shorter lines

Author: tobydriscoll

src/chapter08.jl

@@ -1,8 +1,9 @@
 """
-    poweriter(A,numiter)
+poweriter(A,numiter)
 
-Perform `numiter` power iterations with the matrix `A`, starting from a random vector,
-and return a vector of eigenvalue estimates and the final eigenvector approximation.
+Perform `numiter` power iterations with the matrix `A`, starting
+from a random vector, and return a vector of eigenvalue estimates
+and the final eigenvector approximation.
 """
 function poweriter(A,numiter)
     n = size(A,1)
@@ -19,11 +20,11 @@ function poweriter(A,numiter)
 end
 
 """
-    inviter(A,s,numiter)
+inviter(A,s,numiter)
 
-    Perform `numiter` inverse iterations with the matrix `A` and shift `s`, starting
-    from a random vector, and return a vector of eigenvalue estimates and the final
-    eigenvector approximation.
+Perform `numiter` inverse iterations with the matrix `A` and shift
+`s`, starting from a random vector, and return a vector of
+eigenvalue estimates and the final eigenvector approximation.
 """
 function inviter(A,s,numiter)
     n = size(A,1)
@@ -41,11 +42,11 @@ function inviter(A,s,numiter)
 end
 
 """
-    arnoldi(A,u,m)
+arnoldi(A,u,m)
 
-Perform the Arnoldi iteration for `A` starting with vector `u`, out to the Krylov
-subspace of degree `m`. Return the orthonormal basis (`m`+1 columns) and the upper
-Hessenberg `H` of size `m`+1 by `m`.
+Perform the Arnoldi iteration for `A` starting with vector `u`, out
+to the Krylov subspace of degree `m`. Return the orthonormal basis
+(`m`+1 columns) and the upper Hessenberg `H` of size `m`+1 by `m`.
 """
 function arnoldi(A,u,m)
     n = length(u)
@@ -69,11 +70,11 @@ function arnoldi(A,u,m)
 end
 
 """
-    arngmres(A,b,m)
+arngmres(A,b,m)
 
-Do `m` iterations of GMRES for the linear system `A`*x=`b`. Return the final solution
-estimate x and a vector with the history of residual norms. (This function is for
-demo only, not practical use.)
+Do `m` iterations of GMRES for the linear system `A`*x=`b`. Return
+the final solution estimate x and a vector with the history of
+residual norms. (This function is for demo only, not practical use.)
 """
 function arngmres(A,b,m)
     n = length(b)
@@ -105,10 +106,13 @@ function arngmres(A,b,m)
     return x,residual
 end
 
+# These are not part of the text proper, but replace a MATLAB
+# function to generate random sparse symmetric matrices.
+
 function sprandsym(n,density,lambda::Vector)
     function randjr(A)
         # Random Jacobi rotation similarity transformation.
-        theta = 2pi*rand()
+        theta = 2π*rand()
         c,s = cos(theta),sin(theta)
         i,j = rand(1:n,2)
         A[[i,j],:] = [c s;-s c]*A[[i,j],:]

src/chapter09.jl

@@ -1,8 +1,8 @@
 """
-    polyinterp(t,y)
+polyinterp(t,y)
 
-Return a callable polynomial interpolant through the points in vectors `t`,`y`. Uses
-the barycentric interpolation formula.
+Return a callable polynomial interpolant through the points in
+vectors `t`,`y`. Uses the barycentric interpolation formula.
 """
 function polyinterp(t,y)
     n = length(t)-1
@@ -33,9 +33,10 @@ function polyinterp(t,y)
 end
 
 """
-    triginterp(t,y)
+triginterp(t,y)
 
-Return trigonometric interpolant for points defined by vectors `t` and `y`.
+Return trigonometric interpolant for points defined by vectors `t`
+and `y`.
 """
 function triginterp(t,y)
     N = length(t)
@@ -59,10 +60,11 @@ function triginterp(t,y)
 end
 
