Next: Package lbfgs, Previous: Package interpol [Contents][Index]
Next: Functions and Variables for lapack, Previous: Package lapack, Up: Package lapack [Contents][Index]
lapack is a Common Lisp translation (via the program f2cl) of the Fortran library LAPACK,
as obtained from the SLATEC project.
Previous: Introduction to lapack, Up: Package lapack [Contents][Index]
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.
To make use of this function, you must load the LaPack package via
load("lapack").
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
$$ \mathbf{A} v_j = \lambda_j v_j $$where \(\lambda_j\) is the corresponding eigenvalue. The left eigenvector \(u_j\) (the j-th column of the left eigenvector matrix) satisfies
$$ u_j^\mathbf{H} \mathbf{A} = \lambda_j u_j^\mathbf{H} $$where \(u_j^\mathbf{H}\) denotes the conjugate transpose of \(u_j.\) For a Maxima function to compute the conjugate transpose, see ctranspose.
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);
[ - 0.666642 - 0.515792 ]
(%o5) [[7.54331, 12.4067], [ ],
[ 0.745378 - 0.856714 ]
[ - 0.856714 - 0.745378 ]
[ ]]
[ 0.515792 - 0.666642 ]
(%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 ]
Computes the QR decomposition of the matrix A. All elements of A must be integer or floating point numbers. A may or may not have the same number of rows and columns.
To make use of this function, you must load the LaPack package via
load("lapack").
The real square matrix \(\mathbf{A}\) can be decomposed as
$$ \mathbf{A} = \mathbf{Q}\mathbf{R} $$where \({\bf Q}\) is a square orthogonal matrix with the same number of rows as \(\mathbf{A}\) and \({\bf R}\) is an upper triangular matrix and is the same size as \({\bf A}.\)
A list of two items is returned. The first item is the matrix \({\bf Q}.\) The second item is the matrix \({\bf R},\) The product Q . R, where "." is the noncommutative multiplication operator, is equal to A (ignoring floating point round-off errors).
(%i1) load ("lapack")$
(%i2) fpprintprec : 6; (%o2) 6
(%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]);
[ 1 - 3.2 8 ]
(%o3) [ ]
[ - 11 2.7 5.9 ]
(%i4) [q, r] : dgeqrf (M);
[ - 0.0905357 0.995893 ]
(%o4) [[ ],
[ 0.995893 0.0905357 ]
[ - 11.0454 2.97863 5.15148 ]
[ ]]
[ 0.0 - 2.94241 8.50131 ]
(%i5) q . r - M;
[ - 7.77156e-16 1.77636e-15 - 8.88178e-16 ]
(%o5) [ ]
[ 0.0 - 1.33227e-15 8.88178e-16 ]
(%i6) mat_norm (%, 1); (%o6) 3.10862e-15
Computes the solution x of the linear equation \({\bf A} x = b,\) where \({\bf A}\) is a square matrix, and b is a matrix of the same number of rows as \({\bf A}\) and any number of columns. The return value x is the same size as b.
To make use of this function, you must load the LaPack package via
load("lapack").
The elements of A and b must evaluate to real floating point numbers via float;
thus elements may be any numeric type, symbolic numerical constants, or expressions which evaluate to floats.
The elements of x are always floating point numbers.
All arithmetic is carried out as floating point operations.
dgesv computes the solution via the LU decomposition of A.
Examples:
dgesv computes the solution of the linear equation A x = b.
(%i1) A : matrix ([1, -2.5], [0.375, 5]);
[ 1 - 2.5 ]
(%o1) [ ]
[ 0.375 5 ]
(%i2) b : matrix ([1.75], [-0.625]);
[ 1.75 ]
(%o2) [ ]
[ - 0.625 ]
(%i3) x : dgesv (A, b);
[ 1 - 2.5 ] [ 1.75 ]
(%o3) dgesv([ ], [ ])
[ 0.375 5 ] [ - 0.625 ]
(%i4) dlange (inf_norm, b - A . x);
[ 1.75 ]
(%o4) dlange(inf_norm, [ ]
[ - 0.625 ]
[ 1 - 2.5 ] [ 1 - 2.5 ] [ 1.75 ]
- [ ] . dgesv([ ], [ ]))
[ 0.375 5 ] [ 0.375 5 ] [ - 0.625 ]
b is a matrix with the same number of rows as A and any number of columns. x is the same size as b.
