The Historic MATLAB Users’ Guide 5

Posted by Cleve Moler,

In the 1970s and early 1980s, while I was working on the LINPACK and EISPACK projects that I discussed in two previous posts, I was a Professor of Mathematics and then of Computer Science at the University of New Mexico in Albuquerque. I was teaching courses in Linear Algebra and Numerical Analysis. I wanted my students to have easy access to LINPACK and EISPACK without writing Fortran programs. By "easy access" I meant not going through the remote batch processing and the repeated edit-compile-link-load-execute process that was ordinarily required on the campus central mainframe computer.

So, I studied Niklaus Wirth's book Algorithms + Data Structures = Programs and learned how to parse programming languages. Wirth calls his example language PL/0. (A zinger aimed at PL/I, the monolithic, committee-designed language being promoted by IBM and the U.S. Defense Department at the time.) PL/0 was a pedagogical subset of Wirth's Pascal, which, in turn, was his response to Algol. Quoting Wirth, "In the realm of data types, however, PL/0 adheres to the demand of simplicity without compromise: integers are its only data type."

Following Wirth's approach, I wrote the first MATLAB -- an acronym for Matrix Laboratory -- in Fortran, as a dialect of PL/0 with matrix as the only data type. The project was a kind of hobby, a new aspect of programming for me to learn and something for my students to use. There was never any formal outside support and certainly no business plan. MathWorks was a few years in the future.

This first MATLAB was not a programming language; it was just a simple interactive matrix calculator. There were no m-files, no toolboxes, no graphics. And no ODEs or FFTs. This snapshot of the start-up screen shows all the reserved words and functions. There are only 71 of them. If you wanted to add another function, you would get the source code from me, write a Fortran subroutine, add your new name to the parse table, and recompile MATLAB.

Here is the first Users' Guide, in its entirety, from 1981.

Contents

Users' Guide

                         MATLAB Users' Guide
                              May, 1981
                             Cleve Moler
                   Department of Computer Science
                      University of New Mexico
       ABSTRACT.  MATLAB is an  interactive  computer  program
       that   serves   as   a   convenient   "laboratory"  for
       computations  involving  matrices.   It  provides  easy
       access  to matrix software developed by the LINPACK and
       EISPACK projects.  The program is  written  in  Fortran
       and  is  designed  to  be  readily  installed under any
       operating system which permits interactive execution of
       Fortran programs.
       MATLAB is an interactive computer program that serves  as  a
  convenient  "laboratory" for computations involving matrices.  It
  provides easy access to matrix software developed by the  LINPACK
  and EISPACK projects [1-3].  The capabilities range from standard
  tasks such as solving simultaneous linear equations and inverting
  matrices, through symmetric and nonsymmetric eigenvalue problems,
  to fairly sophisticated matrix tools such as the  singular  value
  decomposition.
       It is expected that one of MATLAB's primary uses will be  in
  the  classroom.   It  should be useful in introductory courses in
  applied linear algebra, as  well  as  more  advanced  courses  in
  numerical analysis, matrix theory, statistics and applications of
  matrices to other disciplines.  In nonacademic  settings,  MATLAB
  can  serve as a "desk calculator" for the quick solution of small
  problems involving matrices.
       The program is written in Fortran  and  is  designed  to  be
  readily  installed  under  any  operating  system  which  permits
  interactive  execution  of  Fortran  programs.    The   resources
  required  are  fairly  modest.  There are less than 7000 lines of
  Fortran  source  code,  including   the   LINPACK   and   EISPACK
  subroutines  used.   With  proper use of overlays, it is possible
  run the system on a minicomputer with only 32K bytes of memory.
       The size of the matrices  that  can  be  handled  in  MATLAB
  depends  upon  the  amount  of storage that is set aside when the
  system is compiled on a particular machine.  We have  found  that
  an  allocation of 5000 words for matrix elements is usually quite
  satisfactory.  This provides room for several 20 by 20  matrices,
  for  example.   One  implementation  on  a  virtual memory system
  provides 100,000 elements.  Since most  of  the  algorithms  used
  access  memory  in  a  sequential  fashion,  the  large amount of
  allocated storage causes no difficulties.
       In some ways, MATLAB  resembles  SPEAKEASY  [4]  and,  to  a
  lesser  extent, APL.  All are interactive terminal languages that
  ordinarily accept single-line  commands  or  statements,  process
  them  immediately,  and  print  the  results.  All have arrays or
  matrices as principal data types.  But for MATLAB, the matrix  is
  the  only  data  type  (although  scalars,  vectors  and text are
  special cases), the underlying system is  portable  and  requires
  fewer resources, and the supporting subroutines are more powerful
  and, in some cases, have better numerical properties.
       Together, LINPACK and EISPACK represent the state of the art
  in software for matrix computation.  EISPACK is a package of over
  70 Fortran subroutines for various matrix eigenvalue computations
  that are based for the most part on Algol procedures published by
  Wilkinson, Reinsch  and  their  colleagues  [5].   LINPACK  is  a
  package  of  40  Fortran subroutines (in each of four data types)
  for solving  and  analyzing  simultaneous  linear  equations  and
  related matrix problems.  Since MATLAB is not primarily concerned
  with either execution time  efficiency  or  storage  savings,  it
  ignores  most  of  the special matrix properties that LINPACK and
  EISPACK subroutines  use  to  advantage.   Consequently,  only  8
  subroutines   from  LINPACK  and  5  from  EISPACK  are  actually
  involved.
       In  more  advanced  applications,  MATLAB  can  be  used  in
  conjunction  with other programs in several ways.  It is possible
  to define new MATLAB functions and add them to the system.   With
  most  operating  systems,  it  is  possible to use the local file
  system to  pass  matrices  between  MATLAB  and  other  programs.
  MATLAB  command  and statement input can be obtained from a local
  file  instead  of  from  the  terminal.   The  most   power   and
  flexibility  is obtained by using MATLAB as a subroutine which is
  called by other programs.
       This document first gives an overview  of  MATLAB  from  the
  user's  point  of  view. Several extended examples involving data
  fitting, partial differential equations,  eigenvalue  sensitivity
  and other topics are included.  A formal definition of the MATLAB
  language and an brief description of the parser  and  interpreter
  are   given.   The  system  was  designed  and  programmed  using
  techniques described by Wirth [6], implemented  in  nonrecursive,
  portable  Fortran.   There  is  a brief discussion of some of the
  matrix algorithms and of their numerical properties.   The  final
  section  describes  how  MATLAB  can be used with other programs.
  The appendix includes the HELP documentation available on-line.

1. Elementary operations

       MATLAB works with essentially only one  kind  of  object,  a
  rectangular matrix with complex elements.  If the imaginary parts
  of the elements are all zero, they  are  not  printed,  but  they
  still  occupy  storage.   In  some situations, special meaning is
  attached to 1 by 1 matrices, that is scalars, and to 1 by n and m
  by 1 matrices, that is row and column vectors.
       Matrices can be introduced into  MATLAB  in  four  different
  ways:
          --  Explicit list of elements,
          --  Use of FOR and WHILE statements,
          --  Read from an external file,
          --  Execute an external Fortran program.
       The explicit list is surrounded by angle brackets,  '<'  and
  '>', and uses the semicolon ';' to indicate the ends of the rows.
  For example, the input line
     A = <1 2 3; 4 5 6; 7 8 9>
  will result in the output
     A     =
         1.    2.   3.
         4.    5.   6.
         7.    8.   9.
  The matrix A  will  be  saved  for  later  use.   The  individual
  elements  are separated by commas or blanks and can be any MATLAB
  expressions, for example
     x = < -1.3, 4/5, 4*atan(1) >
  results in
     X     =
       -1.3000   0.8000   3.1416
  The elementary functions available include sqrt, log,  exp,  sin,
  cos, atan, abs, round, real, imag, and conjg.
       Large matrices can be spread  across  several  input  lines,
  with  the  carriage  returns replacing the semicolons.  The above
  matrix could also have been produced by
     A = < 1 2 3
           4 5 6
           7 8 9 >
       Matrices can be input from the local  file  system.   Say  a
  file named 'xyz' contains five lines of text,
     A = <
     1 2 3
     4 5 6
     7 8 9
     >;
  then the  MATLAB  statement  EXEC('xyz')  reads  the  matrix  and
  assigns it to A .
       The FOR statement allows the generation  of  matrices  whose
  elements  are  given  by  simple formulas.  Our example matrix  A
  could also have been produced by
     for i = 1:3, for j = 1:3, a(i,j) = 3*(i-1)+j;
  The semicolon at the end of the  line  suppresses  the  printing,
  which  in  this  case  would  have  been  nine versions of A with
  changing elements.
       Several statements may be given  on  a  line,  separated  by
  semicolons or commas.
       Two  consecutive  periods  anywhere  on  a   line   indicate
  continuation.   The  periods  and  any  following  characters are
  deleted, then another line is input  and  concatenated  onto  the
  previous line.
       Two  consecutive  slashes  anywhere  on  a  line  cause  the
  remainder  of  the  line  to  be  ignored.   This  is  useful for
  inserting comments.
       Names of variables are formed by a letter, followed  by  any
  number of letters and digits, but only the first 4 characters are
  remembered.
       The special character  prime  (')  is  used  to  denote  the
  transpose of a matrix, so
     x = x'
  changes the row vector above into the column vector
     X     =
       -1.3000
        0.8000
        3.1416
       Individual matrix elements may be  referenced  by  enclosing
  their  subscripts  in  parentheses.  When any element is changed,
  the entire matrix is reprinted.  For  example,  using  the  above
  matrix,
     a(3,3) = a(1,3) + a(3,1)
  results in
     A     =
         1.    2.    3.
         4.    5.    6.
         7.    8.   10.
       Addition, subtraction and  multiplication  of  matrices  are
  denoted  by  +, -, and * .  The operations are performed whenever
  the matrices have the proper dimensions.  For example,  with  the
  above  A  and  x,  the  expressions  A  + x and x*A are incorrect
  because A is 3 by 3 and x is now 3 by 1.  However,
     b = A*x
  is correct and results in the output
     B     =
        9.7248
       17.6496
       28.7159
  Note that both upper and lower case letters are allowed for input
  (on  those  systems  which  have  both),  but  that lower case is
  converted to upper case.
       There are two "matrix division" symbols in MATLAB, \ and / .
  (If  your  terminal  does not have a backslash, use $ instead, or
  see CHAR.) If A and B are matrices, then A\B and  B/A  correspond
  formally  to left and right multiplication of B by the inverse of
  A , that is inv(A)*B and B*inv(A), but  the  result  is  obtained
  directly  without  the computation of the inverse.  In the scalar
  case, 3\1 and 1/3 have the  same  value,  namely  one-third.   In
  general,  A\B  denotes the solution X to the equation A*X = B and
  B/A denotes the solution to X*A = B.
       Left division, A\B, is defined whenever B has as  many  rows
  as   A  .   If  A  is  square,  it  is  factored  using  Gaussian
  elimination.   The  factors  are  used  to  solve  the  equations
  A*X(:,j) = B(:,j) where B(:,j) denotes the j-th column of B.  The
  result is a matrix X with the same dimensions  as  B.   If  A  is
  nearly  singular  (according  to the LINPACK condition estimator,
  RCOND), a warning message is printed.  If A is not square, it  is
  factored   using   Householder   orthogonalization   with  column
  pivoting.   The  factors  are  used  to  solve  the   under-   or
  overdetermined equations in a least squares sense.  The result is
  an m by n matrix X where m is the number of columns of A and n is
  the  number  of  columns  of  B .  Each column of X has at most k
  nonzero components, where k is the effective rank of A .
       Right division,  B/A,  can  be  defined  in  terms  of  left
  division by  B/A = (A'\B')'.
       For example, since our vector  b   was computed as  A*x, the
  statement
     y = A\b
  results in
     Y     =
       -1.3000
        0.8000
        3.1416
  Of course,  y  is  not  exactly  equal  to   x   because  of  the
  roundoff  errors involved in both  A*x  and  A\b , but we are not
  printing enough digits to see the difference.  The result of  the
  statement
     e = x - y
  depends upon the particular computer being used.  In one case  it
  produces
     E     =
        1.0e-15 *
          .3053
         -.2498
          .0000
  The quantity 1.0e-15 is a scale factor which multiplies  all  the
  components  which  follow.  Thus our vectors  x  and  y  actually
  agree to about 15 decimal places on this computer.
       It   is   also   possible   to   obtain   element-by-element
  multiplicative  operations.  If A and B have the same dimensions,
  then A .* B denotes the matrix  whose  elements  are  simply  the
  products  of the individual elements of A and B . The expressions
  A ./ B and A .\ B give the quotients of the individual elements.
       There are several possible output formats.  The statement
     long, x
  results in
     X     =
        -1.300000000000000
          .800000000000000
         3.141592653589793
  The statement
     short
  restores the original format.
       The expression A**p means  A  to  the  p-th  power.   It  is
  defined  if  A  is a square matrix and p is a scalar.  If p is an
  integer greater than one,  the  power  is  computed  by  repeated
  multiplication.   For  other values of p the calculation involves
  the eigenvalues and eigenvectors of A.
       Previously defined matrices and matrix  expressions  can  be
  used inside brackets to generate larger matrices, for example
     C = <A, b; <4 2 0>*x, x'>
  results in
     C     =
        1.0000   2.0000   3.0000   9.7248
        4.0000   5.0000   6.0000  17.6496
        7.0000   8.0000  10.0000  28.7159
       -3.6000  -1.3000   0.8000   3.1416
       There are four predefined variables,  EPS,  FLOP,  RAND  and
  EYE.  The variable EPS is used as a tolerance is determining such
  things as near singularity and rank.  Its initial  value  is  the
  distance  from  1.0  to the next largest floating point number on
  the particular computer being used.  The user may reset  this  to
  any other value, including zero. EPS is changed by CHOP, which is
  described in section 12.
       The value of RAND is a random variable, with a choice  of  a
  uniform or a normal distribution.
       The name EYE is used  in  place  of  I  to  denote  identity
  matrices  because  I is often used as a subscript or as sqrt(-1).
  The dimensions of EYE are determined by context.  For example,
     B = A + 3*EYE
  adds 3 to the diagonal elements of A and
     X = EYE/A
  is one of several ways in MATLAB to invert a matrix.
       FLOP provides a  count  of  the  number  of  floating  point
  operations, or "flops", required for each calculation.
       A statement may consist of an  expression  alone,  in  which
  case a variable named ANS is created and the result stored in ANS
  for possible future use.  Thus
     A\A - EYE
  is the same as
     ANS = A\A - EYE
  (Roundoff error usually causes this result  to  be  a  matrix  of
  "small" numbers, rather than all zeros.)
       All computations are done  using  either  single  or  double
  precision  real  arithmetic,  whichever  is  appropriate  for the
  particular computer.  There  is  no  mixed-precision  arithmetic.
  The  Fortran  COMPLEX  data type is not used because many systems
  create  unnecessary  underflows  and   overflows   with   complex
  operations and because some systems do not allow double precision
  complex arithmetic.