 """
-    ccint(f,n)
+ccint(f,n)
 
-Perform Clenshaw-Curtis integration for the function `f` on `n`+1 nodes in [-1,1]. Return
-integral and a vector of the nodes used. Note: `n` must be even.
+Perform Clenshaw-Curtis integration for the function `f` on `n`+1
+nodes in [-1,1]. Return integral and a vector of the nodes used.
+Note: `n` must be even.
 """
 function ccint(f,n)
     # Find Chebyshev extreme nodes.
@@ -82,10 +84,10 @@ function ccint(f,n)
 end
 
 """
-    glint(f,n)
+glint(f,n)
 
-Perform Gauss-Legendre integration for the function `f` on `n` nodes in (-1,1). Return
-integral and a vector of the nodes used.
+Perform Gauss-Legendre integration for the function `f` on `n` nodes
+in (-1,1). Return integral and a vector of the nodes used.
 """
 function glint(f,n)
     # Nodes and weights are found via a tridiagonal eigenvalue problem.
@@ -102,11 +104,11 @@ function glint(f,n)
 end
 
 """
-    intde(f,h,M)
+intde(f,h,M)
 
-Perform doubly-exponential integration of function `f` over (-Inf,Inf), using
-discretization size `h` and truncation point `M`. Return integral and a vector of the
-nodes used.
+Perform doubly-exponential integration of function `f` over
+(-Inf,Inf), using discretization size `h` and truncation point `M`.
+Return integral and a vector of the nodes used.
 """
 function intde(f,h,M)
     # Find where to truncate the trapezoid sum.
@@ -122,11 +124,11 @@ function intde(f,h,M)
 end
 
 """
-    intsing(f,h,delta)
+intsing(f,h,delta)
 
-Integrate function `f` (possibly singular at 1 and -1) over [-1+`delta`,1-`delta`]
-using discretization size `h`. Return integral and a vector of the
-nodes used.
+Integrate function `f` (possibly singular at 1 and -1) over
+[-1+`delta`,1-`delta`] using discretization size `h`. Return
+integral and a vector of the nodes used.
 """
 function intsing(f,h,delta)
     # Find where to truncate the trapezoid sum.

src/chapter10.jl

@@ -1,15 +1,17 @@
 """
 shoot(phi,xspan,lval,lder,rval,rder,init)
 
-Use the shooting method to solve a two-point boundary value problem. The ODE is
-u'' = `phi`(x,u,u') for x in `xspan`. Specify a function value or derivative at
-the left endpoint using `lval` and `lder`, respectively, and similarly for the
-right endpoint  using `rval` and `rder`. (Use an empty array to denote an
-unknown quantity.) The value `init` is an initial guess for whichever value is
-missing at the left endpoint.
-
-Return vectors for the nodes, the values of u, and the values of u'.
-    """
+Use the shooting method to solve a two-point boundary value problem.
+The ODE is u'' = `phi`(x,u,u') for x in `xspan`. Specify a function
+value or derivative at the left endpoint using `lval` and `lder`,
+respectively, and similarly for the right endpoint  using `rval` and
+`rder`. (Use an empty array to denote an unknown quantity.) The
+value `init` is an initial guess for whichever value is missing at
+the left endpoint.
+
+Returns vectors for the nodes, the values of u, and the values of
+u'.
+"""
 function shoot(phi,xspan,lval,lder,rval,rder,init)
 
     # Tolerances for IVP solver and rootfinder.
@@ -46,11 +48,11 @@ function shoot(phi,xspan,lval,lder,rval,rder,init)
 end
 
 """
-    diffmat2(n,xspan)
+diffmat2(n,xspan)
 
