docs(matrices): update matrices doc

Author: AnzhiZhang

Diff for: src/matrices/binomial.jl

@@ -3,12 +3,10 @@ Binomial Matrix
 ===============
 The matrix is a multiple of an involutory matrix.
 
-*Input options:*
-
-+ dim: the dimension of the matrix.
-
-*References:*
+# Input Options
+- dim: the dimension of the matrix.
 
+# References
 **G. Boyd, C.A. Micchelli, G. Strang and D.X. Zhou**,
 Binomial matrices, Adv. in Comput. Math., 14 (2001), pp 379-391.
 """

Diff for: src/matrices/cauchy.jl

@@ -4,18 +4,13 @@ Cauchy Matrix
 Given two vectors `x` and `y`, the `(i,j)` entry of the Cauchy matrix is
 `1/(x[i]+y[j])`.
 
-*Input options*:
-
- + x: an integer, as vectors 1:x and 1:x.
-
- + x, y: two integers, as vectors 1:x and 1:y.
-
- + x: a vector. `y` defaults to `x`.
-
- + x, y: two vectors.
-
-*References:*
+# Input Options
+- x: an integer, as vectors 1:x and 1:x.
+- x, y: two integers, as vectors 1:x and 1:y.
+- x: a vector. `y` defaults to `x`.
+- x, y: two vectors.
 
+# References
 **N. J. Higham**, Accuracy and Stability of Numerical Algorithms,
 second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA,
 2002; sec. 28.1

Diff for: src/matrices/chebspec.jl

@@ -6,15 +6,12 @@ If `k = 0`,the generated matrix is nilpotent and a vector with
         matrix is nonsingular and well-conditioned. Its eigenvalues
         have negative real parts.
 
-*Input options:*
-
-+ dim, k: `dim` is the dimension of the matrix and
+# Input Options
+- dim, k: `dim` is the dimension of the matrix and
         `k = 0 or 1`.
+- dim: `k=0`.
 
-+ dim: `k=0`.
-
-*References:*
-
+# References
 **L. N. Trefethen and M. R. Trummer**, An instability
         phenomenon in spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023.
 """

Diff for: src/matrices/chow.jl

@@ -3,16 +3,13 @@ Chow Matrix
 ===========
 The Chow matrix is a singular Toeplitz lower Hessenberg matrix.
 
-*Input options:*
-
-+ dim, alpha, delta: `dim` is dimension of the matrix.
+# Input Options
+- dim, alpha, delta: `dim` is dimension of the matrix.
             `alpha`, `delta` are scalars such that `A[i,i] = alpha + delta` and
             `A[i,j] = alpha^(i + 1 -j)` for `j + 1 <= i`.
+- dim: `alpha = 1`, `delta = 0`.
 
-+ dim: `alpha = 1`, `delta = 0`.
-
-*References:*
-
+# References
 **T. S. Chow**, A class of Hessenberg matrices with known
                 eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395.
 """

Diff for: src/matrices/circulant.jl

@@ -5,14 +5,11 @@ A circulant matrix has the property that each row is obtained
 by cyclically permuting the entries of the previous row one
 step forward.
 
-*Input options:*
-
-+ vec: a vector.
-
-+ dim: an integer, as vector 1:dim.
-
-*References:*
+# Input Options
+- vec: a vector.
+- dim: an integer, as vector 1:dim.
 
+# References
 **P. J. Davis**, Circulant Matrices, John Wiley, 1977.
 """
 struct Circulant{T<:Number} <: AbstractMatrix{T}

Diff for: src/matrices/clement.jl

@@ -4,16 +4,13 @@ Clement Matrix
 The Clement matrix is a tridiagonal matrix with zero
         diagonal entries. If k = 1, the matrix is symmetric.
 
-*Input options:*
-
-+ dim, k: `dim` is the dimension of the matrix.
+# Input Options
+- dim, k: `dim` is the dimension of the matrix.
         If `k = 0`, the matrix is of type `Tridiagonal`.
         If `k = 1`, the matrix is of type `SymTridiagonal`.
+- dim: `k = 0`.
 
-+ dim: `k = 0`.
-
-*References:*
-
+# References
 **P. A. Clement**, A class of triple-diagonal
         matrices for test purposes, SIAM Review, 1 (1959), pp. 50-52.
 """

Diff for: src/matrices/companion.jl

@@ -8,13 +8,10 @@ The companion matrix to a monic polynomial
     is the n-by-n matrix with ones on the subdiagonal and
     the last column given by the coefficients of `a(x)`.
 