2. MATLAB functions

       Much of MATLAB's computational power comes from the  various
  matrix functions available.  The current list includes:
     INV(A)          - Inverse.
     DET(A)          - Determinant.
     COND(A)         - Condition number.
     RCOND(A)        - A measure of nearness to singularity.
     EIG(A)          - Eigenvalues and eigenvectors.
     SCHUR(A)        - Schur triangular form.
     HESS(A)         - Hessenberg or tridiagonal form.
     POLY(A)         - Characteristic polynomial.
     SVD(A)          - Singular value decomposition.
     PINV(A,eps)     - Pseudoinverse with optional tolerance.
     RANK(A,eps)     - Matrix rank with optional tolerance.
     LU(A)           - Factors from Gaussian elimination.
     CHOL(A)         - Factor from Cholesky factorization.
     QR(A)           - Factors from Householder orthogonalization.
     RREF(A)         - Reduced row echelon form.
     ORTH(A)         - Orthogonal vectors spanning range of A.
     EXP(A)          - e to the A.
     LOG(A)          - Natural logarithm.
     SQRT(A)         - Square root.
     SIN(A)          - Trigonometric sine.
     COS(A)          - Cosine.
     ATAN(A)         - Arctangent.
     ROUND(A)        - Round the elements to nearest integers.
     ABS(A)          - Absolute value of the elements.
     REAL(A)         - Real parts of the elements.
     IMAG(A)         - Imaginary parts of the elements.
     CONJG(A)        - Complex conjugate.
     SUM(A)          - Sum of the elements.
     PROD(A)         - Product of the elements.
     DIAG(A)         - Extract or create diagonal matrices.
     TRIL(A)         - Lower triangular part of A.
     TRIU(A)         - Upper triangular part of A.
     NORM(A,p)       - Norm with p = 1, 2 or 'Infinity'.
     EYE(m,n)        - Portion of identity matrix.
     RAND(m,n)       - Matrix with random elements.
     ONES(m,n)       - Matrix of all ones.
     MAGIC(n)        - Interesting test matrices.
     HILBERT(n)      - Inverse Hilbert matrices.
     ROOTS(C)        - Roots of polynomial with coefficients C.
     DISPLAY(A,p)    - Print base p representation of A.
     KRON(A,B)       - Kronecker tensor product of A and B.
     PLOT(X,Y)       - Plot Y as a function of X .
     RAT(A)          - Find "simple" rational approximation to A.
     USER(A)         - Function defined by external program.
       Some of these functions have different interpretations  when
  the  argument  is  a  matrix  or  a  vector and some of them have
  additional optional arguments.  Details are  given  in  the  HELP
  document in the appendix.
       Several of these functions can  be  used  in  a  generalized
  assignment statement with two or three variables on the left hand
  side.  For example
     <X,D> = EIG(A)
  stores the eigenvectors of A in  the  matrix  X  and  a  diagonal
  matrix containing the eigenvalues in the matrix D.  The statement
     EIG(A)
  simply computes the eigenvalues and stores them in ANS.
       Future versions of MATLAB will probably  include  additional
  functions, since they can easily be added to the system.

3. Rows, columns and submatrices

       Individual elements of a matrix can be  accessed  by  giving
  their subscripts in parentheses, eg. A(1,2), x(i), TAB(ind(k)+1).
  An expression used as a  subscript  is  rounded  to  the  nearest
  integer.
       Individual rows and columns can be accessed  using  a  colon
  ':' (or a '|') for the free subscript. For example, A(1,:) is the
  first row of A and A(:,j) is the j-th column.  Thus
     A(i,:) = A(i,:) + c*A(k,:)
  adds c times the k-th row of A to the i-th row.
       The colon is used in several other ways in MATLAB,  but  all
  of the uses are based on the following definition.
     j:k    is the same as  <j, j+1, ..., k>
     j:k    is empty if  j > k .
     j:i:k  is the same as  <j, j+i, j+2i, ..., k>
     j:i:k  is empty if  i > 0 and j > k or if i < 0 and j < k .
  The colon is usually used with integers, but it  is  possible  to
  use arbitrary real scalars as well.  Thus
     1:4  is the same as  <1, 2, 3, 4>
     0: 0.1: 0.5 is the same as <0.0, 0.1, 0.2, 0.3, 0.4, 0.5>
       In general, a subscript can be a vector.  If  X  and  V  are
  vectors, then X(V) is <X(V(1)), X(V(2)), ..., X(V(n))> . This can
  also be used with matrices.  If V has m components and  W  has  n
  components,  then  A(V,W)  is  the  m by n matrix formed from the
  elements of A whose subscripts are  the  elements  of  V  and  W.
  Combinations  of the colon notation and the indirect subscripting
  allow manipulation of various submatrices. For example,
     A(<1,5>,:) = A(<5,1>,:)  interchanges rows 1 and 5 of A.
     A(2:k,1:n)  is the submatrix formed from rows 2 through k
        and columns 1 through n of A .
     A(:,<3 1 2>)  is a permutation of the first three columns.
       The notation A(:) has a special meaning.  On the right  hand
  side  of  an assignment statement, it denotes all the elements of
  A, regarded as a single column.  When an expression  is  assigned
  to  A(:),  the  current  dimensions  of  A,  rather  than  of the
  expression, are used.

4. FOR, WHILE and IF

       The FOR clause allows statements to be repeated  a  specific
  number of times.  The general form is
     FOR variable = expr,  statement, ..., statement, END
  The END and the comma before it may be omitted.  In general,  the
  expression  may be a matrix, in which case the columns are stored
  one at a time in the variable and the following statements, up to
  the  END or the end of the line, are executed.  The expression is
  often of the form j:k, and its "columns" are simply  the  scalars
  from j to k.  Some examples (assume n has already been assigned a
  value):
     for i = 1:n, for j = 1:n, A(i,j) = 1/(i+j-1);
  generates the Hilbert matrix.
     for j = 2:n-1, for i = j:n-1, ...
        A(i,j) = 0; end; A(j,j) = j; end; A
  changes all but the "outer edge" of the lower triangle  and  then
  prints the final matrix.
     for h = 1.0: -0.1: -1.0, (<h, cos(pi*h)>)
  prints a table of cosines.
     <X,D> = EIG(A); for v = X, v, A*v
  displays eigenvectors, one at a time.
       The  WHILE  clause  allows  statements  to  be  repeated  an
  indefinite number of times.  The general form is
     WHILE expr relop expr,   statement,..., statement, END
  where relop is =, <,  >,  <=,  >=,  or  <>  (not  equal)  .   The
  statements  are  repeatedly  executed  as  long  as the indicated
  comparison between the real parts of the first components of  the
  two  expressions  is true.  Here are two examples.  (Exercise for
  the reader: What do these segments do?)
     eps = 1;
     while 1 + eps > 1, eps = eps/2;
     eps = 2*eps
     E = 0*A;  F = E + EYE; n = 1;
     while NORM(E+F-E,1) > 0, E = E + F; F = A*F/n; n = n + 1;
     E
       The IF clause allows conditional  execution  of  statements.
  The general form is
     IF expr relop expr,   statement, ..., statement,
        ELSE statement, ..., statement
  The first group of statements are executed  if  the  relation  is
  true  and the second group are executed if the relation is false.
  The ELSE and the statements following it  may  be  omitted.   For
  example,
     if abs(i-j) = 2, A(i,j) = 0;

5. Commands, text, files and macros.

       MATLAB has several commands which control the output  format
  and the overall execution of the system.
       The HELP command allows on-line access to short portions  of
  text   describing   various  operations,  functions  and  special
  characters.   The  entire  HELP  document  is  reproduced  in  an
  appendix.
       Results are usually printed in a scaled fixed  point  format
  that shows 4 or 5 significant figures.  The commands SHORT, LONG,
  SHORT E, LONG E and LONG Z alter the output format,  but  do  not
  alter the precision of the computations or the internal storage.
       The WHO, WHAT and WHY commands provide information about the
  functions and variables that are currently defined.
       The CLEAR command erases all variables,  except  EPS,  FLOP,
  RAND  and  EYE.  The  statement  A = <> indicates that a "0 by 0"
  matrix is to be stored in A.  This causes A to be erased so  that
  its storage can be used for other variables.
       The RETURN and EXIT commands cause return to the  underlying
  operating system through the Fortran RETURN statement.
       MATLAB has a limited facility for handling text.  Any string
  of characters delineated by quotes (with two quotes used to allow
  one quote within the string) is saved  as  a  vector  of  integer
  values  with '1' = 1, 'A' = 10, ' ' = 36, etc. (The complete list
  is in the appendix under CHAR.) For example
     '2*A + 3'  is the same as  <2 43 10 36 41 36 3>
  It is possible,  though  seldom  very  meaningful,  to  use  such
  strings  in matrix operations.  More frequently, the text is used
  as a special argument to various functions.
     NORM(A,'inf')    computes the infinity norm of A .
     DISPLAY(T)       prints the text stored in T .
     EXEC('file')     obtains MATLAB input from an external file.
     SAVE('file')     stores all the current variables in a file.
     LOAD('file')     retrieves all the variables from a file.
     PRINT('file',X)  prints X on a file.
     DIARY('file')    makes a copy of the complete MATLAB session.
       The text can also be used in a limited  string  substitution
  macro  facility.   If a variable, say T, contains the source text
  for a MATLAB statement or expression, then the construction
     > T <
  causes T to be executed or evaluated.  For example
     T = '2*A + 3';
     S = 'B = >T< + 5'
     A = 4;
     > S <
  produces
     B     =
        16.
  Some other examples are given under MACRO in the appendix.   This
  facility  is  useful for fairly short statements and expressions.
  More complicated MATLAB "programs" should use the EXEC facility.
       The operations which access external files cannot be handled
  in  a  completely  machine-independent manner by portable Fortran
  code.  It  is  necessary  for  each  particular  installation  to
  provide  a  subroutine  which associates external text files with
  Fortran logical unit numbers.