-Compute 2nd-order-accurate differentiation matrices on `n`+1 points in the
-interval `xspan`. Return a vector of nodes, and the matrices for the first
-and second derivatives.
+Compute 2nd-order-accurate differentiation matrices on `n`+1 points
+in the interval `xspan`. Return a vector of nodes, and the matrices
+for the first and second derivatives.
 """
 function diffmat2(n,xspan)
     a,b = xspan
@@ -79,11 +81,11 @@ function diffmat2(n,xspan)
 end
 
 """
-    diffcheb(n,xspan)
+diffcheb(n,xspan)
 
 Compute Chebyshev differentiation matrices on `n`+1 points in the
-interval `xspan`. Return a vector of nodes, and the matrices for the first
-and second derivatives.
+interval `xspan`. Return a vector of nodes, and the matrices for the
+first and second derivatives.
 """
 function diffcheb(n,xspan)
     x = [ -cos( k*pi/n ) for k=0:n ]    # nodes in [-1,1]
@@ -110,14 +112,14 @@ function diffcheb(n,xspan)
 end
 
 """
-     bvplin(p,q,r,xspan,lval,rval,n)
+bvplin(p,q,r,xspan,lval,rval,n)
 
-Use finite differences to solve a linear bopundary value problem. The ODE is
-u''+`p`(x)u'+`q`(x)u = `r`(x) on the interval `xspan`, with endpoint function
-values given as `lval` and `rval`. There will be `n`+1 equally spaced nodes,
-including the endpoints.
+Use finite differences to solve a linear bopundary value problem.
+The ODE is u''+`p`(x)u'+`q`(x)u = `r`(x) on the interval `xspan`,
+with endpoint function values given as `lval` and `rval`. There will
+be `n`+1 equally spaced nodes, including the endpoints.
 
-Return vectors of the nodes and the solution values.
+Returns vectors of the nodes and the solution values.
 """
 function bvplin(p,q,r,xspan,lval,rval,n)
     x,Dx,Dxx = diffmat2(n,xspan)
@@ -138,16 +140,17 @@ function bvplin(p,q,r,xspan,lval,rval,n)
 end
 
 """
-    bvp(phi,xspan,lval,lder,rval,rder,init)
+bvp(phi,xspan,lval,lder,rval,rder,init)
 
-Use finite differences to solve a two-point boundary value problem. The ODE is
-u'' = `phi`(x,u,u') for x in `xspan`. Specify a function value or derivative at
-the left endpoint using `lval` and `lder`, respectively, and similarly for the
-right endpoint  using `rval` and `rder`. (Use an empty array to denote an
-unknown quantity.) The value `init` is an initial guess for whichever value is
-missing at the left endpoint.
+Use finite differences to solve a two-point boundary value problem.
+The ODE is u'' = `phi`(x,u,u') for x in `xspan`. Specify a function
+value or derivative at the left endpoint using `lval` and `lder`,
+respectively, and similarly for the right endpoint  using `rval` and
+`rder`. (Use an empty array to denote an unknown quantity.) The
+value `init` is an initial guess for whichever value is missing at
+the left endpoint.
 
-Return vectors for the nodes and the values of u.
+Returns vectors for the nodes and the values of u.
 """
 function bvp(phi,xspan,lval,lder,rval,rder,init)
     n = length(init) - 1
@@ -172,12 +175,12 @@ function bvp(phi,xspan,lval,lder,rval,rder,init)
 end
 
 """
-    fem(c,s,f,a,b,n)
+fem(c,s,f,a,b,n)
 
-Use a piecewise linear finite element method to solve a two-point boundary
-value problem. The ODE is (`c`(x)u')' + `s`(x)u = `f`(x) on the interval
-[`a`,`b`], and the boundary values are zero. The discretization uses `n` equal
-subintervals.
+Use a piecewise linear finite element method to solve a two-point
+boundary value problem. The ODE is (`c`(x)u')' + `s`(x)u = `f`(x) on
+the interval [`a`,`b`], and the boundary values are zero. The
+discretization uses `n` equal subintervals.
 
