Appendix.  The HELP document

NEWS MATLAB NEWS dated 3/06/91.

 This is a maintenance release of Amiga Matlab.  This release has been
 reorganized and debugged, and should prove much more solid than previous
 releases.  This release was reorganized by Mike Broida.  Bug fixes run
 in the hundreds. This release has also been compiled for A2500 and A3000.
 This release will do 3D Plots, X-Y plots, Contour plots, Polar plots, 
 and Histograms.  The Plot package is based on DigLib, written for PDP-11
 and VAX by Hal Brand, ported to Amiga by Dr. Craig Wuest of LLNL,
 debugged and enhanced by Jim Locker and Mike Broida.
 
 If you would like a copy of MATLAB source, Diglib source, and the users 
 guide, send two 3.5 disks and $5.00 american to:

  Jim Locker
  4443 N. Hyland Ave.
  Dayton Oh 45424
 
 If you are already on my list of contributors, the $5.00 is waived, just 
 send the disks.

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 Jim Locker.
 See the HELP NEWS section.

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,'filename') enters a menu-driven plot routine.
 X is any M by N matrix.  'filename' is optional.  Filename is
 the name of a file which contains stored plot definitions. If
 'filename' is present, then PLOT will operate in batch mode,
 automatically loading and executing the stored plot definitions
 with the matrix X.  As of the 19 Jan 1989 release, the PLOT batch
 feature does not work, for reasons unknown.  For further info-
 rmation, see the Matlab Plot manual.

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.