(%i1) A : matrix ([1, -0.15], [1.82, 2]);
[ 1 - 0.15 ]
(%o1) [ ]
[ 1.82 2 ]
(%i2) b : matrix ([3.7, 1, 8], [-2.3, 5, -3.9]);
[ 3.7 1 8 ]
(%o2) [ ]
[ - 2.3 5 - 3.9 ]
(%i3) x : dgesv (A, b);
[ 1 - 0.15 ] [ 3.7 1 8 ]
(%o3) dgesv([ ], [ ])
[ 1.82 2 ] [ - 2.3 5 - 3.9 ]
(%i4) dlange (inf_norm, b - A . x);
[ 3.7 1 8 ]
(%o4) dlange(inf_norm, [ ]
[ - 2.3 5 - 3.9 ]
[ 1 - 0.15 ] [ 1 - 0.15 ]
- [ ] . dgesv([ ],
[ 1.82 2 ] [ 1.82 2 ]
[ 3.7 1 8 ]
[ ]))
[ - 2.3 5 - 3.9 ]
The elements of A and b must evaluate to real floating point numbers.
(%i1) A : matrix ([5, -%pi], [1b0, 11/17]);
[ 5 - %pi ]
[ ]
(%o1) [ 11 ]
[ 1.0b0 -- ]
[ 17 ]
(%i2) b : matrix ([%e], [sin(1)]);
[ %e ]
(%o2) [ ]
[ sin(1) ]
(%i3) x : dgesv (A, b);
[ 5 - %pi ]
[ ] [ %e ]
(%o3) dgesv([ 11 ], [ ])
[ 1.0b0 -- ] [ sin(1) ]
[ 17 ]
(%i4) dlange (inf_norm, b - A . x);
[ %e ]
(%o4) dlange(inf_norm, [ ]
[ sin(1) ]
[ 5 - %pi ] [ 5 - %pi ]
[ ] [ ] [ %e ]
- [ 11 ] . dgesv([ 11 ], [ ]))
[ 1.0b0 -- ] [ 1.0b0 -- ] [ sin(1) ]
[ 17 ] [ 17 ]
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).
To make use of this function, you must load the LaPack package via
load("lapack").
Let m be the number of rows, and n the number of columns of A. The singular value decomposition of \(\mathbf{A}\) comprises three matrices, \(\mathbf{U},\) \(\mathbf{\Sigma},\) and \(\mathbf{V},\) such that
$$ \mathbf{A} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T $$where \(\mathbf{U}\) is an m-by-m unitary matrix, \(\mathbf{\Sigma}\) is an m-by-n diagonal matrix, and \(\mathbf{V}\) is an n-by-n unitary matrix.
Let
\(\mathbf{\sigma}_i\)
be a diagonal element of
\(\mathbf{\Sigma},\)
that is,
\(\mathbf{\Sigma}_{ii} = \sigma_i.\)
The elements
\(\sigma_i\)
are the so-called singular values of
\(\mathbf{A};\)
these are real and nonnegative, and returned in descending order.
The first
\(\min(m, n)\)
columns of
\(\mathbf{U}\)
and
\(\mathbf{V}\)
are the left and right singular vectors of
\(\mathbf{A}.\)
Note that dgesvd returns the transpose of
\(\mathbf{V},\)
not
\(\mathbf{V}\)
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, 0.452547], false, false]
(%i5) [sigma, U, VT] : dgesvd (M, true, true); (%o5) [[14.4744, 6.38637, 0.452547], [ - 0.256731 0.00816168 0.959029 - 0.119523 ] [ ] [ - 0.526456 0.672116 - 0.206236 - 0.478091 ] [ ], [ 0.107997 - 0.532278 - 0.0708315 - 0.83666 ] [ ] [ - 0.803287 - 0.514659 - 0.180867 0.239046 ] [ - 0.374486 - 0.538209 - 0.755044 ] [ ] [ 0.130623 - 0.836799 0.5317 ]] [ ] [ - 0.917986 0.100488 0.383672 ]
(%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 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 ]
Computes a norm or norm-like function of the matrix A. If
A is a real matrix, use dlange. For a matrix with
complex elements, use zlange.
