Description of maxima_unicode_display in reference manual,

[maxima.git] / share / lapack / eigensys.lisp

(in-package :maxima)

(defun maxima-matrix-dims (a)
  (let ((row (second a)))
    ;; Should we check that all rows have the same length?
    (values (length (cdr a)) (length (cdr row)))))

(defun complex-maxima-matrix-p (a)
  (dolist (row (cdr a))
    (dolist (col (cdr row))
      (unless (eql ($imagpart col) 0)
        (return-from complex-maxima-matrix-p t))))
  nil)

(defun lapack-lispify-matrix (a nrow ncol &optional (assume-complex-maxima-matrix-p nil))
  "Convert a Maxima matrix A of dimension NROW and NCOL to Lisp matrix
  suitable for use with LAPACK"
  (setq a ($float a))
  (let* ((array-type (if (or assume-complex-maxima-matrix-p (complex-maxima-matrix-p a))
                         '(complex flonum)
                         'flonum))
         (mat (make-array (* nrow ncol)
                          :element-type array-type))
         (mat-2d (make-array (list ncol nrow)
                             :element-type array-type
                             :displaced-to mat))
         (r 0))
    (dolist (row (cdr a))
      (let ((c 0))
        (dolist (col (cdr row))
          ;; Fortran matrices are in column-major order!
          (setf (aref mat-2d c r) (if (eql array-type 'flonum)
                                      (coerce col 'flonum)
                                      (coerce (complex ($realpart col) ($imagpart col))
                                              '(complex flonum))
                                      ))
          (incf c)))
      (incf r))
    mat))

(defun lapack-maxify-matrix (nrow ncol a)
  "Convert an LAPACK matrix of dimensions NROW and NCOL into a Maxima
matrix (list of lists)"
  (let ((2d (make-array (list ncol nrow) :element-type (array-element-type a)
                        :displaced-to a)))
    (let (res)
      (dotimes (r nrow)
        (let (row)
          (dotimes (c ncol)
            ;; Fortran arrays are column-major order!
            (let ((v (aref 2d c r)))
              (push (add (realpart v) (mul '$%i (imagpart v))) row)))
          (push `((mlist) ,@(nreverse row)) res)))
      `(($matrix) ,@(nreverse res)))))

(defun $dgeev (a &optional right-vec-p left-vec-p)
  "
DGEEV computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.

The right eigenvector v(j) of A satisfies
                 A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
              u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.

A list of three items is returned.  The first item is a list of the
eigenvectors.  The second item is false or the matrix of right
eigenvectors.  The last itme is false or the matrix of left
eigenvectors."
  (flet ((make-eigval (wr wi)
           `((mlist) ,@(map 'list #'(lambda (r i)
                                      (add r (mul '$%i i)))
                            wr wi)))
         (make-eigvec (n vr wi)
           ;; dgeev stores the eigen vectors in a special way.  Undo
           ;; that.  For simplicity, we create a 2-D matrix and store
           ;; the eigenvectors there.  Then we convert that matrix
           ;; into a form that maxima wants.  Somewhat inefficient.
           (let ((evec (make-array (list n n))))
             (do ((col 0 (incf col))
                  (posn 0))
                 ((>= col n))
               (cond ((zerop (aref wi col))
                      (dotimes (row n)
                        (setf (aref evec row col) (aref vr posn))
                        (incf posn)))
                     (t
                      (dotimes (row n)
                        (let* ((next-posn (+ posn n))
                               (val+ (add (aref vr posn)
                                          (mul '$%i (aref vr next-posn))))
                               (val- (sub (aref vr posn)
                                          (mul '$%i (aref vr next-posn)))))
                          (setf (aref evec row col) val+)
                          (setf (aref evec row (1+ col)) val-)
                          (incf posn)))
                      ;; Skip over the next column, which we've already used
                      (incf col)
                      (incf posn n))))
             ;; Now convert this 2-D Lisp matrix into a maxima matrix
             (let (res)
               (dotimes (r n)
                 (let (row)
                   (dotimes (c n)
                     (push (aref evec r c) row))
                   (push `((mlist) ,@(nreverse row)) res)))
               `(($matrix) ,@(nreverse res)))
             )))
    