-*Input options:*
-
-+ vec: `vec` is a vector of coefficients.
-
-+ dim: `vec = [1:dim;]`. `dim` is the dimension of the matrix.
-
-+ polynomial: `polynomial` is a polynomial. vector will be appropriate values from coefficients.
+# Input Options
+- vec: `vec` is a vector of coefficients.
+- dim: `vec = [1:dim;]`. `dim` is the dimension of the matrix.
+- polynomial: `polynomial` is a polynomial. vector will be appropriate values from coefficients.
 """
 struct Companion{T<:Number} <: AbstractMatrix{T}
     n::Integer

Diff for: src/matrices/dingdong.jl

@@ -5,12 +5,10 @@ The Dingdong matrix is a symmetric Hankel matrix invented
 by DR. F. N. Ris of IBM, Thomas J Watson Research Centre.
 The eigenvalues cluster around `π/2` and `-π/2`.
 
-*Input options:*
-
-+ dim: the dimension of the matrix.
-
-*References:*
+# Input Options
+- dim: the dimension of the matrix.
 
+# References
 **J. C. Nash**, Compact Numerical Methods for
 Computers: Linear Algebra and Function Minimisation,
 second edition, Adam Hilger, Bristol, 1990 (Appendix 1).

Diff for: src/matrices/fiedler.jl

@@ -4,14 +4,11 @@ Fiedler Matrix
 The Fiedler matrix is symmetric matrix with a dominant
       positive eigenvalue and all the other eigenvalues are negative.
 
-*Input options:*
-
-+ vec: a vector.
-
-+ dim: `dim` is the dimension of the matrix. `vec=[1:dim;]`.
-
-*References:*
+# Input Options
+- vec: a vector.
+- dim: `dim` is the dimension of the matrix. `vec=[1:dim;]`.
 
+# References
 **G. Szego**, Solution to problem 3705, Amer. Math.
             Monthly, 43 (1936), pp. 246-259.
 

Diff for: src/matrices/forsythe.jl

@@ -4,12 +4,10 @@ Forsythe Matrix
 The Forsythe matrix is a n-by-n perturbed Jordan block.
 This generator is adapted from N. J. Higham's Test Matrix Toolbox.
 
-*Input options:*
-
-+ dim, alpha, lambda: `dim` is the dimension of the matrix.
+# Input Options
+- dim, alpha, lambda: `dim` is the dimension of the matrix.
     `alpha` and `lambda` are scalars.
-
-+ dim: `alpha = sqrt(eps(type))` and `lambda = 0`.
+- dim: `alpha = sqrt(eps(type))` and `lambda = 0`.
 """
 struct Forsythe{T<:Number} <: AbstractMatrix{T}
     n::Integer

Diff for: src/matrices/frank.jl

@@ -5,15 +5,12 @@ The Frank matrix is an upper Hessenberg matrix with
 determinant 1. The eigenvalues are real, positive and
 very ill conditioned.
 
-*Input options:*
-
-+ dim, k: `dim` is the dimension of the matrix, `k = 0 or 1`.
+# Input Options
+- dim, k: `dim` is the dimension of the matrix, `k = 0 or 1`.
     If `k = 1` the matrix reflect about the anti-diagonal.
+- dim: the dimension of the matrix.
 
-+ dim: the dimension of the matrix.
-
-*References:*
-
+# References
 **W. L. Frank**, Computing eigenvalues of complex matrices
     by determinant evaluation and by methods of Danilewski and Wielandt,
     J. Soc. Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388).

Diff for: src/matrices/golub.jl

@@ -5,12 +5,10 @@ Golub matrix is the product of two random unit lower and upper
     triangular matrices respectively. LU factorization without pivoting
     fails to reveal that such matrices are badly conditioned.
 
-*Input options:*
-
-+ dim: the dimension of the matrix.
-
-*References:*
+# Input Options
+- dim: the dimension of the matrix.
 
+# References
 **D. Viswanath and N. Trefethen**. Condition Numbers of
     Random Triangular Matrices, SIAM J. Matrix Anal. Appl. 19, 564-581,
     1998.

Diff for: src/matrices/grcar.jl

@@ -4,15 +4,12 @@ Grcar Matrix
 The Grcar matrix is a Toeplitz matrix with sensitive
 eigenvalues.
 
-*Input options:*
-
-+ dim, k: `dim` is the dimension of the matrix and
+# Input Options
+- dim, k: `dim` is the dimension of the matrix and
     `k` is the number of superdiagonals.
+- dim: the dimension of the matrix.
 
-+ dim: the dimension of the matrix.
-
-*References:*
-
+# References
 **J. F. Grcar**, Operator coefficient methods
     for linear equations, Report SAND89-8691, Sandia National
     Laboratories, Albuquerque, New Mexico, 1989 (Appendix 2).

Diff for: src/matrices/hadamard.jl

@@ -5,12 +5,10 @@ The Hadamard matrix is a square matrix whose entries are
 1 or -1. It was named after Jacques Hadamard. The rows of
 a Hadamard matrix are orthogonal.
 
-*Input options:*
-
-+ dim: the dimension of the matrix, `dim` is a power of 2.
-
-*References:*
+# Input Options
+- dim: the dimension of the matrix, `dim` is a power of 2.
 
