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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.