    (let* ((n (maxima-matrix-dims a))
           (a-mat (lapack-lispify-matrix a n n))
           (wr (make-array n :element-type 'flonum))
           (wi (make-array n :element-type 'flonum))
           (vl (make-array (if left-vec-p (* n n) 0)
                           :element-type 'flonum))
           (vr (make-array (if right-vec-p (* n n) 0)
                           :element-type 'flonum)))
      ;; XXX: FIXME: We need to do more error checking in the calls to
      ;; dgeev!
      (multiple-value-bind (z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                                    z-ldvl z-vr z-ldvr z-work z-lwork info)
          ;; Figure out how much space we need in the work array.
          (lapack:dgeev (if left-vec-p "V" "N")
                        (if right-vec-p "V" "N")
                        n a-mat n wr wi vl n vr n wr -1 0)
        (declare (ignore z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                         z-ldvl z-vr z-ldvr z-work z-lwork info))
        (let* ((opt-lwork (truncate (aref wr 0)))
               (work (make-array opt-lwork :element-type 'flonum)))
          ;; Now do the work with the optimum size of the work space.
          (multiple-value-bind (z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                                        z-ldvl z-vr z-ldvr z-work z-lwork info)
              (lapack:dgeev (if left-vec-p "V" "N")
                            (if right-vec-p "V" "N")
                            n a-mat n wr wi vl n vr n work opt-lwork 0)
            (declare (ignore z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                             z-ldvl z-vr z-ldvr z-work z-lwork))
            (cond ((< info 0)
                   (merror "DGEEV: invalid arguments: ~D" info))
                  ((> info 0)
                   (merror "DGEEV: failed to converge: ~D" info)))
            ;; Convert wr+%i*wi to maxima form
            #+nil
            (progn
              (format t "info = ~A~%" info)
              (format t "lwork = ~A~%" (aref work 0))
              (format t "vr = ~A~%" vr))
            (let ((e-val (make-eigval wr wi))
                  (e-vec-right (if right-vec-p
                                   (make-eigvec n vr wi)
                                   nil))
                  (e-vec-left (if left-vec-p
                                  (make-eigvec n vl wi)
                                  nil)))
              `((mlist) ,e-val ,e-vec-right ,e-vec-left))))))))

(defun $dgesvd (a &optional jobu jobvt)
  "
DGESVD computes the singular value decomposition (SVD) of a real
M-by-N matrix A, optionally computing the left and/or right singular
vectors. The SVD is written

     A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its
min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
are the singular values of A; they are real and non-negative, and
are 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 the routine returns V**T, not V.

A list of three items is returned.  The first is a list containing the
non-zero elements of SIGMA.  If jobu is not false, The second element
is the matrix U.  Otherwise it is false.  Similarly, the third element
is V**T or false, depending on jobvt."
  
