | 13 | if __name__ == "__main__": 14 | setup( 15 | cmdclass={"build_ext": build_ext}, 16 | ext_modules=ext_modules, 17 | zip_safe=False, 18 | ) 19 | -------------------------------------------------------------------------------- /accupy/__init__.py: -------------------------------------------------------------------------------- 1 | from .dot import fdot, kdot 2 | from .ill_cond import ( 3 | cond, 4 | generate_ill_conditioned_dot_product, 5 | generate_ill_conditioned_sum, 6 | ) 7 | from .sums import decker_sum, distill, fsum, kahan_sum, knuth_sum, ksum 8 | 9 | __all__ = [ 10 | "cond", 11 | "kdot", 12 | "fdot", 13 | "generate_ill_conditioned_sum", 14 | "generate_ill_conditioned_dot_product", 15 | "knuth_sum", 16 | "decker_sum", 17 | "distill", 18 | "ksum", 19 | "fsum", 20 | "kahan_sum", 21 | ] 22 | -------------------------------------------------------------------------------- /justfile: -------------------------------------------------------------------------------- 1 | version := `python3 -c "from configparser import ConfigParser; p = ConfigParser(); p.read('setup.cfg'); print(p['metadata']['version'])"` 2 | name := `python3 -c "from configparser import ConfigParser; p = ConfigParser(); p.read('setup.cfg'); print(p['metadata']['name'])"` 3 | 4 | 5 | default: 6 | @echo "\"just publish\"?" 7 | 8 | tag: 9 | @if [ "$(git rev-parse --abbrev-ref HEAD)" != "main" ]; then exit 1; fi 10 | curl -H "Authorization: token `cat ~/.github-access-token`" -d '{"tag_name": "{{version}}"}' https://api.github.com/repos/nschloe/{{name}}/releases 11 | 12 | upload: clean 13 | @if [ "$(git rev-parse --abbrev-ref HEAD)" != "main" ]; then exit 1; fi 14 | # https://stackoverflow.com/a/58756491/353337 15 | python3 -m build --sdist --wheel . 16 | twine upload dist/*.tar.gz 17 | 18 | publish: tag upload 19 | 20 | clean: 21 | @find . | grep -E "(__pycache__|\.pyc|\.pyo$)" | xargs rm -rf 22 | @rm -rf src/*.egg-info/ build/ dist/ .tox/ 23 | 24 | format: 25 | isort . 26 | black . 27 | blacken-docs README.md 28 | 29 | lint: 30 | black --check . 31 | pylint . 32 | -------------------------------------------------------------------------------- /.github/workflows/ci.yml: -------------------------------------------------------------------------------- 1 | name: ci 2 | 3 | on: 4 | push: 5 | branches: 6 | - main 7 | pull_request: 8 | branches: 9 | - main 10 | 11 | jobs: 12 | lint: 13 | runs-on: ubuntu-latest 14 | steps: 15 | - name: Check out repo 16 | uses: actions/checkout@v2 17 | - name: Set up Python 18 | uses: actions/setup-python@v2 19 | - name: Run pre-commit 20 | uses: pre-commit/action@v2.0.3 21 | 22 | build: 23 | runs-on: ubuntu-latest 24 | strategy: 25 | matrix: 26 | python-version: [3.7, 3.8, 3.9] 27 | steps: 28 | - uses: actions/setup-python@v2 29 | with: 30 | python-version: ${{ matrix.python-version }} 31 | - uses: actions/checkout@v2 32 | - name: Install system dependencies 33 | run: sudo apt-get install -y libeigen3-dev 34 | - name: Test with tox 35 | run: | 36 | pip install tox 37 | tox -- --cov accupy --cov-report xml --cov-report term 38 | - uses: codecov/codecov-action@v1 39 | if: ${{ matrix.python-version == '3.9' }} 40 | -------------------------------------------------------------------------------- /setup.cfg: -------------------------------------------------------------------------------- 1 | [metadata] 2 | name = accupy 3 | version = 0.3.6 4 | author = Nico Schlömer 5 | author_email = nico.schloemer@gmail.com 6 | description = Accurate sums and dot products for Python 7 | url = https://github.com/nschloe/accupy 8 | project_urls = 9 | Code=https://github.com/nschloe/accupy 10 | Issues=https://github.com/nschloe/accupy/issues 11 | Funding=https://github.com/sponsors/nschloe 12 | long_description = file: README.md 13 | long_description_content_type = text/markdown 14 | license = GPL-3.0-or-later 15 | classifiers = 16 | Development Status :: 4 - Beta 17 | License :: OSI Approved :: GNU General Public License v3 or later (GPLv3+) 18 | Operating System :: OS Independent 19 | Programming Language :: Python 20 | Programming Language :: Python :: 3 21 | Programming Language :: Python :: 3.7 22 | Programming Language :: Python :: 3.8 23 | Programming Language :: Python :: 3.9 24 | Topic :: Scientific/Engineering 25 | Topic :: Scientific/Engineering :: Mathematics 26 | keywords = 27 | mathematics 28 | physics 29 | engineering 30 | cgal 31 | mesh 32 | mesh generation 33 | 34 | [options] 35 | packages = find: 36 | install_requires = 37 | mpmath 38 | numpy >= 1.20.0 39 | pyfma 40 | python_requires = >=3.7 41 | -------------------------------------------------------------------------------- /accupy/dot.py: -------------------------------------------------------------------------------- 1 | import _accupy 2 | import numpy as np 3 | from numpy.typing import ArrayLike 4 | 5 | from .sums import fsum, ksum 6 | 7 | # def dot2(x, y, prod2=prod2_fma): 8 | # '''Algorithm 5.3. Dot product in twice the working precision. 9 | # in . 10 | # ''' 11 | # p, s = prod2(x[0], y[0]) 12 | # n = len(x) 13 | # for k in range(1, n): 14 | # h, r = prod2(x[k], y[k]) 15 | # p, q = knuth_sum(p, h) 16 | # s += q+r 17 | # return p + s 18 | 19 | 20 | def kdot(x: ArrayLike, y: ArrayLike, K: int = 2) -> float: 21 | """Algorithm 5.10. Dot product algorithm in K-fold working precision, K >= 3.""" 22 | x = np.asarray(x) 23 | y = np.asarray(y) 24 | 25 | xx = x.reshape(-1, x.shape[-1]) 26 | yy = y.reshape(y.shape[0], -1) 27 | 28 | xx = np.ascontiguousarray(xx) 29 | yy = np.ascontiguousarray(yy) 30 | 31 | r = _accupy.kdot_helper(xx, yy).reshape((-1,) + x.shape[:-1] + y.shape[1:]) 32 | return ksum(r, K - 1) 33 | 34 | 35 | def fdot(x: ArrayLike, y: ArrayLike) -> float: 36 | """Algorithm 5.10. Dot product algorithm in K-fold working precision, K >= 3.""" 37 | x = np.asarray(x) 38 | y = np.asarray(y) 39 | 40 | xx = x.reshape(-1, x.shape[-1]) 41 | yy = y.reshape(y.shape[0], -1) 42 | 43 | xx = np.ascontiguousarray(xx) 44 | yy = np.ascontiguousarray(yy) 45 | 46 | r = _accupy.kdot_helper(xx, yy).reshape((-1,) + x.shape[:-1] + y.shape[1:]) 47 | return fsum(r) 48 | -------------------------------------------------------------------------------- /accupy/sums.py: -------------------------------------------------------------------------------- 1 | import math 2 | from typing import Tuple 3 | 4 | import _accupy 5 | import numpy as np 6 | from numpy.typing import ArrayLike 7 | 8 | 9 | def knuth_sum(a: float, b: float) -> Tuple[float, float]: 10 | """Error-free transformation of the sum of two floating point numbers according to 11 | 12 | D.E. Knuth. 13 | The Art of Computer Programming: Seminumerical Algorithms, volume 2. 14 | Addison Wesley, Reading, Massachusetts, second edition, 1981. 15 | 16 | The underlying problem is that the exact sum a+b of two floating point number a and 17 | b is not necessarily a floating point number; for example if you add a very large 18 | and a very small number. It is however known that the difference between the best 19 | floating point approximation of a+b and the exact a+b is again a floating point 20 | number. This routine returns the sum and the error. 21 | 22 | Algorithm 3.1 in . 23 | """ 24 | x = a + b 25 | z = x - a 26 | y = (a - (x - z)) + (b - z) 27 | return x, y 28 | 29 | 30 | def decker_sum(a: float, b: float) -> Tuple[float, float]: 31 | """Computationally equivalent to knuth_sum, but formally a bit cheaper. 32 | Only works for floats though (and not arrays), and the branch make it in 33 | fact less favorable in terms of actual speed. 34 | """ 35 | x = a + b 36 | y = b - (x - a) if abs(a) > abs(b) else a - (x - b) 37 | return x, y 38 | 39 | 40 | def distill(p: ArrayLike, K: int) -> np.ndarray: 41 | """Algorithm 4.3. Error-free vector transformation for summation. 42 | 43 | The vector p is transformed without changing the sum, and p_n is replaced 44 | by float(sum(p)). Kahan [21] calls this a 'distillation algorithm.' 45 | """ 46 | p = np.asarray(p) 47 | 48 | q = p.reshape(p.shape[0], -1) 49 | for _ in range(K): 50 | _accupy.distill(q) 51 | return q.reshape(p.shape) 52 | 53 | 54 | def ksum(p: ArrayLike, K: int = 2) -> float: 55 | """From 56 | 57 | T. Ogita, S.M. Rump, and S. Oishi. 