6. Census example

       Our  first  extended   example   involves   predicting   the
  population  of  the  United States in 1980 using extrapolation of
  various fits to the census data from 1900  through  1970.   There
  are eight observations, so we begin with the MATLAB statement
     n = 8
  The values of the dependent variable, the population in millions,
  can be entered with
     y = < 75.995   91.972  105.711  123.203   ...
          131.669  150.697  179.323  203.212>'
  In order to produce a reasonably scaled matrix,  the  independent
  variable,  time,  is transformed from the interval [1900,1970] to
  [-1.00,0.75].  This can be accomplished directly with
     t = -1.0:0.25:0.75
  or in a fancier, but perhaps clearer, way with
     t = 1900:10:1970;   t = (t - 1940*ones(t))/40
  Either of these is equivalent to
     t = <-1 -.75 -.50 -.25 0 .25 .50 .75>
       The interpolating polynomial of  degree   n-1   involves  an
  Vandermonde  matrix  of  order   n   with  elements that might be
  generated by
     for i = 1:n, for j = 1:n, a(i,j) = t(i)**(j-1);
  However, this results in an error caused by 0**0  when  i = 5 and
  j = 1 .  The preferable approach is
     A = ones(n,n);
     for i = 1:n, for j = 2:n, a(i,j) = t(i)*a(i,j-1);
  Now the statement
     cond(A)
  produces the output
     ANS   =
        1.1819E+03
  which indicates that transformation  of  the  time  variable  has
  resulted in a reasonably well conditioned matrix.
       The statement
     c = A\y
  results in
     C     =
       131.6690
        41.0406
       103.5396
       262.4535
      -326.0658
      -662.0814
       341.9022
       533.6373
  These are the coefficients in the interpolating polynomial
                            n-1
        c  + c t + ... + c t
         1    2           n
  Our transformation of the time variable has resulted in   t  =  1
  corresponding  to  the year 1980.  Consequently, the extrapolated
  population is simply the sum of the coefficients.   This  can  be
  computed by
     p = sum(c)
  The result is
     P     =
       426.0950
  which indicates a 1980 population of over 426 million.   Clearly,
  using  the seventh degree interpolating polynomial to extrapolate
  even a fairly short distance beyond the end of the data  interval
  is not a good idea.
       The coefficients in least squares  fits  by  polynomials  of
  lower  degree can be computed using fewer than  n  columns of the
  matrix.
     for k = 1:n, c = A(:,1:k)\y,  p = sum(c)
  would produce the coefficients of these  fits,  as  well  as  the
  resulting  extrapolated  population.   If we do not want to print
  all the coefficients, we can simply generate  a  small  table  of
  populations  predicted  by  polynomials  of  degrees zero through
  seven.  We also compute the maximum deviation between the  fitted
  and observed values.
     for k = 1:n, X = A(:,1:k);  c = X\y;  ...
        d(k) = k-1;  p(k) = sum(c);  e(k) = norm(X*c-y,'inf');
     <d, p, e>
  The resulting output is
        0   132.7227  70.4892
        1   211.5101   9.8079
        2   227.7744   5.0354
        3   241.9574   3.8941
        4   234.2814   4.0643
        5   189.7310   2.5066
        6   118.3025   1.6741
        7   426.0950   0.0000
  The zeroth degree fit, 132.7 million, is the result of fitting  a
  constant  to  the  data  and  is simply the average.  The results
  obtained with polynomials of degree one through four  all  appear
  reasonable.   The  maximum  deviation  of  the degree four fit is
  slightly greater than the degree three, even though  the  sum  of
  the  squares  of the deviations is less.  The coefficients of the
  highest powers in the fits of degree five and six turn out to  be
  negative  and  the predicted populations of less than 200 million
  are probably unrealistic.  The hopefully absurd prediction of the
  interpolating polynomial concludes the table.
       We  wish  to  emphasize  that  roundoff   errors   are   not
  significant  here.  Nearly identical results would be obtained on
  other computers, or with other algorithms.   The  results  simply
  indicate   the  difficulties  associated  with  extrapolation  of
  polynomial fits of even modest degree.
       A stabilized fit by  a  seventh  degree  polynomial  can  be
  obtained  using  the  pseudoinverse,  but  it  requires  a fairly
  delicate choice of a tolerance. The statement
     s = svd(A)
  produces the singular values
     S     =
        3.4594
        2.2121
        1.0915
        0.4879
        0.1759
        0.0617
        0.0134
        0.0029
  We see that the last three singular values are less  than  0.1  ,
  consequently,   A   can be approximately by a matrix of rank five
  with an error less than 0.1 .  The Moore-Penrose pseudoinverse of
  this  rank  five  matrix  is  obtained  from  the  singular value
  decomposition with the following statements
     c = pinv(A,0.1)*y, p = sum(c), e = norm(a*c-y,'inf')
  The output is
     C     =
      134.7972
       67.5055
       23.5523
        9.2834
        3.0174
        2.6503
       -2.8808
        3.2467
     P     =
      241.1720
     E     =
        3.9469
  The resulting seventh degree polynomial  has  coefficients  which
  are much smaller than those of the interpolating polynomial given
  earlier.  The predicted population and the maximum deviation  are
  reasonable.   Any  choice  of the tolerance between the fifth and
  sixth singular values would produce the same results, but choices
  outside this range result in pseudoinverses of different rank and
  do not work as well.
       The one term exponential approximation
       y(t) = k exp(pt)
  can  be  transformed  into  a  linear  approximation  by   taking
  logarithms.
       log(y(t)) = log k + pt
                 = c  + c t
                    1    2
  The following segment makes use of the fact that a function of  a
  vector is the function applied to the individual components.
     X = A(:,1:2);
     c = X\log(y)
     p = exp(sum(c))
     e = norm(exp(X*c)-y,'inf')
  The resulting output is
     C     =
        4.9083
        0.5407
     P     =
      232.5134
     E     =
        4.9141
  The   predicted   population   and   maximum   deviation   appear
  satisfactory  and  indicate  that  the  exponential  model  is  a
  reasonable one to consider.
       As a curiousity, we return to  the  degree  six  polynomial.
  Since  the coefficient of the high order term is negative and the
  value of the polynomial at t = 1 is positive, it must have a root
  at some value of  t  greater than one.  The statements
     X = A(:,1:7);
     c = X\y;
     c = c(7:-1:1);  //reverse the order of the coefficients
     z = roots(c)
  produce
     Z     =
        1.1023-  0.0000*i
        0.3021+  0.7293*i
       -0.8790+  0.6536*i
       -1.2939-  0.0000*i
       -0.8790-  0.6536*i
        0.3021-  0.7293*i
  There is only one real, positive root.  The corresponding time on
  the original scale is
     1940 + 40*real(z(1))
       =  1984.091
  We conclude that the United States population should become  zero
  early in February of 1984.

7. Partial differential equation example

       Our second extended example is a boundary value problem  for
  Laplace's equation.  The underlying physical problem involves the
  conductivity of a  medium  with  cylindrical  inclusions  and  is
  considered by Keller and Sachs [7].
       Find a function  u(x,y)  satisfying Laplace's equation
                 u   + u   = 0
                  xx    yy
  The domain is a unit square with a quarter circle of  radius  rho
  removed from one corner.  There are Neumann conditions on the top
  and bottom edges and Dirichlet conditions on the remainder of the
  boundary.
                           u  = 0
                            n
                       -------------
                      |             .
                      |             .
                      |              .
                      |               .  u = 1
                      |                 .
                      |                    .
                      |                       .
               u = 0  |                        |
                      |                        |
                      |                        |
                      |                        |  u = 1
                      |                        |
                      |                        |
                      |                        |
                       ------------------------
                                u  = 0
                                 n
  The effective conductivity of an medium  is  then  given  by  the
  integral along the left edge,
                              1
                   sigma = integral  u (0,y) dy
                             0        n
  It is of interest to study the relation between  the  radius  rho
  and  the  conductivity  sigma.   In particular, as rho approaches
  one, sigma becomes infinite.
       Keller and Sachs use a finite difference approximation.  The
  following  technique  makes  use of the fact that the equation is
  actually Laplace's equation and leads to a  much  smaller  matrix
  problem to solve.
       Consider an approximate solution of the form
                   n      2j-1
             u =  sum  c r    cos(2j-1)t
                  j=1   j
  where  r,t  are polar coordinates (t is theta).  The coefficients
  are to be determined.  For any set of coefficients, this function
  already satisfies the differential  equation  because  the  basis
  functions  are  harmonic;  it  satisfies  the  normal  derivative
  boundary condition on the bottom edge of the  domain  because  we
  used   cos  t   in  preference  to   sin t ; and it satisfies the
  boundary condition on the left edge of the domain because we  use
  only odd multiples of  t .
       The computational task is to find coefficients  so that  the
  boundary  conditions on the remaining edges are satisfied as well
  as possible.  To accomplish this, pick  m  points  (r,t)  on  the
  remaining edges.  It is desirable to have  m > n  and in practice
  we usually choose m  to be two or three times as large  as   n  .
  Typical  values  of  n  are 10 or 20 and of  m  are 20 to 60.  An
  m  by  n  matrix  A  is generated.  The  i,j  element is the j-th
  basis  function,  or its normal derivative, evaluated at the i-th
  boundary point.  A right hand side with  m   components  is  also
  generated.   In this example, the elements of the right hand side
  are either zero or one.   The  coefficients  are  then  found  by
  solving the overdetermined set of equations
              Ac = b
  in a least squares sense.
       Once the coefficients have been determined, the  approximate
  solution  is  defined  everywhere  on  the  domain.   It  is then
  possible to compute the effective conductivity sigma .  In  fact,
  a very simple formula results,
                       n       j-1
             sigma =  sum  (-1)   c
                      j=1          j
       To use MATLAB for this problem, the following  "program"  is
  first  stored  in  the  local computer file system, say under the
  name "PDE".
  //Conductivity example.
  //Parameters ---
     rho       //radius of cylindrical inclusion
     n         //number of terms in solution
     m         //number of boundary points
  //initialize operation counter
     flop = <0 0>;
  //initialize variables
     m1 = round(m/3);   //number of points on each straight edge
     m2 = m - m1;       //number of points with Dirichlet conditions
     pi = 4*atan(1);
  //generate points in Cartesian coordinates
     //right hand edge
     for i = 1:m1, x(i) = 1; y(i) = (1-rho)*(i-1)/(m1-1);
     //top edge
     for i = m2+1:m, x(i) = (1-rho)*(m-i)/(m-m2-1); y(i) = 1;
     //circular edge
     for i = m1+1:m2, t = pi/2*(i-m1)/(m2-m1+1); ...
        x(i) = 1-rho*sin(t);  y(i) = 1-rho*cos(t);
  //convert to polar coordinates
     for i = 1:m-1, th(i) = atan(y(i)/x(i));  ...
        r(i) = sqrt(x(i)**2+y(i)**2);
     th(m) = pi/2;  r(m) = 1;
  //generate matrix
     //Dirichlet conditions
     for i = 1:m2, for j = 1:n, k = 2*j-1; ...
        a(i,j) = r(i)**k*cos(k*th(i));
     //Neumann conditions
     for i = m2+1:m, for j = 1:n, k = 2*j-1; ...
        a(i,j) = k*r(i)**(k-1)*sin((k-1)*th(i));
  //generate right hand side
     for i = 1:m2, b(i) = 1;
     for i = m2+1:m, b(i) = 0;
  //solve for coefficients
     c = A\b
  //compute effective conductivity
     c(2:2:n) = -c(2:2:n);
     sigma = sum(c)
  //output total operation count
     ops = flop(2)
       The program can be used within MATLAB by setting  the  three
  parameters and then accessing the file.  For example,
     rho = .9;
     n = 15;
     m = 30;
     exec('PDE')
  The resulting output is
     RHO   =
        .9000
     N     =
      15.
     M     =
      30.
     C     =
        2.2275
       -2.2724
        1.1448
        0.1455
       -0.1678
       -0.0005
       -0.3785
        0.2299
        0.3228
       -0.2242
       -0.1311
        0.0924
        0.0310
       -0.0154
       -0.0038
     SIGM  =
        5.0895
     OPS   =
        16204.
       A total of 16204 floating point operations were necessary to
  set  up  the  matrix,  solve for the coefficients and compute the
  conductivity.  The operation count  is  roughly  proportional  to
  m*n**2.   The  results obtained for sigma as a function of rho by
  this approach are essentially the same as those obtained  by  the
  finite   difference  technique  of  Keller  and  Sachs,  but  the
  computational effort involved is much less.

8. Eigenvalue sensitivity example

       In this example, we construct a matrix whose eigenvalues are
  moderately  sensitive  to  perturbations  and  then  analyze that
  sensitivity. We begin with the statement
     B = <3 0 7; 0 2 0; 0 0 1>
  which produces
     B     =
         3.    0.    7.
         0.    2.    0.
         0.    0.    1.
       Obviously, the eigenvalues of B are 1, 2 and 3 .   Moreover,
  since   B  is  not  symmetric,  these  eigenvalues  are  slightly
  sensitive to perturbation.  (The value b(1,3) = 7 was  chosen  so
  that the elements of the matrix A below are less than 1000.)
       We now generate a similarity transformation to disguise  the
  eigenvalues and make them more sensitive.
     L = <1 0 0; 2 1 0; -3 4 1>, M = L\L'
     L     =
         1.    0.    0.
         2.    1.    0.
        -3.    4.    1.
     M     =
         1.0000    2.0000   -3.0000
        -2.0000   -3.0000   10.0000
        11.0000   18.0000  -48.0000
  The matrix M has determinant equal to 1 and is  moderately  badly
  conditioned.  The similarity transformation is
     A = M*B/M
     A     =
       -64.0000   82.0000   21.0000
       144.0000 -178.0000  -46.0000
      -771.0000  962.0000  248.0000
  Because  det(M) = 1 , the elements of  A  would be exact integers
  if there were no roundoff.  So,
     A = round(A)
     A     =
       -64.   82.   21.
       144. -178.  -46.
      -771.  962.  248.
       This, then, is our test matrix.  We can now  forget  how  it
  was generated and analyze its eigenvalues.
     <X,D> = eig(A)
     D     =
         3.0000    0.0000    0.0000
         0.0000    1.0000    0.0000
         0.0000    0.0000    2.0000
     X     =
         -.0891    3.4903   41.8091
          .1782   -9.1284  -62.7136
         -.9800   46.4473  376.2818
  Since A is similar to B, its eigenvalues are also  1,  2  and  3.
  They  happen  to  be  computed  in  another  order by the EISPACK
  subroutines.  The fact that the  columns  of  X,  which  are  the
  eigenvectors,  are  so  far  from  being orthonormal is our first
  indication that  the  eigenvalues  are  sensitive.  To  see  this
  sensitivity, we display more figures of the computed eigenvalues.
     long, diag(D)
     ANS   =
        2.999999999973599
        1.000000000015625
        2.000000000011505
  We see that, on this computer, the last five significant  figures
  are  contaminated  by  roundoff  error.  A  somewhat  superficial
  explanation of this is provided by
     short,  cond(X)
     ANS   =
        3.2216e+05
  The condition number of X gives an upper bound for  the  relative
  error  in  the  computed  eigenvalues.   However,  this condition
  number is affected by scaling.
     X = X/diag(X(3,:)),  cond(X)
     X     =
          .0909     .0751     .1111
         -.1818    -.1965    -.1667
         1.0000    1.0000    1.0000
     ANS   =
        1.7692e+03
       Rescaling the eigenvectors so that their last components are
  all  equal  to  one  has  two consequences. The condition of X is
  decreased by over two orders of magnitude.  (This  is  about  the
  minimum condition that can be obtained by such diagonal scaling.)
  Moreover, it is now apparent  that  the  three  eigenvectors  are
  nearly parallel.
       More  detailed  information  on  the  sensitivity   of   the
  individual eigenvalues involves the left eigenvectors.
     Y = inv(X'),  Y'*A*X
     Y     =
      -511.5000  259.5000  252.0000
       616.0000 -346.0000 -270.0000
       159.5000  -86.5000  -72.0000
     ANS   =
         3.0000     .0000     .0000
          .0000    1.0000     .0000
          .0000     .0000    2.0000
  We are now in a position to  compute  the  sensitivities  of  the
  individual eigenvalues.
     for j = 1:3, c(j) = norm(Y(:,j))*norm(X(:,j)); end,  C
     C     =
       833.1092
       450.7228
       383.7564
  These three numbers are the reciprocals of  the  cosines  of  the
  angles  between the left and right eigenvectors.  It can be shown
  that  perturbation  of  the  elements  of  A  can  result  in   a
  perturbation of the j-th eigenvalue which is c(j) times as large.
  In  this  example,  the  first   eigenvalue   has   the   largest
  sensitivity.
       We now proceed to show that A is close to a  matrix  with  a
  double eigenvalue.  The direction of the required perturbation is
  given by
     E = -1.e-6*Y(:,1)*X(:,1)'
     E     =
        1.0e-03 *
          .0465    -.0930     .5115
         -.0560     .1120    -.6160
         -.0145     .0290    -.1595
  With some trial and error which we do not show,  we  bracket  the
  point  where  two  eigenvalues of a perturbed A coalesce and then
  become complex.
     eig(A + .4*E),  eig(A + .5*E)
     ANS   =
         1.1500
         2.5996
         2.2504
     ANS   =
        2.4067 +  .1753*i
        2.4067 -  .1753*i
        1.1866 + 0.0000*i
  Now, a bisecting search, driven by the imaginary part of  one  of
  the eigenvalues, finds the point where two eigenvalues are nearly
  equal.
     r = .4;  s = .5;
     while s-r > 1.e-14, t = (r+s)/2; d = eig(A+t*E); ...
       if imag(d(1))=0, r = t; else, s = t;
     long,  T
     T     =
          .450380734134507
       Finally, we display the perturbed matrix, which is obviously
  close  to the original, and its pair of nearly equal eigenvalues.
  (We have dropped a few digits from the long output.)
     A+t*E,  eig(A+t*E)
     A
      -63.999979057   81.999958114   21.000230369
      143.999974778 -177.999949557  -46.000277434
     -771.000006530  962.000013061  247.999928164
     ANS   =
        2.415741150
        2.415740621
        1.168517777
       The  first  two  eigenvectors  of  A  +   t*E   are   almost
  indistinguishable  indicating that the perturbed matrix is almost
  defective.
     <X,D> = eig(A+t*E);  X = X/diag(X(3,:))
     X     =
         .096019578     .096019586    .071608466
        -.178329614    -.178329608   -.199190520
        1.000000000    1.000000000   1.000000000
     short,  cond(X)
     ANS   =
        3.3997e+09