  (flet ((maxify-vector (v)
           `((mlist) ,@(coerce v 'list)))
         (fixup-jobu (arg)
           (if arg "All columns of U" "No columns of U"))
         (fixup-jobvt (arg)
           (if arg "All columns of V^T" "No columns of V^T")))
    (multiple-value-bind (nrow ncol)
        (maxima-matrix-dims a)  
      (let* ((a-mat (lapack-lispify-matrix a nrow ncol))
             (s (make-array (min nrow ncol) :element-type 'flonum))
             (u (make-array (* nrow nrow) :element-type 'flonum))
             (u1 (make-array (list nrow nrow) :element-type 'flonum
                             :displaced-to u))
             (vt (make-array (* ncol ncol)
                             :element-type 'flonum))
             (vt1 (make-array (list ncol ncol) :element-type 'flonum
                              :displaced-to vt))
             (wr (make-array 1 :element-type 'flonum)))
        ;; XXX: FIXME: We need to do more error checking in the calls to
        ;; dgesvd!
        (multiple-value-bind (z-jobu z-jobvt z-m z-n z-a z-lda z-s z-u z-ldu
                                     z-vt z-ldvt z-work z-lwork info)
            ;; Figure out the optimum size for the work array
            (lapack::dgesvd (fixup-jobu jobu)
                            (fixup-jobvt jobvt)
                            nrow ncol a-mat nrow
                            s u nrow
                            vt ncol
                            wr -1
                            0)
          (declare (ignore z-jobu z-jobvt z-m z-n z-a z-lda z-s z-u z-ldu
                           z-vt z-ldvt z-work z-lwork info))
          (let* ((opt-lwork (truncate (aref wr 0)))
                 (work (make-array opt-lwork :element-type 'flonum)))
            ;; Allocate the optimum work array and do the requested
            ;; computation.
            (multiple-value-bind (z-jobu z-jobvt z-m z-n z-a z-lda z-s z-u
                                         z-ldu z-vt z-ldvt z-work z-lwork info)
                (lapack::dgesvd (fixup-jobu jobu)
                                (fixup-jobvt jobvt)
                                nrow ncol a-mat nrow
                                s u nrow
                                vt ncol
                                work opt-lwork
                                0)
              (declare (ignore z-jobu z-jobvt z-m z-n z-a z-lda z-s z-u z-ldu
                               z-vt z-ldvt z-work z-lwork))
              (cond ((< info 0)
                     (merror "DGESVD: invalid arguments: ~D" info))
                    ((> info 0)
                     (merror "DGESVD: failed to converge: ~D" info)))
              (let ((u-max (if jobu
                               (lapack-maxify-matrix nrow nrow u1)
                               nil))
                    (vt-max (if jobvt
                                (lapack-maxify-matrix ncol ncol vt1)
                                nil))
                    (s-max (maxify-vector s)))
                `((mlist) ,s-max ,u-max ,vt-max)))))))))


\f
(defun $dlange (norm a)
  "
DLANGE returns the value

   DLANGE = ( max(abs(A(i,j))), NORM = '$max
            (
            ( norm1(A),         NORM = '$one_norm
            (
            ( normI(A),         NORM = '$inf_norm
            (
            ( normF(A),         NORM = '$frobenius

where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm."

  ;; Norm should be '$max, '$one_norm, '$inf_norm, '$frobenius
  (multiple-value-bind (nrows ncols)
      (maxima-matrix-dims a)
    (let* ((a-mat (lapack-lispify-matrix a nrows ncols))
           (norm-type (ecase norm
                        ($max "M")
                        ($one_norm "O")
                        ($inf_norm "I")
                        ($frobenius "F")))
           (work (make-array (if (equal norm-type "I") nrows 0)
                             :element-type 'flonum)))
      (lapack::dlange norm-type nrows ncols a-mat nrows work))))


(defun $zlange (norm a)
  "
DLANGE returns the value

   DLANGE = ( max(abs(A(i,j))), NORM = '$max
            (
            ( norm1(A),         NORM = '$one_norm
            (
            ( normI(A),         NORM = '$inf_norm
            (
            ( normF(A),         NORM = '$frobenius

where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm."

  ;; Norm should be '$max, '$one_norm, '$inf_norm, '$frobenius
  (multiple-value-bind (nrows ncols)
      (maxima-matrix-dims a)
    (let* ((a-mat (lapack-lispify-matrix a nrows ncols))
           (norm-type (ecase norm
                        ($max "M")
                        ($one_norm "O")
                        ($inf_norm "I")
                        ($frobenius "F")))
           (work (make-array (if (equal norm-type "I") nrows 0)
                             :element-type 'flonum)))
      (lapack::zlange norm-type nrows ncols a-mat nrows work))))
      
(defun $zgeev (a &optional right-vec-p left-vec-p)
  "
ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.

The right eigenvector v(j) of A satisfies
                 A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
              u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.