To make use of this function, you must load the LaPack package via
load("lapack").
norm specifies the kind of norm to be computed:
maxCompute \(\max(|{\bf A}_{ij}|)\) where i and j range over the rows and columns, respectively, of \({\bf A}.\) Note that this function is not a proper matrix norm.
one_normCompute the \(L_1\) norm of \({\bf A},\) that is, the maximum of the sum of the absolute value of elements in each column.
inf_normCompute the \(L_\infty\) norm of \({\bf A},\) that is, the maximum of the sum of the absolute value of elements in each row.
frobeniusCompute the Frobenius norm of \({\bf A},\) that is, the square root of the sum of squares of the matrix elements.
Compute the product of two matrices and optionally add the product to a third matrix.
In the simplest form, dgemm(A, B) computes the
product of the two real matrices, A and B.
To make use of this function, you must load the LaPack package via
load("lapack").
In the second form, dgemm computes
\(\alpha {\bf A} {\bf B} + \beta {\bf C}\)
where
\({\bf A},\)
\({\bf B},\)
and
\({\bf C}\)
are real matrices of the appropriate sizes and
\(\alpha\)
and
\(\beta\)
are real numbers. Optionally,
\({\bf A}\)
and/or
\({\bf B}\)
can
be transposed before computing the product. The extra parameters are
specified by optional keyword arguments: The keyword arguments are
optional and may be specified in any order. They all take the form
key=val. The keyword arguments are:
CThe matrix
\({\bf C}\)
that should be added. The default is false,
which means no matrix is added.
alphaThe product of \({\bf A}\) and \({\bf B}\) is multiplied by this value. The default is 1.
betaIf a matrix \({\bf C}\) is given, this value multiplies \({\bf C}\) before it is added. The default value is 0, which implies that \({\bf C}\) is not added, even if \({\bf C}\) is given. Hence, be sure to specify a non-zero value for \(\beta.\)
transpose_aIf true, the transpose of
\({\bf A}\)
is used instead of
\({\bf A}\)
for the product. The default is false.
transpose_bIf true, the transpose of
\({\bf B}\)
is used instead of
\({\bf B}\)
for the product. The default is false.
(%i1) load ("lapack")$
(%i2) A : matrix([1,2,3],[4,5,6],[7,8,9]);
[ 1 2 3 ]
[ ]
(%o2) [ 4 5 6 ]
[ ]
[ 7 8 9 ]
(%i3) B : matrix([-1,-2,-3],[-4,-5,-6],[-7,-8,-9]);
[ - 1 - 2 - 3 ]
[ ]
(%o3) [ - 4 - 5 - 6 ]
[ ]
[ - 7 - 8 - 9 ]
(%i4) C : matrix([3,2,1],[6,5,4],[9,8,7]);
[ 3 2 1 ]
[ ]
(%o4) [ 6 5 4 ]
[ ]
[ 9 8 7 ]
(%i5) dgemm(A,B);
[ - 30.0 - 36.0 - 42.0 ]
[ ]
(%o5) [ - 66.0 - 81.0 - 96.0 ]
[ ]
[ - 102.0 - 126.0 - 150.0 ]
(%i6) A . B;
[ - 30 - 36 - 42 ]
[ ]
(%o6) [ - 66 - 81 - 96 ]
[ ]
[ - 102 - 126 - 150 ]
(%i7) dgemm(A,B,transpose_a=true);
[ - 66.0 - 78.0 - 90.0 ]
[ ]
(%o7) [ - 78.0 - 93.0 - 108.0 ]
[ ]
[ - 90.0 - 108.0 - 126.0 ]
(%i8) transpose(A) . B;
[ - 66 - 78 - 90 ]
[ ]
(%o8) [ - 78 - 93 - 108 ]
[ ]
[ - 90 - 108 - 126 ]
(%i9) dgemm(A,B,c=C,beta=1);
[ - 27.0 - 34.0 - 41.0 ]
[ ]
(%o9) [ - 60.0 - 76.0 - 92.0 ]
[ ]
[ - 93.0 - 118.0 - 143.0 ]
(%i10) A . B + C;
[ - 27 - 34 - 41 ]
[ ]
(%o10) [ - 60 - 76 - 92 ]
[ ]
[ - 93 - 118 - 143 ]
(%i11) dgemm(A,B,c=C,beta=1, alpha=-1);
[ 33.0 38.0 43.0 ]
[ ]
(%o11) [ 72.0 86.0 100.0 ]
[ ]
[ 111.0 134.0 157.0 ]
(%i12) -A . B + C;
[ 33 38 43 ]
[ ]
(%o12) [ 72 86 100 ]
[ ]
[ 111 134 157 ]
Like dgeev, but the matrix
\({\bf A}\)
is complex.