9. Syntax diagrams

       A formal description of the language acceptable  to  MATLAB,
  as well as a flow chart of the MATLAB program, is provided by the
  syntax diagrams or syntax graphs of Wirth [6].  There are  eleven
  non-terminal symbols in the language:
     line, statement, clause, expression, term,
     factor, number, integer, name, command, text .
  The diagrams define each of the non-terminal  symbols  using  the
  others and the terminal symbols:
     letter -- A through Z,
     digit  -- 0 through 9,
     char   -- ( ) ; : + - * / \ = . , < >
     quote  -- '
  line
         |-----> statement >----|
         |                      |
         |-----> clause >-------|
         |                      |
  -------|-----> expr >---------|------>
       | |                      | |
       | |-----> command >------| |
       | |                      | |
       | |-> > >-> expr >-> < >-| |
       | |                      | |
       | |----------------------| |
       |                          |
       |        |-< ; <-|         |
       |--------|       |---------|
                |-< , <-|
  statement
       |-> name >--------------------------------|
       |          |                              |
       |          |         |--> : >---|         |
       |          |         |          |         |
       |          |-> ( >---|-> expr >-|---> ) >-|
       |                  |              |       |
  -----|                  |-----< , <----|       |--> = >--> expr >--->
       |                                         |
       |       |--< , <---|                      |
       |       |          |                      |
       |-> < >---> name >---> > >----------------|
  clause
       |---> FOR   >---> name >---> = >---> expr >--------------|
       |                                                        |
       | |-> WHILE >-|                                          |
       |-|           |-> expr >----------------------           |
       | |-> IF    >-|          |   |   |   |   |   |           |
  -----|                        <   <=  =   <>  >=  >           |---->
       |                        |   |   |   |   |   |           |
       |                        ----------------------> expr >--|
       |                                                        |
       |---> ELSE  >--------------------------------------------|
       |                                                        |
       |---> END   >--------------------------------------------|
  expr
         |-> + >-|
         |       |
  -------|-------|-------> term >---------->
         |       |    |             |
         |-> - >-|    |  |-< + <-|  |
                      |  |       |  |
                      |--|-< - <-|--|
                         |       |
                         |-< : <-|
  term
  ---------------------> factor >---------------------->
          |                                   |
          |             |-< * <-|             |
          |  |-------|  |       |  |-------|  |
          |--|       |--|-< / <-|--|       |--|
             |-< . <-|  |       |  |-< . <-|
                        |-< \ <-|
  factor
       |----------------> number >---------------|
       |                                         |
       |-> name >--------------------------------|
       |          |                              |
       |          |         |--> : >---|         |
       |          |         |          |         |
       |          |-> ( >---|-> expr >-|---> ) >-|
       |                  |              |       |
       |                  |-----< , <----|       |
       |                                         |
  -----|------------> ( >-----> expr >-----> ) >-|-|-------|----->
       |                                         | |       | |
       |                  |--------------|       | |-> ' >-| |
       |                  |              |       |           |
       |------------> < >-|---> expr >---|-> > >-|           |
       |                    |          |         |           |
       |                    |--<   <---|         |           |
       |                    |          |         |           |
       |                    |--< ; <---|         |           |
       |                    |          |         |           |
       |                    |--< , <---|         |           |
       |                                         |           |
       |------------> > >-----> expr >-----> < >-|           |
       |                                         |           |
       |-----> factor >---> ** >---> factor >----|           |
       |                                                     |
       |------------> ' >-----> text >-----> ' >-------------|
  number
      |----------|                          |-> + >-|
      |          |                          |       |
  -----> int >-----> . >---> int >-----> E >---------> int >---->
               |                   | |      |       |        |
               |                   | |      |-> - >-|        |
               |                   | |                       |
               |---------------------------------------------|
  int
  ------------> digit >----------->
            |           |
            |-----------|
  name
                    |--< letter <--|
                    |              |
  ------> letter >--|--------------|----->
                    |              |
                    |--< digit  <--|
  command
                          |--> name >--|
                          |            |
  --------> name >--------|------------|---->
                          |            |
                          |--> char >--|
                          |            |
                          |---> ' >----|
  text
                  |-> letter >--|
                  |             |
                  |-> digit >---|
  ----------------|             |-------------->
              |   |-> char >----|   |
              |   |             |   |
              |   |-> ' >-> ' >-|   |
              |                     |
              |---------------------|

10. The parser-interpreter

       The structure of the parser-interpreter is similar  to  that
  of  Wirth's  compiler  [6] for his simple language, PL/0 , except
  that MATLAB  is  programmed  in  Fortran,  which  does  not  have
  explicit recursion.  The interrelation of the primary subroutines
  is shown in the following diagram.
        MAIN
          |
        MATLAB    |--CLAUSE
          |       |    |
        PARSE-----|--EXPR----TERM----FACTOR
                  |    |       |       |
                  |    |-------|-------|
                  |    |       |       |
                  |  STACK1  STACK2  STACKG
                  |
                  |--STACKP--PRINT
                  |
                  |--COMAND
                  |
                  |
                  |          |--CGECO
                  |          |
                  |          |--CGEFA
                  |          |
                  |--MATFN1--|--CGESL
                  |          |
                  |          |--CGEDI
                  |          |
                  |          |--CPOFA
                  |
                  |
                  |          |--IMTQL2
                  |          |
                  |          |--HTRIDI
                  |          |
                  |--MATFN2--|--HTRIBK
                  |          |
                  |          |--CORTH
                  |          |
                  |          |--COMQR3
                  |
                  |
                  |--MATFN3-----CSVDC
                  |
                  |
                  |          |--CQRDC
                  |--MATFN4--|
                  |          |--CQRSL
                  |
                  |
                  |          |--FILES
                  |--MATFN5--|
                             |--SAVLOD
       Subroutine  PARSE  controls  the  interpretation   of   each
  statement.    It  calls  subroutines  that  process  the  various
  syntactic  quantities  such  as  command,  expression,  term  and
  factor.   A  fairly  simple  program stack mechanism allows these
  subroutines to recursively "call"  each  other  along  the  lines
  allowed  by  the  syntax  diagrams.   The  four STACK subroutines
  manage the variable memory  and  perform  elementary  operations,
  such as matrix addition and transposition.
       The  four  subroutines  MATFN1  though  MATFN4  are   called
  whenever  "serious"  matrix  computations are required.  They are
  interface routines which call the  various  LINPACK  and  EISPACK
  subroutines.  MATFN5 primarily handles the file access tasks.
       Two large real arrays, STKR and STKI, are used to store  all
  the  matrices.   Four integer arrays are used to store the names,
  the row and column dimensions, and the  pointers  into  the  real
  stacks.  The following diagram illustrates this storage scheme.
  TOP         IDSTK     MSTK NSTK LSTK               STKR       STKI
   --      -- -- -- --   --   --   --              --------   --------
  |  |--->|  |  |  |  | |  | |  | |  |----------->|        | |        |
   --      -- -- -- --   --   --   --              --------   --------
          |  |  |  |  | |  | |  | |  |            |        | |        |
           -- -- -- --   --   --   --              --------   --------
                .         .    .    .                  .          .
                .         .    .    .                  .          .
                .         .    .    .                  .          .
           -- -- -- --   --   --   --              --------   --------
  BOT     |  |  |  |  | |  | |  | |  |            |        | |        |
   --      -- -- -- --   --   --   --              --------   --------
  |  |--->| X|  |  |  | | 2| | 1| |  |----------->|  3.14  | |  0.00  |
   --      -- -- -- --   --   --   --              --------   --------
          | A|  |  |  | | 2| | 2| |  |---------   |  0.00  | |  1.00  |
           -- -- -- --   --   --   --          \   --------   --------
          | E| P| S|  | | 1| | 1| |  |-------   ->| 11.00  | |  0.00  |
           -- -- -- --   --   --   --        \     --------   --------
          | F| L| O| P| | 1| | 2| |  |------  \   | 21.00  | |  0.00  |
           -- -- -- --   --   --   --       \  \   --------   --------
          | E| Y| E|  | |-1| |-1| |  |---    \ |  | 12.00  | |  0.00  |
           -- -- -- --   --   --   --    \   | |   --------   --------
          | R| A| N| D| | 1| | 1| |  |-   \  | |  | 22.00  | |  0.00  |
           -- -- -- --   --   --   --  \  |  \ \   --------   --------
                                       |  \   \ ->| 1.E-15 | |  0.00  |
                                       \   \   \   --------   --------
                                        \   \   ->|  0.00  | |  0.00  |
                                         \   \     --------   --------
                                          \   \   |  0.00  | |  0.00  |
                                           \   \   --------   --------
                                            \   ->|  1.00  | |  0.00  |
                                             \     --------   --------
                                              --->| URAND  | |  0.00  |
                                                   --------   --------
       The top portion of the stack is used for temporary variables
  and the bottom portion for saved variables.  The figure shows the
  situation after the line
     A = <11,12; 21,22>,  x = <3.14, sqrt(-1)>'
  has been processed.  The four permanent names,  EPS,  FLOP,  RAND
  and  EYE,  occupy the last four positions of the variable stacks.
  RAND has dimensions 1 by 1, but whenever its value is  requested,
  a random number generator is used instead.  EYE has dimensions -1
  by -1 to indicate that the actual dimensions must  be  determined
  later by context.  The two saved variables have dimensions 2 by 2
  and 2 by 1 and so take up a total of 6 locations.
       Subsequent statements involving  A  and  x  will  result  in
  temporary  copies  being  made in the top of the stack for use in
  the actual calculations.  Whenever the top of the  stack  reaches
  the  bottom,  a  message  indicating  memory has been exceeded is
  printed, but the current variables are not affected.
       This modular structure makes it possible to implement MATLAB
  on a system with a limited amount of memory.  The object code for
  the MATFN's and the LINPACK-EISPACK subroutines is rarely needed.
  Although  it  is  not  standard,  many  Fortran operating systems
  provide some overlay mechanism so that this code is brought  into
  the  main memory only when required.  The variables, which occupy
  a relatively small portion of the memory, remain in place,  while
  the subroutines which process them are loaded a few at a time.