58 | Accurate Sum and Dot Product, 59 | SIAM J. Sci. Comput., 26(6), 1955–1988 (34 pages). 60 | . 61 | 62 | Algorithm 4.8. Summation as in K-fold precision by (K−1)-fold error-free 63 | vector transformation. 64 | """ 65 | # Don't override the input data. 66 | p = np.asarray(p) 67 | q = p.copy() 68 | distill(q, K - 1) 69 | return np.sum(q[:-1], axis=0) + q[-1] 70 | 71 | 72 | _math_fsum_vec = np.vectorize(math.fsum, signature="(m)->()") 73 | 74 | 75 | def fsum(p: ArrayLike) -> float: 76 | p = np.asarray(p) 77 | return _math_fsum_vec(p.T).T 78 | 79 | 80 | def kahan_sum(p: ArrayLike) -> float: 81 | """Kahan summation 82 | . 83 | """ 84 | p = np.asarray(p) 85 | q = p.reshape(p.shape[0], -1) 86 | s = _accupy.kahan(q) 87 | return s.reshape(p.shape[1:]) 88 | -------------------------------------------------------------------------------- /accupy/ill_cond.py: -------------------------------------------------------------------------------- 1 | import math 2 | from typing import Optional, Tuple 3 | 4 | import numpy as np 5 | import pyfma 6 | from mpmath import mp 7 | from numpy.typing import ArrayLike 8 | 9 | from .dot import fdot, fsum 10 | 11 | 12 | def cond( 13 | x: ArrayLike, y: Optional[ArrayLike] = None, dps: Optional[int] = None 14 | ) -> float: 15 | """Compute the condition number of a sum (if only x is given) or a dot-product (if 16 | both x and y are given). 17 | """ 18 | if dps is None: 19 | sum_exact = fsum 20 | dot_exact = fdot 21 | else: 22 | mp.dps = dps 23 | sum_exact = mp.fsum 24 | dot_exact = mp.fdot 25 | 26 | if y is None: 27 | return sum_exact(np.abs(x)) / np.abs(sum_exact(x)) 28 | 29 | return 2 * dot_exact(np.abs(x), np.abs(y)) / abs(dot_exact(x, y)) 30 | 31 | 32 | def generate_ill_conditioned_sum( 33 | n: int, c: float, dps: int = 100 34 | ) -> Tuple[np.ndarray, float, float]: 35 | # From : 36 | # Ill-conditioned sums of length 2n are generated from dot products of 37 | # length n using Algorithm 3.3 (TwoProduct) and randomly permuting the 38 | # summands. 39 | x, y, _, C = generate_ill_conditioned_dot_product(n, c, dps) 40 | 41 | prod = x * y 42 | err = pyfma.fma(x, y, -prod) 43 | res = np.array([prod, err]) 44 | 45 | out = np.random.permutation(res.flatten()) 46 | 47 | mp.dps = dps 48 | sum_exact = mp.fsum 49 | 50 | exact = sum_exact(out) 51 | 52 | # condition = fsum(np.abs(out)) / abs(exact) 53 | condition = C / 2 54 | 55 | return out, exact, condition 56 | 57 | 58 | def generate_ill_conditioned_dot_product( 59 | n: int, c: float, dps: int = 100 60 | ) -> Tuple[np.ndarray, np.ndarray, float, float]: 61 | """n ... length of vector 62 | c ... target condition number 63 | """ 64 | # Algorithm 6.1 from 65 | # 66 | # ACCURATE SUM AND DOT PRODUCT, 67 | # TAKESHI OGITA, SIEGFRIED M. RUMP, AND SHIN'ICHI OISHI. 68 | assert n >= 6 69 | n2 = round(n / 2) 70 | x = np.zeros(n) 71 | y = np.zeros(n) 72 | 73 | b = math.log2(c) 74 | # vector of exponents between 0 and b/2: 75 | e = np.rint(np.random.rand(n2) * b / 2).astype(int) 76 | # make sure exponents b/2 and 0 actually occur in e 77 | # vectors x,y 78 | e[0] = round(b / 2) + 1 79 | e[-1] = 0 80 | 81 | # generate first half of vectors x, y 82 | rx, ry = np.random.rand(2, n2) 83 | x[:n2] = (2 * rx - 1) * 2 ** e 84 | y[:n2] = (2 * ry - 1) * 2 ** e 85 | 86 | mp.dps = dps 87 | dot_exact = mp.fdot 88 | 89 | # for i=n2+1:n and v=1:i, 90 | # generate x_i, y_i such that (*) x(v)’*y(v) ~ 2^e(i-n2) 91 | # generate exponents for second half 92 | e = np.rint(np.linspace(b / 2, 0, n - n2)).astype(int) 93 | rx, ry = np.random.rand(2, n2) 94 | for i in range(n2, n): 95 | # x_i random with generated exponent 96 | x[i] = (2 * rx[i - n2] - 1) * 2 ** e[i - n2] 97 | # y_i according to (*) 98 | y[i] = ( 99 | (2 * ry[i - n2] - 1) * 2 ** e[i - n2] - dot_exact(x[: i + 1], y[: i + 1]) 100 | ) / x[i] 101 | 102 | x, y = np.random.permutation((x, y)) 103 | # the true dot product rounded to nearest floating point 104 | d = dot_exact(x, y) 105 | # the actual condition number 106 | C = 2 * dot_exact(abs(x), abs(y)) / abs(d) 107 | 108 | return x, y, d, C 109 | -------------------------------------------------------------------------------- /tests/test_dot.py: -------------------------------------------------------------------------------- 1 | import dufte 2 | import matplotlib.pyplot as plt 3 | import numpy as np 4 | import perfplot 5 | import pytest 6 | 7 | import accupy 8 | 9 | 10 | def test_cond(): 11 | cond = accupy.cond([np.pi, np.e], [23225 / 8544, -355 / 113]) 12 | print(cond) 13 | ref = 4.852507317687677e7 14 | assert abs(cond - ref) < 1.0e-13 * abs(ref) 15 | 16 | 17 | @pytest.mark.parametrize("cond", [1.0, 1.0e15]) 18 | def test_kdot2(cond): 19 | x, y, ref, _ = accupy.generate_ill_conditioned_dot_product(100, cond) 20 | assert abs(accupy.kdot(x, y, K=2) - ref) < 1.0e-13 * abs(ref) 21 | 22 | 23 | @pytest.mark.parametrize("cond", [1.0, 1.0e15, 1.0e30]) 24 | def test_kdot3(cond): 25 | x, y, ref, _ = accupy.generate_ill_conditioned_dot_product(100, cond) 26 | assert abs(accupy.kdot(x, y, K=3) - ref) < 1.0e-13 * abs(ref) 27 | 28 | 29 | @pytest.mark.parametrize("cond", [1.0, 1.0e15, 1.0e30, 1.0e38]) 30 | def test_fdot(cond): 31 | x, y, ref, _ = accupy.generate_ill_conditioned_dot_product(100, cond) 32 | assert abs(accupy.fdot(x, y) - ref) < 1.0e-13 * abs(ref) 33 | 34 | 35 | def test_accuracy_comparison_illcond(target_cond=None): 36 | plt.style.use(dufte.style) 37 | 38 | if target_cond is None: 39 | target_cond = [10 ** k for k in range(2)] 40 | 41 | kernels = [ 42 | np.dot, 43 | lambda x, y: accupy.kdot(x, y, K=2), 44 | lambda x, y: accupy.kdot(x, y, K=3), 45 | accupy.fdot, 46 | ] 47 | labels = ["np.dot", "accupy.kdot[2]", "accupy.kdot[3]", "accupy.fdot"] 48 | data = np.empty((len(target_cond), len(kernels))) 49 | condition_numbers = np.empty(len(target_cond)) 50 | np.random.seed(0) 51 | for k, tc in enumerate(target_cond): 52 | x, y, ref, C = accupy.generate_ill_conditioned_dot_product(1000, tc) 53 | condition_numbers[k] = C 54 | data[k] = [abs(kernel(x, y) - ref) / abs(ref) for kernel in kernels] 55 | 56 | # sort 57 | s = np.argsort(condition_numbers) 58 | condition_numbers = condition_numbers[s] 59 | data = data[s] 60 | 61 | for label, d in zip(labels, data.T): 62 | plt.loglog(condition_numbers, d, label=label) 63 | 64 | dufte.legend() 65 | plt.xlabel("condition number") 66 | dufte.ylabel("relative error") 67 | 68 | 69 | def test_speed_comparison1(n_range=None): 70 | plt.style.use(dufte.style) 71 | 72 | if n_range is None: 73 | n_range = [2 ** k for k in range(2)] 74 | 75 | np.random.seed(0) 76 | perfplot.plot( 77 | setup=lambda n: (np.random.rand(n, 100), np.random.rand(100, n)), 78 | kernels=[ 79 | lambda xy: np.dot(*xy), 80 | lambda xy: accupy.kdot(*xy, K=2), 81 | lambda xy: accupy.kdot(*xy, K=3), 82 | lambda xy: accupy.fdot(*xy), 83 | ], 84 | labels=["np.dot", "accupy.kdot[2]", "accupy.kdot[3]", "accupy.fdot"], 85 | n_range=n_range, 86 | xlabel="n", 87 | ) 88 | plt.title("dot(random(n, 100), random(100, n))") 89 | 90 | 91 | def test_speed_comparison2(n_range=None): 92 | if n_range is None: 93 | n_range = [2 ** k for k in range(2)] 94 | 95 | np.random.seed(0) 96 | perfplot.plot( 97 | setup=lambda n: (np.random.rand(100, n), np.random.rand(n, 100)), 98 | kernels=[ 99 | lambda xy: np.dot(*xy), 100 | lambda xy: accupy.kdot(*xy, K=2), 101 | lambda xy: accupy.kdot(*xy, K=3), 102 | lambda xy: accupy.fdot(*xy), 103 | ], 104 | labels=["np.dot", "accupy.kdot[2]", "accupy.kdot[3]", "accupy.fdot"], 105 | n_range=n_range, 106 | xlabel="n", 107 | logx=True, 108 | logy=True, 109 | ) 110 | plt.title("dot(random(100, n), random(n, 100))") 111 | 112 | 113 | def test_discontiguous(): 114 | x = np.random.rand(3, 10) 115 | y = np.random.rand(3, 10) 116 | accupy.kdot(x.T, y) 117 | accupy.fdot(x.T, y) 118 | 119 | 120 | if __name__ == "__main__": 121 | test_accuracy_comparison_illcond([10 ** k for k in range(0, 37, 1)]) 122 | plt.savefig("accuracy-dot.svg", transparent=True, bbox_inches="tight") 123 | plt.close() 124 | 125 | test_speed_comparison1(n_range=[2 ** k for k in range(8)]) 126 | plt.savefig("speed-comparison-dot1.svg", transparent=True, bbox_inches="tight") 127 | plt.close() 128 | 129 | test_speed_comparison2(n_range=[2 ** k for k in range(8)]) 130 | plt.savefig("speed-comparison-dot2.svg", transparent=True, bbox_inches="tight") 131 | plt.close() 132 | -------------------------------------------------------------------------------- /src/pybind11.cpp: -------------------------------------------------------------------------------- 1 | #include 2 | 3 | #include 4 | #include 5 | #include 6 | 7 | #include 8 | 9 | namespace py = pybind11; 10 | using namespace pybind11::literals; 11 | 12 | 13 | // ============================================================================= 14 | // sum.h 15 | // ============================================================================= 16 | // void 17 | // distill(py::array_t p) { 18 | // auto r = p.mutable_unchecked<2>(); 19 | // for (ssize_t i = 1; i < r.shape(0); i++) { 20 | // for (ssize_t j = 0; j < r.shape(1); j++) { 21 | // double x = r(i, j) + r(i-1, j); 22 | // double z = x - r(i, j); 23 | // double y = (r(i, j) - (x-z)) + (r(i-1, j) - z); 24 | // r(i, j) = x; 25 | // r(i-1, j) = y; 26 | // } 27 | // } 28 | // } 29 | 30 | using RowMatrixXd = Eigen::Matrix; 31 | 32 | // Algorithm 4.3. Error-free vector transformation for summation. 33 | // 34 | // The vector p is transformed without changing the sum, and p_n is replaced 35 | // by float(sum(p)). Kahan [21] calls this a "distillation algorithm". 36 | void 37 | distill(Eigen::Ref r) { 38 | for (int i = 1; i < r.rows(); i++) { 39 | auto x = r.row(i) + r.row(i-1); 40 | auto z = x - r.row(i); 41 | for (int j = 0; j < r.cols(); j++) { 42 | const double xj = x(j); 43 | r(i-1, j) = (r(i, j) - (x(j) - z(j))) + (r(i-1, j) - z(j)); 44 | r(i, j) = xj; 45 | } 46 | } 47 | } 48 | 49 | 50 | py::array_t 51 | kahan(py::array_t p) { 52 | // Kahan summation. 53 | // See for 54 | // details. 55 | auto buf_p = p.request(); 56 | if (buf_p.ndim != 2) 57 | throw std::runtime_error("Number of dimensions must be two"); 58 | 59 | const ssize_t m = buf_p.shape[0]; 60 | const ssize_t n = buf_p.shape[1]; 61 | 62 | auto s = py::array_t(n); 63 | auto buf_s = s.request(); 64 | 65 | auto c = py::array_t(n); 66 | auto buf_c = c.request(); 67 | 68 | double *ptr_p = (double *) buf_p.ptr; 69 | double *ptr_c = (double *) buf_c.ptr; 70 | double *ptr_s = (double *) buf_s.ptr; 71 | 72 | // zero out c and s 73 | std::fill(ptr_c, ptr_c+n, 0.0); 74 | std::fill(ptr_s, ptr_s+n, 0.0); 75 | 76 | // Kahan 77 | for (ssize_t i = 0; i < m; i++) { 78 | for (ssize_t j = 0; j < n; j++) { 79 | double y = ptr_p[i*n + j] - ptr_c[j]; 80 | double t = ptr_s[j] + y; 81 | ptr_c[j] = (t - ptr_s[j]) - y; 82 | ptr_s[j] = t; 83 | } 84 | } 85 | return s; 86 | } 87 | // ============================================================================= 88 | // dot.h 89 | // Headers aren't automatically installed though; 90 | // . 91 | // ============================================================================= 92 | using RowMatrixXd = Eigen::Matrix; 93 | 94 | py::array_t 95 | kdot_helper(Eigen::Ref x, Eigen::Ref y) { 96 | // Algorithm 5.10. Dot product algorithm in K-fold working precision, K >= 3. 97 | if (x.cols() != y.rows()) 98 | throw std::runtime_error("Input shapes must match"); 99 | 100 | const int n = x.cols(); 101 | 102 | // Use Eigen::Tensor to avoid stack overflows with native C arrays. 103 | Eigen::Tensor result(2, n, x.rows(), y.cols()); 104 | 105 | // After the loop, p will hold the naive values of the dot product, result(0) 106 | // the multiplication errors and result(1) the addition errors. 107 | auto p = RowMatrixXd(x.rows(), y.cols()); 108 | p.setZero(); 109 | 110 | // Use ikj ordering for speed; see, e.g., 111 | // 112 | for (int i=0; i < x.rows(); i++) { 113 | for (int k=0; k < n; k++) { 114 | for (int j=0; j < y.cols(); j++) { 115 | // product with exact error 116 | double h = x(i, k) * y(k, j); 117 | result(0, k, i, j) = fma(x(i, k), y(k, j), -h); 118 | // Knuth sum: p+h with exact error z2 119 | double z0 = p(i, j) + h; 120 | double z1 = z0 - p(i, j); 121 | double z2 = (p(i, j) - (z0-z1)) + (h-z1); 122 | p(i, j) = z0; 123 | result(1, k, i, j) = z2; 124 | } 125 | } 126 | } 127 | 128 | // Override the meaningless first addition error; it's exactly 0.0 anyways. 