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55. lapack


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55.1 Introduction to lapack

lapack is a Common Lisp translation (via the program f2c) of the Fortran library LAPACK, as obtained from the SLATEC project.


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55.2 Functions and Variables for lapack

Function: dgeev (A)
Function: dgeev (A, right_p, left_p)

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      ]
Function: dgeqrf (A)

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.

A list of two items is returned. The first item is the matrix Q, which is a square, orthonormal matrix which has the same number of rows as A. The second item is the matrix R, which is the same size as A, and which has all elements equal to zero below the diagonal. The product Q . R, where "." is the noncommutative multiplication operator, is equal to A (ignoring floating point round-off errors).

Examples:

(%i1) load (lapack) $
(%i2) fpprintprec : 6 $
(%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]) $
(%i4) [q, r] : dgeqrf (M);
       [ - .0905357  .995893  ]
(%o4) [[                      ], 
       [  .995893    .0905357 ]
                               [ - 11.0454   2.97863   5.15148 ]
                               [                               ]]
                               [     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
Function: dgesv (A, b)

Computes the solution x of the linear equation A x = b, where A is a square matrix, and b is a matrix of the same number of rows as A and any number of columns. The return value x is the same size as b.

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);
                            [  1.210526315789474  ]
(%o3)                       [                     ]
                            [ - 0.215789473684211 ]
(%i4) dlange (inf_norm, b - A.x);
(%o4)                                 0.0

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);
      [  3.103827540695117  1.20985481742191    6.781786185657722 ]
(%o3) [                                                           ]
      [ -3.974483062032557  1.399032116146062  -8.121425428948527 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4)                       1.1102230246251565E-15

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);
                             [ 0.690375643155986 ]
(%o3)                        [                   ]
                             [ 0.233510982552952 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4)                        2.220446049250313E-16
Function: dgesvd (A)
Function: dgesvd (A, left_p, right_p)

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      ]
Function: dlange (norm, A)
Function: zlange (norm, A)

Computes a norm or norm-like function of the matrix A.

max

Compute 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_norm

Compute the L[1] norm of A,that is, the maximum of the sum of the absolute value of elements in each column.

inf_norm

Compute the L[inf] norm of A, that is, the maximum of the sum of the absolute value of elements in each row.

frobenius

Compute the Frobenius norm of A, that is, the square root of the sum of squares of the matrix elements.

Function: dgemm (A, B)
Function: dgemm (A, B, options)

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.

In the second form, dgemm computes the alpha * A * B + beta * C where A, B, C are real matrices of the appropriate sizes and alpha and beta are real numbers. Optionally, A and/or B can be transposed before computing the product. The extra parameters are specifed 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:

C

The matrix C that should be added. The default is false, which means no matrix is added.

alpha

The product of A and B is multiplied by this value. The default is 1.

beta

If a matrix C is given, this value multiplies C before it is added. The default value is 0, which implies that C is not added, even if C is given. Hence, be sure to specify a non-zero value for beta.

transpose_a

If true, the transpose of A is used instead of A for the product. The default is false.

transpose_b

If true, the transpose of B is used instead of B for the product. The default is false.

Examples:

(%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 ]


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