11. The numerical algorithms

       The algorithms underlying the  basic  MATLAB  functions  are
  described  in the LINPACK and EISPACK guides [1-3]. The following
  list gives the subroutines used by these functions.
     INV(A)          - CGECO,CGEDI
     DET(A)          - CGECO,CGEDI
     LU(A)           - CGEFA
     RCOND(A)        - CGECO
     CHOL(A)         - CPOFA
     SVD(A)          - CSVDC
     COND(A)         - CSVDC
     NORM(A,2)       - CSVDC
     PINV(A,eps)     - CSVDC
     RANK(A,eps)     - CSVDC
     QR(A)           - CQRDC,CQRSL
     ORTH(A)         - CQRDC,CSQSL
     A\B and B/A     - CGECO,CGESL if A is square.
                     - CQRDC,CQRSL if A is not square.
     EIG(A)          - HTRIDI,IMTQL2,HTRIBK if A is Hermitian.
                     - CORTH,COMQR2         if A is not Hermitian.
     SCHUR(A)        - same as EIG.
     HESS(A)         - same as EIG.
       Minor modifications were made to all these subroutines.  The
  LINPACK  routines  were  changed  to  replace the Fortran complex
  arithmetic with explicit references to real and imaginary  parts.
  Since  most of the floating point arithmetic is concentrated in a
  few low-level subroutines which perform  vector  operations  (the
  Basic  Linear  Algebra  Subprograms),  this  was not an extensive
  change.  It also facilitated implementation of the FLOP and  CHOP
  features  which count and optionally truncate each floating point
  operation.
       The EISPACK subroutine COMQR2 was modified to  allow  access
  to  the  Schur  triangular  form, ordinarily just an intermediate
  result.   IMTQL2  was  modified  to  make  computation   of   the
  eigenvectors   optional.    Both  subroutines  were  modified  to
  eliminate the machine-dependent accuracy parameter  and  all  the
  EISPACK subroutines were changed to include FLOP and CHOP.
       The algorithms employed for the  POLY  and  ROOTS  functions
  illustrate  an  interesting  aspect  of  the  modern  approach to
  eigenvalue computation.   POLY(A)  generates  the  characteristic
  polynomial  of  A  and  ROOTS(POLY(A))  finds  the  roots of that
  polynomial, which are, of course, the eigenvalues of A . But both
  POLY  and  ROOTS  use  EISPACK eigenvalues subroutines, which are
  based on similarity transformations.  So the  classical  approach
  which  characterizes  eigenvalues  as roots of the characteristic
  polynomial is actually reversed.
       If A is an n by n matrix, POLY(A) produces the  coefficients
  C(1) through C(n+1), with C(1) = 1, in
        DET(z*EYE-A) = C(1)*z**n + ... + C(n)*z + C(n+1) .
  The algorithm can be expressed compactly using MATLAB:
        Z = EIG(A);
        C = 0*ONES(n+1,1);  C(1) = 1;
        for j = 1:n, C(2:j+1) = C(2:j+1) - Z(j)*C(1:j);
        C
  This recursion is easily derived by expanding the product
        (z - z(1))*(z - z(2))* ... * (z-z(n)) .
  It is possible to prove that POLY(A) produces the coefficients in
  the  characteristic  polynomial of a matrix within roundoff error
  of  A .  This is true even if the  eigenvalues  of  A  are  badly
  conditioned.    The  traditional  algorithms  for  obtaining  the
  characteristic polynomial which do not use the eigenvalues do not
  have such satisfactory numerical properties.
       If C is a vector with n+1  components,  ROOTS(C)  finds  the
  roots of the polynomial of degree n ,
         p(z) = C(1)*z**n + ... + C(n)*z + C(n+1) .
  The algorithm simply involves computing the  eigenvalues  of  the
  companion matrix:
        A = 0*ONES(n,n)
        for j = 1:n, A(1,j) = -C(j+1)/C(1);
        for i = 2:n, A(i,i-1) = 1;
        EIG(A)
  It is possible to prove that the results produced are  the  exact
  eigenvalues  of  a  matrix within roundoff error of the companion
  matrix A, but this does not mean that they are the exact roots of
  a  polynomial with coefficients within roundoff error of those in
  C .  There are more accurate, more efficient methods for  finding
  polynomial  roots,  but  this  approach has the crucial advantage
  that it does not require very much additional code.
       The elementary functions EXP, LOG, SQRT, SIN, COS  and  ATAN
  are  applied  to  square  matrices  by  diagonalizing the matrix,
  applying the functions to the  individual  eigenvalues  and  then
  transforming back.  For example, EXP(A) is computed by
        <X,D> = EIG(A);
        for j = 1:n, D(j,j) = EXP(D(j,j));
        X*D/X
  This is essentially method number 14  out  of  the  19  'dubious'
  possibilities described in [8].  It is dubious because it doesn't
  always work.  The matrix of eigenvectors  X  can  be  arbitrarily
  badly  conditioned  and  all  accuracy lost in the computation of
  X*D/X.  A warning message is printed if RCOND(X) is  very  small,
  but  this  only  catches the extreme cases.  An example of a case
  not detected is when A has a double eigenvalue, but theoretically
  only  one  linearly  independent  eigenvector associated with it.
  The computed eigenvalues will be separated by  something  on  the
  order  of the square root of the roundoff level.  This separation
  will be reflected in RCOND(X) which will probably  not  be  small
  enough to trigger the error message.  The computed EXP(A) will be
  accurate to only half precision.  Better methods  are  known  for
  computing EXP(A), but they do not easily extend to the other five
  functions and would require a considerable amount  of  additional
  code.
       The expression A**p is evaluated by repeated  multiplication
  if p is an integer greater than 1.  Otherwise it is evaluated by
        <X,D> = EIG(A);
        for j = 1:n, D(j,j) = EXP(p*LOG(D(j,j)))
        X*D/X
  This suffers from the same potential loss of  accuracy  if  X  is
  badly conditioned.  It was partly for this reason that the case p
  = 1 is included in the general case.  Comparison of A**1  with  A
  gives some idea of the loss of accuracy for other values of p and
  for the elementary functions.
       RREF, the reduced row echelon form, is of some  interest  in
  theoretical  linear algebra, although it has little computational
  value.  It is included in MATLAB for  pedagogical  reasons.   The
  algorithm  is essentially Gauss-Jordan elimination with detection
  of negligible columns applied to rectangular matrices.
       There are three separate places in MATLAB where the rank  of
  a  matrix  is  implicitly  computed:  in RREF(A), in A\B for non-
  square A, and in  the  pseudoinverse  PINV(A).   Three  different
  algorithms  with  three  different criteria for negligibility are
  used and so it is possible that three different values  could  be
  produced for the same matrix.  With RREF(A), the rank of A is the
  number of nonzero rows.  The elimination algorithm used for  RREF
  is  the  fastest of the three rank-determining algorithms, but it
  is the least sophisticated numerically and  the  least  reliable.
  With  A\B,  the  algorithm  is  essentially  that used by example
  subroutine SQRST  in  chapter  9  of  the  LINPACK  guide.   With
  PINV(A),   the   algorithm   is   based  on  the  singular  value
  decomposition and is described  in  chapter  11  of  the  LINPACK
  guide.   The  SVD  algorithm  is the most time-consuming, but the
  most reliable and is therefore also used for RANK(A).
       The  uniformly  distributed  random  numbers  in  RAND   are
  obtained  from  the  machine-independent  random number generator
  URAND described in [9].  It is possible  to  switch  to  normally
  distributed   random   numbers,   which   are  obtained  using  a
  transformation also described in [9].
       The computation of
                  2    2
            sqrt(a  + b )
  is  required  in  many  matrix  algorithms,  particularly   those
  involving  complex  arithmetic.   A  new approach to carrying out
  this operation is described by Moler and Morrison [10].  It is  a
  cubically  convergent  algorithm  which  starts with  a  and  b ,
  rather than with their squares, and  thereby  avoids  destructive
  arithmetic underflows and overflows.  In MATLAB, the algorithm is
  used for complex modulus, Euclidean vector norm, plane rotations,
  and  the  shift  calculation in the eigenvalue and singular value
  iterations.

12. FLOP and CHOP

       Detailed information about the amount of  work  involved  in
  matrix  calculations  and  the  resulting accuracy is provided by
  FLOP and CHOP.  The basic unit of work is the "flop", or floating
  point operation.  Roughly, one flop is one execution of a Fortran
  statement like
        S = S + X(I)*Y(I)
  or
        Y(I) = Y(I) + T*X(I)
  In other words, it consists of one floating point multiplication,
  together  with  one  floating  point  addition and the associated
  indexing and storage reference operations.
       MATLAB will  print  the  number  of  flops  required  for  a
  particular statement when the statement is terminated by an extra
  comma.  For example, the line
        n = 20;  RAND(n)*RAND(n);,
  ends with an extra comma.  Two  20  by  20  random  matrices  are
  generated  and  multiplied  together.   The result is assigned to
  ANS, but the semicolon suppresses its printing.  The only  output
  is
          8800 flops
  This is  n**3 + 2*n**2  flops,  n**2  for each random matrix  and
  n**3 for the product.
       FLOP is a predefined vector with two components.  FLOP(1) is
  the number of flops used by the most recently executed statement,
  except that statements with zero flops are ignored.  For example,
  after executing the previous statement,
        flop(1)/n**3
  results in
        ANS   =
            1.1000
       FLOP(2) is the cumulative total of all the flops used  since
  the beginning of the MATLAB session.  The statement
        FLOP = <0 0>
  resets the total.
       There are several difficulties  associated  with  keeping  a
  precise  count  of  floating  point  operations.  An  addition or
  subtraction that is not paired with a multiplication  is  usually
  counted as a flop. The same is true of an isolated multiplication
  that is  not  paired  with  an  addition.   Each  floating  point
  division counts as a flop.  But the number of operations required
  by system dependent library functions such as square root  cannot
  be  counted, so most elementary functions are arbitrarily counted
  as using only one flop.
       The  biggest  difficulty  occurs  with  complex  arithmetic.
  Almost  all operations on the real parts of matrices are counted.
  However, the operations on the  complex  parts  of  matrices  are
  counted only when they involve nonzero elements.  This means that
  simple operations on nonreal matrices require only about twice as
  many  flops as the same operations on real matrices.  This factor
  of two is not necessarily an accurate  measure  of  the  relative
  costs of real and complex arithmetic.
       The result of each floating  point  operation  may  also  be
  "chopped" to simulate a computer with a shorter word length.  The
  details of this chopping operation depend upon the format of  the
  floating point word.  Usually, the fraction in the floating point
  word  can  be  regarded  as  consisting  of  several   octal   or
  hexadecimal digits.  The least significant of these digits can be
  set to zero by a logical masking operation.  Thus the statement
        CHOP(p)
  causes the  p  least significant octal or hexadecimal  digits  in
  the  result  of  each floating point operation to be set to zero.
  For example, if the computer being  used  has  an  IBM  360  long
  floating  point  word with 14 hexadecimal digits in the fraction,
  then CHOP(8) results in simulation of  a  computer  with  only  6
  hexadecimal  digits  in the fraction, i.e. a short floating point
  word. On a computer such as the CDC 6600 with  16  octal  digits,
  CHOP(8)  results in about the same accuracy because the remaining
  8 octal digits represent the same number of bits as 6 hexadecimal
  digits.
       Some idea of the effect of CHOP on any particular system can
  be obtained by executing the following statements.
        long,   t = 1/10
        long z, t = 1/10
        chop(8)
        long,   t = 1/10
        long z, t = 1/10
       The following Fortran subprograms illustrate more details of
  FLOP  and CHOP. The first subprogram is a simplified example of a
  system-dependent function used within MATLAB itself.  The  common
  variable  FLP  is essentially the first component of the variable
  FLOP.  The common variable CHP is initially zero, but it  is  set
  to  p  by the statement  CHOP(p).  To shorten the DATA statement,
  we assume there are only 6 hexadecimal digits.  We also assume an
  extension  of  Fortran  that  allows .AND. to be used as a binary
  operation between two real variables.
        REAL FUNCTION FLOP(X)
        REAL X
        INTEGER FLP,CHP
        COMMON FLP,CHP
        REAL MASK(5)
        DATA MASK/ZFFFFFFF0,ZFFFFFF00,ZFFFFF000,ZFFFF0000,ZFFF00000/
        FLP = FLP + 1
        IF (CHP .EQ. 0) FLOP = X
        IF (CHP .GE. 1 .AND. CHP .LE. 5) FLOP = X .AND. MASK(CHP)
        IF (CHP .GE. 6) FLOP = 0.0
        RETURN
        END
       The following subroutine illustrates a typical  use  of  the
  previous  function  within MATLAB.  It is a simplified version of
  the Basic Linear Algebra Subprogram that adds a  scalar  multiple
  of  one  vector  to another.  We assume here that the vectors are
  stored with a memory increment of one.
        SUBROUTINE SAXPY(N,TR,TI,XR,XI,YR,YI)
        REAL TR,TI,XR(N),XI(N),YR(N),YI(N),FLOP
        IF (N .LE. 0) RETURN
        IF (TR .EQ. 0.0 .AND. TI .EQ. 0.0) RETURN
        DO 10 I = 1, N
           YR(I) = FLOP(YR(I) + TR*XR(I) - TI*XI(I))
           YI(I) = YI(I) + TR*XI(I) + TI*XR(I)
           IF (YI(I) .NE. 0.0D0) YI(I) = FLOP(YI(I))
     10 CONTINUE
        RETURN
        END
       The  saxpy  operation  is  perhaps  the   most   fundamental
  operation  within  LINPACK.  It is used in the computation of the
  LU, the QR and the  SVD  factorizations,  and  in  several  other
  places.   We  see  that  adding  a  multiple of one vector with n
  components to another uses n flops if the vectors  are  real  and
  between  n  and  2*n  flops if the vectors have nonzero imaginary
  components.
       The permanent MATLAB variable EPS is reset by the  statement
  CHOP(p).   Its new value is usually the smallest inverse power of
  two that satisfies the Fortran logical test
              FLOP(1.0+EPS) .GT. 1.0
  However, if EPS had been directly reset to a  larger  value,  the
  old value is retained.