 Return vectors for the nodes and the values of u.
 """

src/chapter11.jl

@@ -1,9 +1,10 @@
 """
-    diffper(n,xspan)
+diffper(n,xspan)
 
-Construct 2nd-order differentiation matrices for functions with periodic end
-conditions, using `n` unique nodes in the interval `xspan`. Return a vector of
-nodes and the  matrices for the first and second derivatives.
+Construct 2nd-order differentiation matrices for functions with
+periodic end conditions, using `n` unique nodes in the interval
+`xspan`. Return a vector of nodes and the  matrices for the first
+and second derivatives.
 """
 function diffper(n,xspan)
     a,b = xspan

src/chapter13.jl

@@ -1,20 +1,20 @@
 """
-    ndgrid(x,y,...)
+ndgrid(x,y,...)
 
-For d vector inputs, return d matrices representing the coordinate functions on the
-tensor product grid.
+For d vector inputs, return d matrices representing the coordinate
+functions on the tensor product grid.
 """
 function ndgrid(x...)
     I = CartesianIndices( fill(undef,length.(x)) )
     [ [ x[d][i[d]] for i in I]  for d = 1:length(x) ]
 end
 
 """
-    rectdisc(m,xspan,n,yspan)
+rectdisc(m,xspan,n,yspan)
 
-Create matrices and helpers for finite-difference discretization of a rectangle that is
-the tensor  product of intervals `xspan` and `yspan`, using `m`+1 and `n`+1 points in
-the two coordinates.
+Create matrices and helpers for finite-difference discretization of
+a rectangle that is the tensor  product of intervals `xspan` and
+`yspan`, using `m`+1 and `n`+1 points in the two coordinates.
 """
 function rectdisc(m,xspan,n,yspan)
     # Initialize grid and finite differences.
@@ -40,14 +40,16 @@ function rectdisc(m,xspan,n,yspan)
 end
 
 """
-    poissonfd(f,g,m,xspan,n,yspan)
+poissonfd(f,g,m,xspan,n,yspan)
 
-Solve Poisson's equation on a rectangle by finite differences. Function `f` is the
-forcing function and function `g` gives the  Dirichlet boundary condition. The rectangle
-is the tensor product of intervals `xspan` and `yspan`,  and the discretization uses
-`m`+1 and `n`+1 points in the two coordinates.
+Solve Poisson's equation on a rectangle by finite differences.
+Function `f` is the forcing function and function `g` gives the
+Dirichlet boundary condition. The rectangle is the tensor product of
+intervals `xspan` and `yspan`,  and the discretization uses `m`+1
+and `n`+1 points in the two coordinates.
 
-Return matrices of the solution values, and the coordinate functions, on the grid.
+Returns matrices of the solution values, and the coordinate
+functions, on the grid.
 """
 function poissonfd(f,g,m,xspan,n,yspan)
     # Initialize the rectangle discretization.
@@ -70,13 +72,15 @@ function poissonfd(f,g,m,xspan,n,yspan)
 end
 
 """
-    newtonpde(f,g,m,xspan,n,yspan)
+newtonpde(f,g,m,xspan,n,yspan)
 
-Newton's method with finite differences to solve the PDE `f`(u,x,y,disc)=0 on the
-rectangle `xspan` ``\times`` `yspan`, subject to `g`(x,y)=0 on the boundary. Use `m`+1
-points in x by `n`+1 points in y.
+Newton's method with finite differences to solve the PDE
+`f`(u,x,y,disc)=0 on the rectangle `xspan` ``\times`` `yspan`,
+subject to `g`(x,y)=0 on the boundary. Use `m`+1 points in x by
+`n`+1 points in y.
 
-Return matrices of the solution values, and the coordinate functions, on the grid.
+Returns matrices of the solution values, and the coordinate
+functions, on the grid.
 """
 
 function newtonpde(f,g,m,xspan,n,yspan)