To make use of this function, you must load the LaPack package via
load("lapack").
Like dgeev, but the matrix
\({\bf A}\)
is assumed to be a square
complex Hermitian matrix. If eigvec_p is true, then the
eigenvectors of the matrix are also computed.
To make use of this function, you must load the LaPack package via
load("lapack").
No check is made that the matrix \({\bf A}\) is, in fact, Hermitian.
A list of two items is returned, as in dgeev: a list of
eigenvalues, and false or the matrix of the eigenvectors.
(%i1) load("lapack")$
(%i2) M: matrix(
[9.14 +%i*0.00 , -4.37 -%i*9.22 , -1.98 -%i*1.72 , -8.96 -%i*9.50],
[-4.37 +%i*9.22 , -3.35 +%i*0.00 , 2.25 -%i*9.51 , 2.57 +%i*2.40],
[-1.98 +%i*1.72 , 2.25 +%i*9.51 , -4.82 +%i*0.00 , -3.24 +%i*2.04],
[-8.96 +%i*9.50 , 2.57 -%i*2.40 , -3.24 -%i*2.04 , 8.44 +%i*0.00]);
[ 9.14 ] [ - 9.22 %i - 4.37 ]
[ ] [ ]
[ 9.22 %i - 4.37 ] [ - 3.35 ]
(%o2) Col 1 = [ ] Col 2 = [ ]
[ 1.72 %i - 1.98 ] [ 9.51 %i + 2.25 ]
[ ] [ ]
[ 9.5 %i - 8.96 ] [ 2.57 - 2.4 %i ]
[ - 1.72 %i - 1.98 ] [ - 9.5 %i - 8.96 ]
[ ] [ ]
[ 2.25 - 9.51 %i ] [ 2.4 %i + 2.57 ]
Col 3 = [ ] Col 4 = [ ]
[ - 4.82 ] [ 2.04 %i - 3.24 ]
[ ] [ ]
[ - 2.04 %i - 3.24 ] [ 8.44 ]
(%i3) zheev(M);
(%o3) [[- 16.004746472094734, - 6.764970154793324,
6.6657114535070985, 25.51400517338097], false]
(%i4) E: zheev(M,true)$
(%i5) E[1];
(%o5) [- 16.004746472094737, - 6.764970154793325,
6.665711453507101, 25.514005173380962]
(%i6) E[2];
[ 0.26746505331727455 %i + 0.21754535866650165 ]
[ ]
[ 0.002696730886619885 %i + 0.6968836773391712 ]
(%o6) Col 1 = [ ]
[ - 0.6082406376714117 %i - 0.012106142926979313 ]
[ ]
[ 0.15930818580950368 ]
[ 0.26449374706674444 %i + 0.4773693349937472 ]
[ ]
[ - 0.28523890360316206 %i - 0.14143627420116733 ]
Col 2 = [ ]
[ 0.2654607680986639 %i + 0.44678181171841735 ]
[ ]
[ 0.5750762708542709 ]
[ 0.28106497673059216 %i - 0.13352639282451817 ]
[ ]
[ 0.28663101328695556 %i - 0.4536971347853274 ]
Col 3 = [ ]
[ - 0.29336843237542953 %i - 0.49549724255410565 ]
[ ]
[ 0.5325337537576771 ]
[ - 0.5737316575503476 %i - 0.39661467994277055 ]
[ ]
[ 0.018265026190214573 %i + 0.35305577043870173 ]
Col 4 = [ ]
[ 0.16737009000854253 %i + 0.01476684746229564 ]
[ ]
[ 0.6002632636961784 ]
Next: Package lbfgs, Previous: Package interpol [Contents][Index]