129 | for (int i=0; i < x.rows(); i++) 130 | for (int j=0; j < y.cols(); j++) 131 | result(1, 0, i, j) = p(i, j); 132 | 133 | return py::array_t( 134 | std::vector{2, n, x.rows(), y.cols()}, 135 | result.data() 136 | ); 137 | } 138 | // ============================================================================= 139 | PYBIND11_MODULE(_accupy, m) { 140 | // sum: 141 | m.def("distill", &distill, "r"_a.noconvert()); 142 | m.def("kahan", &kahan, "p"_a.noconvert()); 143 | // dot: 144 | m.def("kdot_helper", &kdot_helper, "x"_a.noconvert(), "y"_a.noconvert()); 145 | } 146 | -------------------------------------------------------------------------------- /tests/test_sums.py: -------------------------------------------------------------------------------- 1 | import dufte 2 | import matplotlib.pyplot as plt 3 | import numpy as np 4 | import perfplot 5 | import pytest 6 | 7 | import accupy 8 | 9 | 10 | def test_cond(): 11 | cond = accupy.cond([1.0, 1.0e-16, -1.0]) 12 | ref = 2.0e16 13 | assert abs(cond - ref) < 1.0e-13 * abs(ref) 14 | 15 | 16 | @pytest.mark.parametrize("cond", [1.0, 1.0e15]) 17 | def test_ksum2(cond): 18 | p, ref, _ = accupy.generate_ill_conditioned_sum(100, cond) 19 | assert abs(accupy.ksum(p, K=2) - ref) < 1.0e-15 * abs(ref) 20 | 21 | 22 | @pytest.mark.parametrize("cond", [1.0, 1.0e15, 1.0e30]) 23 | def test_ksum3(cond): 24 | p, ref, _ = accupy.generate_ill_conditioned_sum(100, cond) 25 | assert abs(accupy.ksum(p, K=3) - ref) < 1.0e-15 * abs(ref) 26 | 27 | 28 | @pytest.mark.parametrize("cond", [1.0, 1.0e15, 1.0e30, 1.0e35]) 29 | def test_fsum(cond): 30 | p, ref, _ = accupy.generate_ill_conditioned_sum(100, cond) 31 | assert abs(accupy.fsum(p) - ref) < 1.0e-15 * abs(ref) 32 | 33 | 34 | def test_accuracy_comparison_illcond(target_conds=None): 35 | plt.style.use(dufte.style) 36 | 37 | if target_conds is None: 38 | target_conds = [10 ** k for k in range(1, 2)] 39 | 40 | kernels = [ 41 | sum, 42 | np.sum, 43 | accupy.kahan_sum, 44 | lambda p: accupy.ksum(p, K=2), 45 | lambda p: accupy.ksum(p, K=3), 46 | accupy.fsum, 47 | ] 48 | labels = [ 49 | "sum", 50 | "np.sum", 51 | "accupy.kahan_sum", 52 | "accupy.ksum[2]", 53 | "accupy.ksum[3]", 54 | "accupy.fsum", 55 | ] 56 | colors = plt.rcParams["axes.prop_cycle"].by_key()["color"][: len(labels)] 57 | 58 | data = np.empty((len(target_conds), len(kernels))) 59 | condition_numbers = np.empty(len(target_conds)) 60 | np.random.seed(0) 61 | for k, target_cond in enumerate(target_conds): 62 | p, ref, C = accupy.generate_ill_conditioned_sum(1000, target_cond) 63 | condition_numbers[k] = C 64 | data[k] = [abs(kernel(p) - ref) / abs(ref) for kernel in kernels] 65 | 66 | # sort 67 | s = np.argsort(condition_numbers) 68 | condition_numbers = condition_numbers[s] 69 | data = data[s] 70 | 71 | for label, color, d in zip(labels, colors, data.T): 72 | plt.loglog(condition_numbers, d, label=label, color=color) 73 | 74 | dufte.legend() 75 | plt.xlabel("condition number") 76 | dufte.ylabel("relative error") 77 | # plt.gca().set_aspect(1.3) 78 | 79 | 80 | def test_speed_comparison1(n_range=None): 81 | plt.style.use(dufte.style) 82 | 83 | if n_range is None: 84 | n_range = [2 ** k for k in range(2)] 85 | 86 | np.random.seed(0) 87 | perfplot.plot( 88 | setup=lambda n: np.random.rand(n, 100), 89 | kernels=[ 90 | sum, 91 | lambda p: np.sum(p, axis=0), 92 | accupy.kahan_sum, 93 | lambda p: accupy.ksum(p, K=2), 94 | lambda p: accupy.ksum(p, K=3), 95 | accupy.fsum, 96 | ], 97 | labels=[ 98 | "sum", 99 | "np.sum", 100 | "accupy.kahan_sum", 101 | "accupy.ksum[2]", 102 | "accupy.ksum[3]", 103 | "accupy.fsum", 104 | ], 105 | n_range=n_range, 106 | xlabel="n", 107 | ) 108 | plt.title("Sum(random(n, 100))") 109 | 110 | 111 | def test_speed_comparison2(n_range=None): 112 | plt.style.use(dufte.style) 113 | 114 | if n_range is None: 115 | n_range = [2 ** k for k in range(2)] 116 | 117 | np.random.seed(0) 118 | perfplot.plot( 119 | setup=lambda n: np.random.rand(100, n), 120 | kernels=[ 121 | sum, 122 | lambda p: np.sum(p, axis=0), 123 | accupy.kahan_sum, 124 | lambda p: accupy.ksum(p, K=2), 125 | lambda p: accupy.ksum(p, K=3), 126 | accupy.fsum, 127 | ], 128 | labels=[ 129 | "sum", 130 | "np.sum", 131 | "accupy.kahan_sum", 132 | "accupy.ksum[2]", 133 | "accupy.ksum[3]", 134 | "accupy.fsum", 135 | ], 136 | n_range=n_range, 137 | xlabel="n", 138 | ) 139 | plt.title("Sum(random(100, n))") 140 | 141 | 142 | def test_knuth_sum(): 143 | a16 = np.float16(1.0e1) 144 | b16 = np.float16(1.0e-1) 145 | x16, y16 = accupy.knuth_sum(a16, b16) 146 | xy = np.float64(x16) + np.float64(y16) 147 | ab = np.float64(a16) + np.float64(b16) 148 | assert abs(xy - ab) < 1.0e-15 * ab 149 | 150 | 151 | def test_decker_sum(): 152 | a16 = np.float16(1.0e1) 153 | b16 = np.float16(1.0e-1) 154 | x16, y16 = accupy.decker_sum(a16, b16) 155 | xy = np.float64(x16) + np.float64(y16) 156 | ab = np.float64(a16) + np.float64(b16) 157 | assert abs(xy - ab) < 1.0e-15 * ab 158 | 159 | 160 | def test_discontiguous(): 161 | x = np.random.rand(3, 10).T 162 | accupy.ksum(x.T) 163 | accupy.fsum(x.T) 164 | 165 | 166 | if __name__ == "__main__": 167 | test_accuracy_comparison_illcond([10 ** k for k in range(0, 37, 1)]) 168 | plt.savefig("accuracy-sum.svg", transparent=True, bbox_inches="tight") 169 | plt.close() 170 | 171 | test_speed_comparison1(n_range=[2 ** k for k in range(15)]) 172 | plt.savefig("speed-comparison1.svg", transparent=True, bbox_inches="tight") 173 | plt.close() 174 | 175 | test_speed_comparison2(n_range=[2 ** k for k in range(15)]) 176 | plt.savefig("speed-comparison2.svg", transparent=True, bbox_inches="tight") 177 | plt.close() 178 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 |

2 | accupy 3 |

Accurate sums and (dot) products for Python.

4 |

5 | 6 | [![PyPi Version](https://img.shields.io/pypi/v/accupy.svg?style=flat-square)](https://pypi.org/project/accupy) 7 | [![PyPI pyversions](https://img.shields.io/pypi/pyversions/accupy.svg?style=flat-square)](https://pypi.org/pypi/accupy/) 8 | [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.1185173.svg?style=flat-square)](https://doi.org/10.5281/zenodo.1185173) 9 | [![GitHub stars](https://img.shields.io/github/stars/nschloe/accupy.svg?style=flat-square&logo=github&label=Stars&logoColor=white)](https://github.com/nschloe/accupy) 10 | [![PyPi downloads](https://img.shields.io/pypi/dm/accupy.svg?style=flat-square)](https://pypistats.org/packages/accupy) 11 | 12 | [![Discord](https://img.shields.io/static/v1?logo=discord&label=chat&message=on%20discord&color=7289da&style=flat-square)](https://discord.gg/hnTJ5MRX2Y) 13 | 14 | [![gh-actions](https://img.shields.io/github/workflow/status/nschloe/accupy/ci?style=flat-square)](https://github.com/nschloe/accupy/actions?query=workflow%3Aci) 15 | [![codecov](https://img.shields.io/codecov/c/github/nschloe/accupy.svg?style=flat-square)](https://codecov.io/gh/nschloe/accupy) 16 | [![Code style: black](https://img.shields.io/badge/code%20style-black-000000.svg?style=flat-square)](https://github.com/psf/black) 17 | 18 | ### Sums 19 | 20 | Summing up values in a list can get tricky if the values are floating point 21 | numbers; digit cancellation can occur and the result may come out wrong. A 22 | classical example is the sum 23 | 24 | ``` 25 | 1.0e16 + 1.0 - 1.0e16 26 | ``` 27 | 28 | The actual result is `1.0`, but in double precision, this will result in `0.0`. 29 | While in this example the failure is quite obvious, it can get a lot more 30 | tricky than that. accupy provides 31 | 32 | ```python 33 | p, exact, cond = accupy.generate_ill_conditioned_sum(100, 1.0e20) 34 | ``` 35 | 36 | which, given a length and a target condition number, will produce an array of 37 | floating point numbers that is hard to sum up. 38 | 39 | Given one or two vectors, accupy can compute the condition of the sum or dot product via 40 | 41 | ```python 42 | accupy.cond(x) 43 | accupy.cond(x, y) 44 | ``` 45 | 46 | accupy has the following methods for summation: 47 | 48 | - `accupy.kahan_sum(p)`: [Kahan 49 | summation](https://en.wikipedia.org/wiki/Kahan_summation_algorithm) 50 | 51 | - `accupy.fsum(p)`: A vectorization wrapper around 52 | [math.fsum](https://docs.python.org/3/library/math.html#math.fsum) (which 53 | uses Shewchuck's algorithm [[1]](#references) (see also 54 | [here](https://code.activestate.com/recipes/393090/))). 