A list of three items is returned.  The first item is a list of the
eigenvectors.  The second item is false or the matrix of right
eigenvectors.  The last itme is false or the matrix of left
eigenvectors."
  (flet ((make-eigval (w)
           `((mlist) ,@(map 'list #'(lambda (z)
                                      (add (realpart z) (mul '$%i (imagpart z))))
                            w))))
    
    (let* ((n (maxima-matrix-dims a))
           (a-mat (lapack-lispify-matrix a n n t))
           (w (make-array n :element-type '(complex flonum)))
           (vl (make-array (if left-vec-p (* n n) 0)
                           :element-type '(complex flonum)))
           (vr (make-array (if right-vec-p (* n n) 0)
                           :element-type '(complex flonum)))
           (rwork (make-array 1 :element-type 'flonum)))
      ;; XXX: FIXME: We need to do more error checking in the calls to
      ;; zgeev!
      (multiple-value-bind (z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                                    z-ldvl z-vr z-ldvr z-work z-lwork info)
          ;; Figure out how much space we need in the work array.
          (lapack:zgeev (if left-vec-p "V" "N")
                        (if right-vec-p "V" "N")
                        n a-mat n w vl n vr n w -1 rwork 0)
        (declare (ignore z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                         z-ldvl z-vr z-ldvr z-work z-lwork info))
        (let* ((opt-lwork (truncate (realpart (aref w 0))))
               (work (make-array opt-lwork :element-type '(complex flonum)))
               (rwork (make-array opt-lwork :element-type 'flonum)))
          ;; Now do the work with the optimum size of the work space.
          (multiple-value-bind (z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                                        z-ldvl z-vr z-ldvr z-work z-lwork info)
              (lapack:zgeev (if left-vec-p "V" "N")
                            (if right-vec-p "V" "N")
                            n a-mat n w vl n vr n work opt-lwork rwork 0)
            (declare (ignore z-jobvl z-jobvr z-n z-a z-lda z-wr z-wi z-vl
                             z-ldvl z-vr z-ldvr z-work z-lwork))
            (cond ((< info 0)
                   (merror "ZGEEV: invalid arguments: ~D" info))
                  ((> info 0)
                   (merror "ZGEEV: failed to converge: ~D" info)))
            (let ((e-val (make-eigval w))
                  (e-vec-right (if right-vec-p
                                   (lapack-maxify-matrix n n vr)
                                   nil))
                  (e-vec-left (if left-vec-p
                                  (lapack-maxify-matrix n n vl)
                                  nil)))
              `((mlist) ,e-val ,e-vec-right ,e-vec-left))))))))

(defun $zheev (a &optional eigen-vector-p)
  (flet ((make-eigval (w)
           `((mlist) ,@(map 'list #'(lambda (z)
                                      (add (realpart z) (mul '$%i (imagpart z))))
                            w))))
    
    (let* ((n (maxima-matrix-dims a))
           (a-mat (lapack-lispify-matrix a n n t))
           (w (make-array n :element-type 'flonum))
           (work (make-array 1 :element-type '(complex flonum)))
           (rwork (make-array (max 1 (- (* 3 n) 2))
                              :element-type 'flonum)))
      ;; XXX: FIXME: We need to do more error checking in the calls to
      ;; zgeev!
      (multiple-value-bind (z-jobz z-uplo z-n z-a z-lda z-w z-work
                            z-lwork z-rwork info)
          ;; Figure out how much space we need in the work array.
          (lapack:zheev (if eigen-vector-p "V" "N")
                        "U"
                        n
                        a-mat
                        n
                        w
                        work
                        -1
                        rwork
                        0)
        (declare (ignore z-jobz z-uplo z-n z-a z-lda z-w z-work
                         z-lwork z-rwork))
        (let* ((opt-lwork (truncate (realpart (aref work 0))))
               (work (make-array opt-lwork :element-type '(complex flonum))))
          ;; Now do the work with the optimum size of the work space.
          (multiple-value-bind (z-jobz z-uplo z-n z-a z-lda z-w z-work
                                z-lwork z-rwork info)
              (lapack:zheev (if eigen-vector-p "V" "N")
                            "U"
                            n
                            a-mat
                            n
                            w
                            work
                            opt-lwork
                            rwork
                            0)
            (declare (ignore z-jobz z-uplo z-n z-a z-lda z-w z-work
                             z-lwork z-rwork))
            (cond ((< info 0)
                   (merror "ZHEEV: invalid arguments: ~D" info))
                  ((> info 0)
                   (merror "ZHEEV: failed to converge: ~D" info)))
            (let ((e-val (make-eigval w))
                  (e-vec (if eigen-vector-p
                                   (lapack-maxify-matrix n n a-mat)
                                   nil)))
              `((mlist) ,e-val ,e-vec))))))))