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| 58.1 Introduction to lapack | ||
| 58.2 Functions and Variables for lapack |
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lapack is a Common Lisp translation (via the program f2c) of the Fortran library LAPACK,
as obtained from the SLATEC project.
@ref{Category: Numerical methods} · @ref{Category: Share packages} · @ref{Category: Package lapack}
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Computes the eigenvalues and, optionally, the eigenvectors of a matrix A. All elements of A must be integer or floating point numbers. A must be square (same number of rows and columns). A might or might not be symmetric.
dgeev(A) computes only the eigenvalues of A.
dgeev(A, right_p, left_p) computes the eigenvalues of A
and the right eigenvectors when right_p = true
and the left eigenvectors when left_p = true.
A list of three items is returned.
The first item is a list of the eigenvalues.
The second item is false or the matrix of right eigenvectors.
The third item is false or the matrix of left eigenvectors.
The right eigenvector v(j) (the j-th column of the right eigenvector matrix) satisfies
A . v(j) = lambda(j) . v(j)
where lambda(j) is the corresponding eigenvalue. The left eigenvector u(j) (the j-th column of the left eigenvector matrix) satisfies
u(j)**H . A = lambda(j) . u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).
The Maxima function ctranspose computes the conjugate transpose.
The computed eigenvectors are normalized to have Euclidean norm equal to 1, and largest component has imaginary part equal to zero.
Example:
(%i1) load (lapack)$
(%i2) fpprintprec : 6;
(%o2) 6
(%i3) M : matrix ([9.5, 1.75], [3.25, 10.45]);
[ 9.5 1.75 ]
(%o3) [ ]
[ 3.25 10.45 ]
(%i4) dgeev (M);
(%o4) [[7.54331, 12.4067], false, false]
(%i5) [L, v, u] : dgeev (M, true, true);
[ - .666642 - .515792 ]
(%o5) [[7.54331, 12.4067], [ ],
[ .745378 - .856714 ]
[ - .856714 - .745378 ]
[ ]]
[ .515792 - .666642 ]
(%i6) D : apply (diag_matrix, L);
[ 7.54331 0 ]
(%o6) [ ]
[ 0 12.4067 ]
(%i7) M . v - v . D;
[ 0.0 - 8.88178E-16 ]
(%o7) [ ]
[ - 8.88178E-16 0.0 ]
(%i8) transpose (u) . M - D . transpose (u);
[ 0.0 - 4.44089E-16 ]
(%o8) [ ]
[ 0.0 0.0 ]
@ref{Category: Package lapack}
Computes the singular value decomposition (SVD) of a matrix A, comprising the singular values and, optionally, the left and right singular vectors. All elements of A must be integer or floating point numbers. A might or might not be square (same number of rows and columns).
Let m be the number of rows, and n the number of columns of A. The singular value decomposition of A comprises three matrices, U, Sigma, and V^T, such that
A = U . Sigma . V^T
where U is an m-by-m unitary matrix, Sigma is an m-by-n diagonal matrix, and V^T is an n-by-n unitary matrix.
Let sigma[i] be a diagonal element of Sigma,
that is, Sigma[i, i] = sigma[i].
The elements sigma[i] are the so-called singular values of A;
these are real and nonnegative, and returned in descending order.
The first min(m, n) columns of U and V are
the left and right singular vectors of A.
Note that dgesvd returns the transpose of V, not V itself.
dgesvd(A) computes only the singular values of A.
dgesvd(A, left_p, right_p) computes the singular values of A
and the left singular vectors when left_p = true
and the right singular vectors when right_p = true.
A list of three items is returned.
The first item is a list of the singular values.
The second item is false or the matrix of left singular vectors.
The third item is false or the matrix of right singular vectors.
Example:
(%i1) load (lapack)$
(%i2) fpprintprec : 6;
(%o2) 6
(%i3) M: matrix([1, 2, 3], [3.5, 0.5, 8], [-1, 2, -3], [4, 9, 7]);
[ 1 2 3 ]
[ ]
[ 3.5 0.5 8 ]
(%o3) [ ]
[ - 1 2 - 3 ]
[ ]
[ 4 9 7 ]
(%i4) dgesvd (M);
(%o4) [[14.4744, 6.38637, .452547], false, false]
(%i5) [sigma, U, VT] : dgesvd (M, true, true);
(%o5) [[14.4744, 6.38637, .452547],
[ - .256731 .00816168 .959029 - .119523 ]
[ ]
[ - .526456 .672116 - .206236 - .478091 ]
[ ],
[ .107997 - .532278 - .0708315 - 0.83666 ]
[ ]
[ - .803287 - .514659 - .180867 .239046 ]
[ - .374486 - .538209 - .755044 ]
[ ]
[ .130623 - .836799 0.5317 ]]
[ ]
[ - .917986 .100488 .383672 ]
(%i6) m : length (U);
(%o6) 4
(%i7) n : length (VT);
(%o7) 3
(%i8) Sigma:
genmatrix(lambda ([i, j], if i=j then sigma[i] else 0),
m, n);
[ 14.4744 0 0 ]
[ ]
[ 0 6.38637 0 ]
(%o8) [ ]
[ 0 0 .452547 ]
[ ]
[ 0 0 0 ]
(%i9) U . Sigma . VT - M;
[ 1.11022E-15 0.0 1.77636E-15 ]
[ ]
[ 1.33227E-15 1.66533E-15 0.0 ]
(%o9) [ ]
[ - 4.44089E-16 - 8.88178E-16 4.44089E-16 ]
[ ]
[ 8.88178E-16 1.77636E-15 8.88178E-16 ]
(%i10) transpose (U) . U;
[ 1.0 5.55112E-17 2.498E-16 2.77556E-17 ]
[ ]
[ 5.55112E-17 1.0 5.55112E-17 4.16334E-17 ]
(%o10) [ ]
[ 2.498E-16 5.55112E-17 1.0 - 2.08167E-16 ]
[ ]
[ 2.77556E-17 4.16334E-17 - 2.08167E-16 1.0 ]
(%i11) VT . transpose (VT);
[ 1.0 0.0 - 5.55112E-17 ]
[ ]
(%o11) [ 0.0 1.0 5.55112E-17 ]
[ ]
[ - 5.55112E-17 5.55112E-17 1.0 ]
@ref{Category: Package lapack}
Computes a norm or norm-like function of the matrix A.
maxCompute max(abs(A(i, j))) where i and j range over the rows and columns, respectively, of A. Note that this function is not a proper matrix norm.
one_normCompute the L[1] norm of A, that is, the maximum of the sum of the absolute value of elements in each column.
inf_normCompute the L[inf] norm of A, that is, the maximum of the sum of the absolute value of elements in each row.
frobeniusCompute the Frobenius norm of A, that is, the square root of the sum of squares of the matrix elements.
@ref{Category: Package lapack}
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