55 | 56 | - `accupy.ksum(p, K=2)`: Summation in K-fold precision (from [[2]](#references)) 57 | 58 | All summation methods sum the first dimension of a multidimensional NumPy array. 59 | 60 | Let's compare them. 61 | 62 | #### Accuracy comparison (sum) 63 | 64 | ![](https://nschloe.github.io/accupy/accuracy-sum.svg) 65 | 66 | As expected, the naive 67 | [sum](https://docs.python.org/3/library/functions.html#sum) performs very badly 68 | with ill-conditioned sums; likewise for 69 | [`numpy.sum`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html) 70 | which uses pairwise summation. Kahan summation not significantly better; [this, 71 | too, is 72 | expected](https://en.wikipedia.org/wiki/Kahan_summation_algorithm#Accuracy). 73 | 74 | Computing the sum with 2-fold accuracy in `accupy.ksum` gives the correct 75 | result if the condition is at most in the range of machine precision; further 76 | increasing `K` helps with worse conditions. 77 | 78 | Shewchuck's algorithm in `math.fsum` always gives the correct result to full 79 | floating point precision. 80 | 81 | #### Runtime comparison (sum) 82 | 83 | ![](https://nschloe.github.io/accupy/speed-comparison1.svg) 84 | 85 | ![](https://nschloe.github.io/accupy/speed-comparison2.svg) 86 | 87 | We compare more and more sums of fixed size (above) and larger and larger sums, 88 | but a fixed number of them (below). In both cases, the least accurate method is 89 | the fastest (`numpy.sum`), and the most accurate the slowest (`accupy.fsum`). 90 | 91 | ### Dot products 92 | 93 | accupy has the following methods for dot products: 94 | 95 | - `accupy.fdot(p)`: A transformation of the dot product of length _n_ into a 96 | sum of length _2n_, computed with 97 | [math.fsum](https://docs.python.org/3/library/math.html#math.fsum) 98 | 99 | - `accupy.kdot(p, K=2)`: Dot product in K-fold precision (from 100 | [[2]](#references)) 101 | 102 | Let's compare them. 103 | 104 | #### Accuracy comparison (dot) 105 | 106 | accupy can construct ill-conditioned dot products with 107 | 108 | ```python 109 | x, y, exact, cond = accupy.generate_ill_conditioned_dot_product(100, 1.0e20) 110 | ``` 111 | 112 | With this, the accuracy of the different methods is compared. 113 | 114 | ![](https://nschloe.github.io/accupy/accuracy-dot.svg) 115 | 116 | As for sums, `numpy.dot` is the least accurate, followed by instanced of `kdot`. 117 | `fdot` is provably accurate up into the last digit 118 | 119 | #### Runtime comparison (dot) 120 | 121 | ![](https://nschloe.github.io/accupy/speed-comparison-dot1.svg) 122 | ![](https://nschloe.github.io/accupy/speed-comparison-dot2.svg) 123 | 124 | NumPy's `numpy.dot` is _much_ faster than all alternatives provided by accupy. 125 | This is because the bookkeeping of truncation errors takes more steps, but 126 | mostly because of NumPy's highly optimized dot implementation. 127 | 128 | ### References 129 | 130 | 1. [Richard Shewchuk, _Adaptive Precision Floating-Point Arithmetic and Fast 131 | Robust Geometric Predicates_, J. Discrete Comput. Geom. (1997), 18(305), 132 | 305–363](https://doi.org/10.1007/PL00009321) 133 | 134 | 2. [Takeshi Ogita, Siegfried M. Rump, and Shin'ichi Oishi, _Accurate Sum and Dot 135 | Product_, SIAM J. Sci. Comput. (2006), 26(6), 1955–1988 (34 136 | pages)](https://doi.org/10.1137/030601818) 137 | 138 | ### Dependencies 139 | 140 | accupy needs the C++ [Eigen 141 | library](http://eigen.tuxfamily.org/index.php?title=Main_Page), provided in 142 | Debian/Ubuntu by 143 | [`libeigen3-dev`](https://packages.ubuntu.com/search?keywords=libeigen3-dev). 144 | 145 | ### Installation 146 | 147 | accupy is [available from the Python Package Index](https://pypi.org/project/accupy/), so with 148 | 149 | ``` 150 | pip install accupy 151 | ``` 152 | 153 | you can install. 154 | 155 | ### Testing 156 | 157 | To run the tests, just check out this repository and type 158 | 159 | ``` 160 | MPLBACKEND=Agg pytest 161 | ``` 162 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------