13. Communicating with other programs

       There  are  four  different  ways  MATLAB  can  be  used  in
  conjunction with other programs:
        -- USER,
        -- EXEC,
        -- SAVE and LOAD,
        -- MATZ, CALL and RETURN .
       Let us illustrate each of  these  by  the  following  simple
  example.
        n = 6
        for i = 1:n, for j = 1:n, a(i,j) = abs(i-j);
        A
        X = inv(A)
       The example  A  could be introduced into MATLAB  by  writing
  the following Fortran subroutine.
           SUBROUTINE USER(A,M,N,S,T)
           DOUBLE PRECISION A(1),S,T
           N = IDINT(A(1))
           M = N
           DO 10 J = 1, N
           DO 10 I = 1, N
              K = I + (J-1)*M
              A(K) = IABS(I-J)
        10 CONTINUE
           RETURN
           END
  This subroutine should be compiled  and  linked  into  MATLAB  in
  place   of  the  original  version  of  USER.   Then  the  MATLAB
  statements
        n = 6
        A = user(n)
        X = inv(A)
  do the job.
       The example A could be generated by  storing  the  following
  text in a file named, say, EXAMPLE .
        for i = 1:n, for j = 1:n, a(i,j) = abs(i-j);
  Then the MATLAB statements
        n = 6
        exec('EXAMPLE',0)
        X = inv(A)
  have the desired effect.  The 0 as the optional second  parameter
  of exec indicates that the text in the file should not be printed
  on the terminal.
       The matrices A and X could also be  stored  in  files.   Two
  separate main programs would be involved.  The first is:
           PROGRAM MAINA
           DOUBLE PRECISION A(10,10)
           N = 6
           DO 10 J = 1, N
           DO 10 I = 1, N
              A(I,J) = IABS(I-J)
        10 CONTINUE
           OPEN(UNIT=1,FILE='A')
           WRITE(1,101) N,N
       101 FORMAT('A   ',2I4)
           DO 20 J = 1, N
              WRITE(1,102) (A(I,J),I=1,N)
        20 CONTINUE
       102 FORMAT(4Z18)
           END
  The OPEN statement may take different forms on different systems.
  It  attaches  Fortran  logical unit number 1 to the file named A.
  The FORMAT  number  102  may  also  be  system  dependent.   This
  particular one is appropriate for hexadecimal computers with an 8
  byte double precision floating point  word.   Check,  or  modify,
  MATLAB subroutine SAVLOD.
       After this program is executed, enter MATLAB  and  give  the
  following statements:
        load('A')
        X = inv(A)
        save('X',X)
  If all goes according to plan, this will read the matrix  A  from
  the  file A, invert it, store the inverse in X and then write the
  matrix X on the file X .  The following program can then access X
  .
           PROGRAM MAINX
           DOUBLE PRECISION X(10,10)
           OPEN(UNIT=1,FILE='X')
           REWIND 1
           READ (1,101) ID,M,N
       101 FORMAT(A4,2I4)
           DO 10 J = 1, N
              READ(1,102) (X(I,J),I=1,M)
        10 CONTINUE
       102 FORMAT(4Z18)
           ...
           ...
       The most elaborate mechanism  involves  using  MATLAB  as  a
  subroutine within another program.  Communication with the MATLAB
  stack is accomplished using subroutine MATZ which is  distributed
  with  MATLAB,  but  which  is  not  used  by  MATLAB itself.  The
  preample of MATZ is:
        SUBROUTINE MATZ(A,LDA,M,N,IDA,JOB,IERR)
        INTEGER LDA,M,N,IDA(1),JOB,IERR
        DOUBLE PRECISION A(LDA,N)
  C
  C     ACCESS MATLAB VARIABLE STACK
  C     A IS AN M BY N MATRIX, STORED IN AN ARRAY WITH
  C         LEADING DIMENSION LDA.
  C     IDA IS THE NAME OF A.
  C         IF IDA IS AN INTEGER K LESS THAN 10, THEN THE NAME IS 'A'K
  C         OTHERWISE, IDA(1:4) IS FOUR CHARACTERS, FORMAT 4A1.
  C     JOB =  0  GET REAL A FROM MATLAB,
  C         =  1  PUT REAL A INTO MATLAB,
  C         = 10  GET IMAG PART OF A FROM MATLAB,
  C         = 11  PUT IMAG PART OF A INTO MATLAB.
  C     RETURN WITH NONZERO IERR AFTER MATLAB ERROR MESSAGE.
  C
  C     USES MATLAB ROUTINES STACKG, STACKP AND ERROR
       The preample of subroutine MATLAB is:
        SUBROUTINE MATLAB(INIT)
  C     INIT = 0 FOR FIRST ENTRY, NONZERO FOR SUBSEQUENT ENTRIES
       To do our example, write the following program:
           DOUBLE PRECISION A(10,10),X(10,10)
           INTEGER IDA(4),IDX(4)
           DATA LDA/10/
           DATA IDA/'A',' ',' ',' '/, IDX/'X',' ',' ',' '/
           CALL MATLAB(0)
           N = 6
           DO 10 J = 1, N
           DO 10 I = 1, N
              A(I,J) = IABS(I-J)
        10 CONTINUE
           CALL MATZ(A,LDA,N,N,IDA,1,IERR)
           IF (IERR .NE. 0) GO TO ...
           CALL MATLAB(1)
           CALL MATZ(X,LDA,N,N,IDX,0,IERR)
           IF (IERR .NE. 0) GO TO ...
           ...
           ...
  When this program is executed, the call to MATLAB(0) produces the
  MATLAB greeting, then waits for input.  The command
           return
  sends control back to our  example  program.   The  matrix  A  is
  generated  by the program and sent to the stack by the first call
  to MATZ.  The call to MATLAB(1) produces the MATLAB prompt.  Then
  the statements
           X = inv(A)
           return
  will invert our matrix, put the result on the stack and  go  back
  to our program.  The second call to MATZ will retrieve X .
       By the way, this matrix  X  is interesting. Take a  look  at
  round(2*(n-1)*X).
  References
  [1]  J. J. Dongarra, J. R. Bunch, C. B. Moler and G. W.  Stewart,
       LINPACK  Users'  Guide,  Society  for Industrial and Applied
       Mathematics, Philadelphia, 1979.
  [2]  B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S.  Garbow,  Y.
       Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines
       -- EISPACK Guide, Lecture Notes in Computer Science,  volume
       6, second edition, Springer-Verlag, 1976.
  [3]  B. S. Garbow, J. M. Boyle, J.  J.  Dongarra,  C.  B.  Moler,
       Matrix  Eigensystem  Routines  --  EISPACK  Guide Extension,
       Lecture Notes in  Computer  Science,  volume  51,  Springer-
       Verlag, 1977.
  [4]  S. Cohen and  S.  Piper,  SPEAKEASY  III  Reference  Manual,
       Speakeasy Computing Corp., Chicago, Ill., 1979.
  [5]  J. H. Wilkinson  and  C.  Reinsch,  Handbook  for  Automatic
       Computation,  volume  II,  Linear  Algebra, Springer-Verlag,
       1971.
  [6]  Niklaus Wirth, Algorithms  +  Data  Structures  =  Programs,
       Prentice-Hall, 1976.
  [7]  H. B. Keller and D. Sachs, "Calculations of the Conductivity
       of  a  Medium Containing Cylindrical Inclusions", J. Applied
       Physics 35, 537-538, 1964.
  [8]  C. B. Moler and C. F. Van Loan,  Nineteen  Dubious  Ways  to
       Compute  the  Exponential  of a Matrix, SIAM Review 20, 801-
       836, 1979.
  [9]  G. E. Forsythe, M. A. Malcolm  and  C.  B.  Moler,  Computer
       Methods for Mathematical Computations, Prentice-Hall, 1977.
  [10] C. B. Moler and D. R. Morrison, "Replacing square  roots  by
       Pythagorean   sums",  University  of  New  Mexico,  Computer
       Science  Department,   technical   report,   submitted   for
       publication, 1980.