+# References
 **S. W. Golomb and L. D. Baumert**, The search for
 Hadamard matrices, Amer. Math. Monthly, 70 (1963) pp. 12-17
 """

Diff for: src/matrices/hankel.jl

@@ -4,15 +4,12 @@ Hankel Matrix
 A Hankel matrix is a matrix that is symmetric and constant
                 across the anti-diagonals.
 
-*Input options:*
-
-+ vc, vr: `vc` and `vc` are the first column and last row of the
+# Input Options
+- vc, vr: `vc` and `vc` are the first column and last row of the
        matrix. If the last element of `vc` differs from the first element
                 of `vr`, the last element of `rc` prevails.
-
-+ v: a vector, as `vc = v` and `vr` will be zeros.
-
-+ dim: `dim` is the dimension of the matrix. `v = [1:dim;]`.
+- v: a vector, as `vc = v` and `vr` will be zeros.
+- dim: `dim` is the dimension of the matrix. `v = [1:dim;]`.
 """
 struct Hankel{T<:Number} <: AbstractMatrix{T}
     m::Integer

Diff for: src/matrices/hilbert.jl

@@ -5,14 +5,11 @@ The Hilbert matrix has `(i,j)` element `1/(i+j-1)`. It is
 notorious for being ill conditioned. It is symmetric
 positive definite and totally positive.
 
-*Input options:*
-
-+ dim: the dimension of the matrix.
-
-+ row\\_dim, col\\_dim: the row and column dimensions.
-
-*References:*
+# Input Options
+- dim: the dimension of the matrix.
+- row\\_dim, col\\_dim: the row and column dimensions.
 
+# References
 **M. D. Choi**, Tricks or treats with the Hilbert matrix,
 Amer. Math. Monthly, 90 (1983), pp. 301-312.
 

Diff for: src/matrices/inversehilbert.jl

@@ -1,12 +1,11 @@
 """
 Inverse of the Hilbert Matrix
 =============================
-*Input options:*
 
-+ dim: the dimension of the matrix.
-
-*References:*
+# Input Options
+- dim: the dimension of the matrix.
 
+# References
 **M. D. Choi**, Tricks or treats with the Hilbert matrix,
     Amer. Math. Monthly, 90 (1983), pp. 301-312.
 

Diff for: src/matrices/involutory.jl

@@ -3,12 +3,10 @@ Involutory Matrix
 =================
 An involutory matrix is a matrix that is its own inverse.
 
-*Input options:*
-
-+ dim: `dim` is the dimension of the matrix.
-
-*References:*
+# Input Options
+- dim: `dim` is the dimension of the matrix.
 
+# References
 **A. S. Householder and J. A. Carpenter**, The
         singular values of involutory and of idempotent matrices,
         Numer. Math. 5 (1963), pp. 234-237.

Diff for: src/matrices/kahan.jl

@@ -6,17 +6,13 @@ The Kahan matrix is an upper trapezoidal matrix, i.e., the
     `θ` is `0 < θ < π`. The diagonal is perturbed by
     `pert*eps()*diagm([n:-1:1;])`.
 
-*Input options:*
-
-+ rowdim, coldim, θ, pert: `rowdim` and `coldim` are the row and column
+# Input Options
+- rowdim, coldim, θ, pert: `rowdim` and `coldim` are the row and column
     dimensions of the matrix. `θ` and `pert` are scalars.
+- dim, θ, pert: `dim` is the dimension of the matrix.
+- dim: `θ = 1.2`, `pert = 25`.
 
-+ dim, θ, pert: `dim` is the dimension of the matrix.
-
-+ dim: `θ = 1.2`, `pert = 25`.
-
-*References:*
-
+# References
 **W. Kahan**, Numerical linear algebra, Canadian Math.
     Bulletin, 9 (1966), pp. 757-801.
 """

Diff for: src/matrices/kms.jl

@@ -2,15 +2,12 @@
 Kac-Murdock-Szego Toeplitz matrix
 =================================
 
-*Input options:*
-
-+ dim, rho: `dim` is the dimension of the matrix, `rho` is a
+# Input Options
+- dim, rho: `dim` is the dimension of the matrix, `rho` is a
     scalar such that `A[i,j] = rho^(abs(i-j))`.
+- dim: `rho = 0.5`.
 
-+ dim: `rho = 0.5`.
-
-*References:*
-
+# References
 **W. F. Trench**, Numerical solution of the eigenvalue
     problem for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis
     and Appl., 10 (1989), pp. 135-146 (and see the references therein).

Diff for: src/matrices/lehmer.jl

@@ -5,12 +5,10 @@ The Lehmer matrix is a symmetric positive definite matrix.
             It is totally nonnegative. The inverse is tridiagonal and
             explicitly known
 
-*Input options:*
-
-+ dim: the dimension of the matrix.
-
-*References:*
+# Input Options
+- dim: the dimension of the matrix.
 
+# References
 **M. Newman and J. Todd**, The evaluation of
             matrix inversion programs, J. Soc. Indust. Appl. Math.,
             6 (1958), pp. 466-476.