Appendix. The HELP document

  NEWS  MATLAB NEWS dated May, 1981.
        This describes recent or local changes.
        The new features added since the November,  1980,  printing
        of the Users' Guide include DIARY, EDIT, KRON, MACRO, PLOT,
        RAT, TRIL, TRIU and six element-by-element operations:
              .*   ./   .\   .*.   ./.   .\.
        Some additional  capabilities  have  been  added  to  EXIT,
        RANDOM, RCOND, SIZE and SVD.
  INTRO Welcome to MATLAB.
        Here are a few sample statements:
        A = <1 2; 3 4>
        b = <5 6>'
        x = A\b
        <V,D> = eig(A),  norm(A-V*D/V)
        help \ , help eig
        exec('demo',7)
        For more information, see the MATLAB Users' Guide which  is
        contained in file ...  or may be obtained from ... .
  HELP  HELP gives assistance.
        HELP HELP obviously prints this message.
        To see all the HELP messages, list the file ... .
  <     < > Brackets used in forming vectors and matrices.
        <6.9  9.64  SQRT(-1)>  is  a  vector  with  three  elements
        separated  by  blanks.   <6.9,  9.64, sqrt(-1)> is the same
        thing.  <1+I 2-I 3>  and  <1 +I 2 -I 3>  are not the  same.
        The first has three elements, the second has five.
        <11 12 13; 21 22 23>  is a 2 by 3 matrix .   The  semicolon
        ends the first row.
        Vectors and matrices can be used inside < > brackets.
        <A B; C>  is allowed if the number of rows  of   A   equals
        the  number  of rows of  B  and the number of columns of  A
        plus the number of columns of   B   equals  the  number  of
        columns  of   C  .   This  rule  generalizes in a hopefully
        obvious way to allow fairly complicated constructions.
        A = < >  stores an empty matrix in  A , thereby removing it
        from the list of current variables.
        For the use of < and > on the left of  the  =  in  multiple
        assignment statements, see LU, EIG, SVD and so on.
        In WHILE and IF clauses, <>  means  less  than  or  greater
        than,  i.e.  not  equal, < means less than, > means greater
        than, <= means less than or equal, >= means greater than or
        equal.
        For the use of > and < to delineate macros, see MACRO.
  >     See < .  Also see MACRO.
  (     ( ) Used to indicate precedence in  arithmetic  expressions
        in  the  usual way.  Used to enclose arguments of functions
        in the usual way.  Used to enclose  subscripts  of  vectors
        and  matrices  in  a  manner somewhat more general than the
        usual way.  If  X   and   V  are  vectors,  then   X(V)  is
        <X(V(1)),  X(V(2)),  ...,  X(V(N))> .  The components of  V
        are rounded to nearest integers and used as subscripts.  An
        error  occurs  if  any  such  subscript  is  less than 1 or
        greater than the dimension of  X .  Some examples:
        X(3)  is the third element of  X .
        X(<1 2 3>)  is the first three elements of  X .  So is
        X(<SQRT(2), SQRT(3), 4*ATAN(1)>)  .
        If  X  has  N  components,  X(N:-1:1) reverses them.
        The same indirect subscripting is used in matrices.  If   V
        has   M  components and  W  has  N  components, then A(V,W)
        is the  M by N  matrix formed from the elements of A  whose
        subscripts are the elements of  V  and  W .  For example...
        A(<1,5>,:) = A(<5,1>,:)  interchanges rows 1 and 5 of  A .
  )     See  ( .
  =     Used in assignment statements and to mean equality in WHILE
        and IF clauses.
  .     Decimal point.  314/100, 3.14  and   .314E1   are  all  the
        same.
        Element-by-element multiplicative operations  are  obtained
        using  .*  ,  ./  , or .\ .  For example, C = A ./ B is the
        matrix with elements  c(i,j) = a(i,j)/b(i,j) .
        Kronecker tensor products and quotients are  obtained  with
        .*. , ./.  and .\. .  See KRON.
        Two or  more  points  at  the  end  of  the  line  indicate
        continuation.    The   total  line  length  limit  is  1024
        characters.
  ,     Used to separate matrix subscripts and function  arguments.
        Used  at  the  end  of  FOR, WHILE and IF clauses.  Used to
        separate statements  in  multi-statement  lines.   In  this
        situation,  it  may  be  replaced  by semicolon to suppress
        printing.
  ;     Used inside brackets to end rows.
        Used after an expression or statement to suppress printing.
        See SEMI.
  \     Backslash or matrix left division.   A\B   is  roughly  the
        same  as   INV(A)*B  , except it is computed in a different
        way.  If  A  is an N by N matrix and  B  is a column vector
        with  N  components, or a matrix with several such columns,
        then X = A\B  is the solution to  the  equation   A*X  =  B
        computed  by  Gaussian  elimination.   A warning message is
        printed if  A is badly scaled or nearly singular.
        A\EYE produces the inverse of  A .
        If  A  is an  M by N  matrix with  M < or > N  and  B  is a
        column vector with  M  components, or a matrix with several
        such columns, then  X = A\B  is the solution in  the  least
        squares  sense  to  the under- or overdetermined system  of
        equations A*X = B .  The  effective  rank,  K,  of   A   is
        determined  from  the  QR  decomposition  with pivoting.  A
        solution  X  is  computed  which  has  at  most  K  nonzero
        components  per column.  If  K < N this will usually not be
        the same solution as PINV(A)*B .
        A\EYE produces a generalized inverse of  A .
        If A and B have the  same  dimensions,  then  A  .\  B  has
        elements a(i,j)\b(i,j) .
        Also, see EDIT.
  /     Slash or matrix right division.  B/A  is roughly  the  same
        as  B*INV(A) .  More precisely,  B/A = (A'\B')' .  See \ .
        IF A and B have the  same  dimensions,  then  A  ./  B  has
        elements a(i,j)/b(i,j) .
        Two or more slashes together on a line indicate  a  logical
        end of line.  Any following text is ignored.
  '     Transpose.  X'  is the complex conjugate transpose of  X  .
        Quote.   'ANY  TEXT'   is a vector whose components are the
        MATLAB internal codes for the characters.  A  quote  within
        the text is indicated by two quotes.  See DISP and FILE .
  +     Addition.  X + Y .  X and Y must have the same dimensions.
  -     Subtraction.  X  -  Y  .   X  and  Y  must  have  the  same
        dimensions.
  *     Matrix multiplication, X*Y .  Any scalar (1  by  1  matrix)
        may multiply anything.  Otherwise, the number of columns of
        X must equal the number of rows of Y .
        Element-by-element multiplication is obtained with X .* Y .
        The Kronecker tensor product is denoted by X .*. Y .
        Powers.  X**p  is  X  to the   p   power.   p   must  be  a
        scalar.  If  X  is a matrix, see  FUN .
  :     Colon.  Used in subscripts,  FOR  iterations  and  possibly
        elsewhere.
        J:K  is the same as  <J, J+1, ..., K>
        J:K  is empty if  J > K .
        J:I:K  is the same as  <J, J+I, J+2I, ..., K>
        J:I:K  is empty if  I > 0 and J > K or if I < 0 and J < K .
        The colon notation can be used to pick out  selected  rows,
        columns and elements of vectors and matrices.
        A(:)  is all the  elements  of  A,  regarded  as  a  single
        column.
        A(:,J)  is the  J-th  column of A
        A(J:K)  is  A(J),A(J+1),...,A(K)
        A(:,J:K)  is  A(:,J),A(:,J+1),...,A(:,K) and so on.
        For the use of the colon in the FOR statement, See FOR .
  ABS   ABS(X)  is the absolute value, or complex modulus,  of  the
        elements of X .
  ANS   Variable created automatically  when  expressions  are  not
        assigned to anything else.
  ATAN  ATAN(X)  is the arctangent of  X .  See FUN .
  BASE  BASE(X,B) is a vector containing the base B  representation
        of   X  .   This is often used in conjunction with DISPLAY.
        DISPLAY(X,B)  is  the  same  as  DISPLAY(BASE(X,B)).    For
        example,    DISP(4*ATAN(1),16)   prints   the   hexadecimal
        representation of pi.
  CHAR  CHAR(K)  requests  an  input  line  containing   a   single
        character  to  replace  MATLAB  character  number  K in the
        following table.  For example, CHAR(45) replaces backslash.
        CHAR(-K) replaces the alternate character number K.
                  K  character alternate name
                0 - 9   0 - 9    0 - 9   digits
               10 - 35  A - Z    a - z   letters
                 36                      blank
                 37       (        (     lparen
                 38       )        )     rparen
                 39       ;        ;     semi
                 40       :        |     colon
                 41       +        +     plus
                 42       -        -     minus
                 43       *        *     star
                 44       /        /     slash
                 45       \        $     backslash
                 46       =        =     equal
                 47       .        .     dot
                 48       ,        ,     comma
                 49       '        "     quote
                 50       <        [     less
                 51       >        ]     great
  CHOL  Cholesky factorization.  CHOL(X)  uses  only  the  diagonal
        and upper triangle of  X .  The lower triangular is assumed
        to be the (complex conjugate) transpose of the  upper.   If
        X   is  positive  definite,  then  R = CHOL(X)  produces an
        upper triangular  R  so that  R'*R = X .   If   X   is  not
        positive definite, an error message is printed.
  CHOP  Truncate arithmetic.  CHOP(P) causes P places to be chopped
        off   after   each   arithmetic   operation  in  subsequent
        computations.  This means  P  hexadecimal  digits  on  some
        computers  and  P octal digits on others.  CHOP(0) restores
        full precision.
  CLEAR Erases all variables, except EPS, FLOP, EYE and RAND.
        X = <>  erases only variable  X .  So does CLEAR X .
  COND  Condition number in 2-norm.  COND(X) is the  ratio  of  the
        largest singular value of  X  to the smallest.
  CONJG CONJG(X)  is the complex conjugate of  X .
  COS   COS(X)  is the cosine of  X .  See FUN .
  DET   DET(X)  is the determinant of the square matrix  X .
  DIAG  If  V  is  a  row  or  column  vector  with  N  components,
        DIAG(V,K)   is a square matrix of order  N+ABS(K)  with the
        elements of  V  on the K-th diagonal.  K = 0  is  the  main
        diagonal,  K  >  0  is above the main diagonal and K < 0 is
        below the main diagonal.  DIAG(V)  simply puts  V   on  the
        main diagonal.
        eg. DIAG(-M:M) + DIAG(ONES(2*M,1),1) + DIAG(ONES(2*M,1),-1)
        produces a tridiagonal matrix of order 2*M+1 .
        IF  X  is a matrix,  DIAG(X,K)  is a column  vector  formed
        from the elements of the K-th diagonal of  X .
        DIAG(X)  is the main diagonal of  X .
        DIAG(DIAG(X))  is a diagonal matrix .
  DIARY DIARY('file') causes a  copy  of  all  subsequent  terminal
        input and most of the resulting output to be written on the
        file. DIARY(0) turns it off.  See FILE.
  DISP  DISPLAY(X) prints X  in  a  compact  format.   If  all  the
        elements  of  X  are  integers  between 0 and 51, then X is
        interpreted  as  MATLAB  text  and   printed   accordingly.
        Otherwise,  +  ,  -   and  blank  are printed for positive,
        negative and zero elements.  Imaginary parts are ignored.
        DISP(X,B) is the same as DISP(BASE(X,B)).
  EDIT  There  are  no   editing   features   available   on   most
        installations and EDIT is not a command.  However, on a few
        systems a command line consisting of a single  backslash  \
        will  cause  the local file editor to be called with a copy
        of the  previous  input  line.   When  the  editor  returns
        control to MATLAB, it will execute the line again.
  EIG   Eigenvalues and eigenvectors.
        EIG(X) is a vector containing the eigenvalues of  a  square
        matrix  X .
        <V,D>  =  EIG(X)   produces  a  diagonal  matrix    D    of
        eigenvalues  and  a  full  matrix  V  whose columns are the
        corresponding eigenvectors so that  X*V = V*D .
  ELSE  Used with IF .
  END   Terminates the scope  of  FOR,  WHILE  and  IF  statements.
        Without  END's,  FOR  and WHILE repeat all statements up to
        the end of the line.  Each END is paired with  the  closest
        previous  unpaired FOR or WHILE and serves to terminate its
        scope.  The line
        FOR I=1:N, FOR J=1:N, A(I,J)=1/(I+J-1); A
        would cause A to be printed  N**2  times, once for each new
        element.  On the other hand, the line
        FOR I=1:N, FOR J=1:N, A(I,J)=1/(I+J-1); END, END, A
        will lead to only the final printing of  A .
        Similar considerations apply to WHILE.
        EXIT terminates execution of loops or of MATLAB itself.
  EPS   Floating point relative  accuracy.   A  permanent  variable
        whose  value is initially the distance from 1.0 to the next
        largest floating point number.  The  value  is  changed  by
        CHOP,  and  other values may be assigned.  EPS is used as a
        default tolerance by PINV and RANK.
  EXEC  EXEC('file',k) obtains  subsequent  MATLAB  input  from  an
        external  file.  The printing of input is controlled by the
        optional parameter k .
        If k = 1 , the input is echoed.
        If k = 2 , the MATLAB prompt <> is printed.
        If k = 4 , MATLAB pauses before each prompt and waits for a
        null line to continue.
        If k = 0 , there is no echo, prompt or pause.  This is  the
        default if the exec command is followed by a semicolon.
        If k = 7 , there will be echos, prompts and pauses. This is
        useful for demonstrations on video terminals.
        If k = 3 , there will be echos and prompts, but no  pauses.
        This is the the default if the exec command is not followed
        by a semicolon.
        EXEC(0) causes subsequent input to  be  obtained  from  the
        terminal. An end-of-file has the same effect.
        EXEC's may be nested, i.e. the text in the file may contain
        EXEC of another file.  EXEC's may also be driven by FOR and
        WHILE loops.
  EXIT  Causes termination of a FOR or WHILE loop.
        If not in a loop, terminates execution of MATLAB.
  EXP   EXP(X)  is the exponential of  X ,  e  to the X .  See  FUN
        .
  EYE   Identity matrix.  EYE(N) is the N  by  N  identity  matrix.
        EYE(M,N)   is an M by N matrix with 1's on the diagonal and
        zeros elsewhere.  EYE(A)  is the same size  as   A  .   EYE
        with  no  arguments is an identity matrix of whatever order
        is appropriate in the context.   For  example,  A  +  3*EYE
        adds  3  to each diagonal element of  A .
  FILE  The EXEC, SAVE, LOAD,  PRINT  and  DIARY  functions  access
        files.   The  'file'  parameter  takes  different forms for
        different operating systems.  On most systems,  'file'  may
        be a string of up to 32 characters in quotes.  For example,
        SAVE('A') or EXEC('matlab/demo.exec') .  The string will be
        used as the name of a file in the local operating system.
        On all systems, 'file' may be a positive integer   k   less
        than  10  which  will  be  used  as  a FORTRAN logical unit
        number. Some systems then automatically access a file  with
        a  name  like  FORT.k  or FORk.DAT. Other systems require a
        file with a name like FT0kF001 to be assigned  to  unit   k
        before  MATLAB  is  executed. Check your local installation
        for details.
  FLOPS Count of floating point operations.
        FLOPS  is  a  permanently  defined  row  vector  with   two
        elements.    FLOPS(1)  is  the  number  of  floating  point
        operations counted during the previous statement.  FLOPS(2)
        is  a  cumulative total.  FLOPS can be used in the same way
        as any other vector.  FLOPS(2) = 0  resets  the  cumulative
        total.   In  addition,  FLOPS(1) will be printed whenever a
        statement is terminated by an extra comma.  For example,
        X = INV(A);,
        or
        COND(A),   (as the last statement on the line).
        HELP FLPS gives more details.
  FLPS  More detail on FLOPS.
        It is not feasible to count absolutely all  floating  point
        operations,  but  most  of  the important ones are counted.
        Each multiply and add in a real vector operation such as  a
        dot  product  or  a 'saxpy' counts one flop.  Each multiply
        and add in a complex vector  operation  counts  two  flops.
        Other additions, subtractions and multiplications count one
        flop each if the result is real and two flops if it is not.
        Real  divisions  count one and complex divisions count two.
        Elementary functions count one if real and two if  complex.
        Some examples.  If A and B are real N by N matrices, then
        A + B  counts N**2 flops,
        A*B    counts N**3 flops,
        A**100 counts 99*N**3 flops,
        LU(A)  counts roughly (1/3)*N**3 flops.
  FOR   Repeat statements a specific number of times.
        FOR variable = expr, statement, ..., statement, END
        The END at the end of a line may  be  omitted.   The  comma
        before  the  END  may  also be omitted.  The columns of the
        expression are stored one at a time  in  the  variable  and
        then the following statements, up to the END, are executed.
        The expression is often of the form X:Y, in which case  its
        columns  are  simply  scalars.  Some examples (assume N has
        already been assigned a value).
        FOR I = 1:N, FOR J = 1:N, A(I,J) = 1/(I+J-1);
        FOR J = 2:N-1, A(J,J) = J; END; A
        FOR S = 1.0: -0.1: 0.0, ...  steps S with increments of -0.1 .
        FOR E = EYE(N), ...   sets  E  to the unit N-vectors.
        FOR V = A, ...   has the same effect as
        FOR J = 1:N, V = A(:,J); ...  except J is also set here.
  FUN   For matrix arguments  X , the  functions  SIN,  COS,  ATAN,
        SQRT,  LOG,  EXP and X**p are computed using eigenvalues  D
        and eigenvectors  V .  If  <V,D> =  EIG(X)   then   f(X)  =
        V*f(D)/V  .   This method may give inaccurate results if  V
        is badly conditioned.  Some idea of  the  accuracy  can  be
        obtained by comparing  X**1  with  X .
        For vector arguments,  the  function  is  applied  to  each
        component.
  HESS  Hessenberg form.  The Hessenberg form of a matrix  is  zero
        below the first subdiagonal.  If the matrix is symmetric or
        Hermitian,  the  form  is  tridiagonal.   <P,H>  =  HESS(A)
        produces  a  unitary  matrix P and a Hessenberg matrix H so
        that A = P*H*P'.  By itself, HESS(A) returns H.
  HILB  Inverse Hilbert matrix.  HILB(N)  is the inverse of  the  N
        by  N   matrix  with elements  1/(i+j-1), which is a famous
        example of a badly conditioned matrix.  The result is exact
        for  N  less than about 15, depending upon the computer.
  IF    Conditionally execute statements.  Simple form...
        IF expression rop expression, statements
        where rop is =, <, >, <=, >=, or  <>  (not  equal)  .   The
        statements  are  executed  once if the indicated comparison
        between the real parts of the first components of  the  two
        expressions  is true, otherwise the statements are skipped.
        Example.
        IF ABS(I-J) = 1, A(I,J) = -1;
        More complicated forms use END in the same way it  is  used
        with FOR and WHILE and use ELSE as an abbreviation for END,
        IF expression not rop expression .  Example
        FOR I = 1:N, FOR J = 1:N, ...
           IF I = J, A(I,J) = 2; ELSE IF ABS(I-J) = 1, A(I,J) = -1; ...
           ELSE A(I,J) = 0;
        An easier way to accomplish the same thing is
        A = 2*EYE(N);
        FOR I = 1:N-1, A(I,I+1) = -1; A(I+1,I) = -1;
  IMAG  IMAG(X)  is the imaginary part of  X .
  INV   INV(X)  is the inverse of the square matrix  X .  A warning
        message  is  printed  if   X   is  badly  scaled  or nearly
        singular.
  KRON  KRON(X,Y) is the Kronecker tensor product of X and Y  .  It
        is  also  denoted by X .*. Y . The result is a large matrix
        formed by taking all possible products between the elements
        of  X  and  those  of Y . For example, if X is 2 by 3, then
        X .*. Y is
              < x(1,1)*Y  x(1,2)*Y  x(1,3)*Y
                x(2,1)*Y  x(2,2)*Y  x(2,3)*Y >
        The five-point discrete Laplacian for an n-by-n grid can be
        generated by
              T = diag(ones(n-1,1),1);  T = T + T';  I = EYE(T);
              A = T.*.I + I.*.T - 4*EYE;
        Just  in  case  they  might  be  useful,  MATLAB   includes
        constructions called Kronecker tensor quotients, denoted by
        X ./. Y and X .\. Y .  They are obtained by  replacing  the
        elementwise multiplications in X .*. Y with divisions.
  LINES An internal count is kept of the number of lines of  output
        since  the  last  input.   Whenever this count approaches a
        limit, the  user  is  asked  whether  or  not  to  suppress
        printing  until the next input.  Initially the limit is 25.
        LINES(N) resets the limit to N .
  LOAD  LOAD('file') retrieves all the variables from  the  file  .
        See  FILE  and  SAVE for more details.  To prepare your own
        file for LOADing, change the READs to WRITEs  in  the  code
        given under SAVE.
  LOG   LOG(X)  is the  natural  logarithm  of   X  .   See  FUN  .
        Complex results are produced if  X  is not positive, or has
        nonpositive eigenvalues.
  LONG  Determine output format.   All  computations  are  done  in
        complex arithmetic and double precision if it is available.
        SHORT and  LONG  merely  switch  between  different  output
        formats.
        SHORT    Scaled fixed point format with about 5 digits.
        LONG     Scaled fixed point format with about 15 digits.
        SHORT E  Floating point format with about 5 digits.
        LONG E   Floating point format with about 15 digits.
        LONG Z   System dependent format, often hexadecimal.
  LU    Factors from Gaussian elimination.  <L,U> = LU(X)  stores a
        upper triangular matrix in  U  and a 'psychologically lower
        triangular matrix', i.e. a product of lower triangular  and
        permutation matrices, in L , so that  X = L*U .  By itself,
        LU(X) returns the output from CGEFA .
  MACRO The macro facility involves text and inward pointing  angle
        brackets.  If  STRING  is  the  source  text for any MATLAB
        expression or statement, then
              t = 'STRING';
        encodes the text as a vector of integers  and  stores  that
        vector in  t .  DISP(t) will print the text and
              >t<
        causes the text to be interpreted, either as a statement or
        as a factor in an expression.  For example
              t = '1/(i+j-1)';
              disp(t)
              for i = 1:n, for j = 1:n, a(i,j) = >t<;
        generates the Hilbert matrix of order n.
        Another example showing indexed text,
              S = <'x = 3            '
                   'y = 4            '
                   'z = sqrt(x*x+y*y)'>
              for k = 1:3, >S(k,:)<
        It is necessary that the strings making up  the  "rows"  of
        the "matrix"  S  have the same lengths.
  MAGIC Magic square.  MAGIC(N) is an N  by  N  matrix  constructed
        from  the integers 1 through N**2 with equal row and column
        sums.
  NORM  For matrices..
        NORM(X)  is the largest singular value of  X .
        NORM(X,1)  is the 1-norm of  X .
        NORM(X,2)  is the same as NORM(X) .
        NORM(X,'INF')  is the infinity norm of  X .
        NORM(X,'FRO')  is the F-norm, i.e.  SQRT(SUM(DIAG(X'*X))) .
        For vectors..
        NORM(V,P) = (SUM(V(I)**P))**(1/P) .
        NORM(V) = NORM(V,2) .
        NORM(V,'INF') = MAX(ABS(V(I))) .
  ONES  All ones.  ONES(N)  is an N by N matrix of ones.  ONES(M,N)
        is an M by N matrix of ones .  ONES(A)  is the same size as
        A  and all ones .
  ORTH  Orthogonalization.   Q  =  ORTH(X)   is   a   matrix   with
        orthonormal  columns,  i.e. Q'*Q = EYE, which span the same
        space as the columns of  X .
  PINV  Pseudoinverse.  X = PINV(A) produces a matrix   X   of  the
        same  dimensions as  A' so that  A*X*A = A , X*A*X = X  and
        AX  and  XA  are Hermitian .  The computation is  based  on
        SVD(A)  and  any  singular values less than a tolerance are
        treated   as    zero.     The    default    tolerance    is
        NORM(SIZE(A),'inf')*NORM(A)*EPS.   This  tolerance  may  be
        overridden with X = PINV(A,tol).  See RANK.
  PLOT  PLOT(X,Y) produces a plot of  the  elements  of  Y  against
        those  of X . PLOT(Y) is the same as PLOT(1:n,Y) where n is
        the  number  of   elements   in   Y   .    PLOT(X,Y,P)   or
        PLOT(X,Y,p1,...,pk)  passes the optional parameter vector P
        or scalars p1 through pk to the plot routine.  The  default
        plot  routine  is a crude printer-plot. It is hoped that an
        interface to local graphics equipment can be provided.
        An interesting example is
              t = 0:50;
              PLOT( t.*cos(t), t.*sin(t) )
  POLY  Characteristic polynomial.
        If  A  is an N by N matrix, POLY(A) is a column vector with
        N+1   elements   which   are   the   coefficients   of  the
        characteristic polynomial,  DET(lambda*EYE - A) .
        If V is a vector, POLY(V) is a vector  whose  elements  are
        the  coefficients  of  the  polynomial  whose roots are the
        elements of V .  For vectors, ROOTS and  POLY  are  inverse
        functions  of  each  other,  up  to  ordering, scaling, and
        roundoff error.
        ROOTS(POLY(1:20)) generates Wilkinson's famous example.
  PRINT PRINT('file',X) prints X on  the  file  using  the  current
        format determined by SHORT, LONG Z, etc.  See FILE.
  PROD  PROD(X)  is the product of all the elements of  X .
  QR    Orthogonal-triangular decomposition.
        <Q,R> = QR(X)  produces an upper triangular  matrix   R  of
        the  same dimension as  X  and a unitary matrix  Q  so that
        X = Q*R .
        <Q,R,E> = QR(X)  produces a  permutation  matrix   E  ,  an
        upper  triangular  R  with decreasing diagonal elements and
        a unitary  Q  so that  X*E = Q*R .
        By itself, QR(X) returns the output of CQRDC .  TRIU(QR(X))
        is R .
  RAND  Random numbers and matrices.  RAND(N)  is an N by N  matrix
        with  random  entries.  RAND(M,N)  is an M by N matrix with
        random entries.  RAND(A)  is the same size as   A  .   RAND
        with no arguments is a scalar whose value changes each time
        it is referenced.
        Ordinarily,  random numbers are  uniformly  distributed  in
        the  interval  (0.0,1.0)  .   RAND('NORMAL')  switches to a
        normal distribution  with  mean  0.0  and  variance  1.0  .
        RAND('UNIFORM')  switches back to the uniform distribution.
        RAND('SEED') returns the current value of the seed for  the
        generator.    RAND('SEED',n)   sets   the   seed   to  n  .
        RAND('SEED',0) resets the seed to 0, its value when  MATLAB
        is first entered.
  RANK  Rank.  K = RANK(X) is the number of singular values  of   X
        that are larger than NORM(SIZE(X),'inf')*NORM(X)*EPS.
        K = RANK(X,tol) is the number of singular values of  X that
        are larger than tol .
  RCOND RCOND(X)   is  an  estimate  for  the  reciprocal  of   the
        condition  of   X   in  the  1-norm obtained by the LINPACK
        condition estimator.  If  X  is well conditioned,  RCOND(X)
        is  near  1.0  .   If  X  is badly conditioned, RCOND(X) is
        near 0.0 .
        <R, Z> = RCOND(A) sets  R  to RCOND(A) and also produces  a
        vector  Z so that
                   NORM(A*Z,1) = R*NORM(A,1)*NORM(Z,1)
        So, if RCOND(A) is small, then  Z  is an  approximate  null
        vector.
  RAT   An experimental  function  which  attempts  to  remove  the
        roundoff   error  from  results  that  should  be  "simple"
        rational numbers.
        RAT(X) approximates each  element  of   X  by  a  continued
        fraction of the form
                  a/b = d1 + 1/(d2 + 1/(d3 + ... + 1/dk))
        with k <= len, integer di and abs(di) <= max .  The default
        values of the parameters are len = 5 and max = 100.
        RAT(len,max) changes the default values.  Increasing either
        len or max increases the number of possible fractions.
        <A,B> = RAT(X) produces integer matrices A and B so that
                  A ./ B  =  RAT(X)
        Some examples:
              long
              T = hilb(6), X = inv(T)
              <A,B> = rat(X)
              H = A ./ B, S = inv(H)
              short e
              d = 1:8,  e = ones(d),  A = abs(d'*e - e'*d)
              X = inv(A)
              rat(X)
              display(ans)
  REAL  REAL(X)  is the real part of  X .
  RETURN  From the terminal, causes return to the operating  system
        or  other  program  which  invoked  MATLAB.  From inside an
        EXEC, causes  return  to  the  invoking  EXEC,  or  to  the
        terminal.
  RREF  RREF(A) is the reduced row echelon form of the  rectangular
        matrix.  RREF(A,B) is the same as RREF(<A,B>) .
  ROOTS Find polynomial roots.  ROOTS(C)  computes the roots of the
        polynomial  whose  coefficients  are  the  elements  of the
        vector  C .  If  C  has  N+1  components, the polynomial is
        C(1)*X**N + ... + C(N)*X + C(N+1) .  See POLY.
  ROUND ROUND(X)  rounds  the  elements  of   X   to  the   nearest
        integers.
  SAVE  SAVE('file') stores all the current variables in a file.
        SAVE('file',X) saves only X .  See FILE .
        The variables may be retrieved later by LOAD('file') or  by
        your  own program using the following code for each matrix.
        The lines involving XIMAG may be eliminated  if  everything
        is known to be real.
              attach lunit to 'file'
              REAL or DOUBLE PRECISION XREAL(MMAX,NMAX)
              REAL or DOUBLE PRECISION XIMAG(MMAX,NMAX)
              READ(lunit,101) ID,M,N,IMG
              DO 10 J = 1, N
                 READ(lunit,102) (XREAL(I,J), I=1,M)
                 IF (IMG .NE. 0) READ(lunit,102) (XIMAG(I,J),I=1,M)
           10 CONTINUE
        The formats used are system dependent.  The  following  are
        typical.     See    SUBROUTINE   SAVLOD   in   your   local
        implementation of MATLAB.
          101 FORMAT(4A1,3I4)
          102 FORMAT(4Z18)
          102 FORMAT(4O20)
          102 FORMAT(4D25.18)
  SCHUR Schur decomposition.  <U,T> = SCHUR(X)  produces  an  upper
        triangular  matrix   T , with the eigenvalues of  X  on the
        diagonal, and a unitary matrix  U so that  X =  U*T*U'  and
        U'*U = EYE .  By itself, SCHUR(X) returns  T .
  SHORT See LONG .
  SEMI  Semicolons at the end of  lines  will  cause,  rather  than
        suppress,  printing.   A  second  SEMI restores the initial
        interpretation.
  SIN   SIN(X)  is the sine of  X .  See FUN .
  SIZE  If X is an M by N matrix, then SIZE(X) is <M, N> .
        Can also be used with a multiple assignment,
              <M, N> = SIZE(X) .
  SQRT  SQRT(X)  is the square root of  X .   See  FUN  .   Complex
        results  are  produced  if   X   is  not  positive,  or has
        nonpositive eigenvalues.
  STOP  Use EXIT instead.
  SUM   SUM(X)   is  the  sum  of  all  the  elements   of    X   .
        SUM(DIAG(X))  is the trace of  X .
  SVD   Singular value decomposition.  <U,S,V> = SVD(X)  produces a
        diagonal  matrix  S , of the same dimension as  X  and with
        nonnegative diagonal  elements  in  decreasing  order,  and
        unitary matrices  U  and  V  so that  X = U*S*V' .
        By itself, SVD(X) returns a vector containing the  singular
        values.
        <U,S,V>   =   SVD(X,0)   produces   the   "economy    size"
        decomposition.   If  X  is m by n with m > n, then only the
        first n columns of U are computed and S is n by n .
  TRIL  Lower triangle.  TRIL(X) is the lower triangular part of X.
        TRIL(X,K) is the elements on and below the K-th diagonal of
        X.  K = 0 is the main diagonal, K > 0  is  above  the  main
        diagonal and K < 0 is below the main diagonal.
  TRIU  Upper triangle.  TRIU(X) is the upper triangular part of X.
        TRIU(X,K) is the elements on and above the K-th diagonal of
        X.  K = 0 is the main diagonal, K > 0  is  above  the  main
        diagonal and K < 0 is below the main diagonal.
  USER  Allows personal  Fortran  subroutines  to  be  linked  into
        MATLAB .  The subroutine should have the heading
                 SUBROUTINE USER(A,M,N,S,T)
                 REAL or DOUBLE PRECISION A(M,N),S,T
        The MATLAB statement  Y = USER(X,s,t)  results in a call to
        the  subroutine with a copy of the matrix  X  stored in the
        argument  A , its column and row dimensions in  M  and  N ,
        and  the scalar parameters  s  and  t  stored in  S  and  T
        . If  s and t  are omitted, they are set to  0.0  .   After
        the  return,   A  is stored in  Y .  The dimensions  M  and
        N  may be reset within the subroutine.  The statement  Y  =
        USER(K)  results in a call with M = 1, N = 1  and  A(1,1) =
        FLOAT(K) .  After the subroutine has been written, it  must
        be compiled and linked to the MATLAB object code within the
        local operating system.
  WHAT  Lists commands and functions currently available.
  WHILE Repeat statements an indefinite number of times.
        WHILE expr rop expr, statement, ..., statement, END
        where rop is =, <, >, <=, >=, or <> (not equal) .  The  END
        at  the end of a line may be omitted.  The comma before the
        END may also be omitted.  The commas  may  be  replaced  by
        semicolons   to   avoid   printing.    The  statements  are
        repeatedly executed as long  as  the  indicated  comparison
        between  the  real parts of the first components of the two
        expressions is true.   Example  (assume  a  matrix   A   is
        already defined).
        E = 0*A; F = E + EYE; N = 1;
        WHILE NORM(E+F-E,1) > 0, E = E + F; F = A*F/N; N = N + 1;
        E
  WHO   Lists current variables.
  WHY   Provides succinct answers to any questions.
  //


Get the MATLAB code

Published with MATLAB® R2018a

5 CommentsOldest to Newest

Vinod Kumar Singla replied on : 3 of 5

Here’s what MATLAB has to say about this…

>> why % not ?
To fool the tall good and smart system manager.

and the fun never ends…

Phou replied on : 4 of 5

Great insight from reading over the blog and learning more about the history of MATLAB.

The list of functions and keywords today has expanded beyond the 1981 version of MATLAB, which 71 keywords and functions were mentioned.

This got me wondering, “What would Cleve put in his top 5 favorite MATLAB functions of all time?”.

I try to make an effort to learn some new functions each week, it is quite an ‘aha moment’ when I discover and use ones I’ve yet to learn.

Add A Comment

Your email address will not be published. Required fields are marked *

Preview: hide