LinAlgebraPacks - Revision history

GethinWilliams: /* LAPACK:direct solutions for dense matrices */

2011-10-31T12:20:22Z

LAPACK:direct solutions for dense matrices

2011-10-31T12:19:44Z

Introduction

2011-10-26T16:47:28Z

Details of the LU Decomposition

2011-10-26T16:37:10Z

PetSc: Iterative solutions and sparse matrices

2011-10-26T16:27:00Z

Solving Parabolic Problems using LAPACK

2011-10-26T16:22:02Z

2011-10-26T16:20:45Z

2011-10-26T16:12:47Z

2011-10-26T16:11:53Z

2011-07-11T14:16:27Z

LAPACK:direct solutions for dense matrices

@@ Line 102: / Line 102: @@
 </pre>
-To see how to solve the above matrix example using LAPACK take a look inside '''dgesv-example.f90'''.  First the declarations:
+To see how to solve the above matrix example using LAPACK, look inside '''dgesv-example.f90'''.  First the declarations:
 <source lang="fortran">

@@ Line 4: / Line 4: @@
 =Introduction=
-This tutorial walks through a number of example programs which solve equations using linear algebra packages.  These packages include LAPACK, PETSc, PLASMA and MAGMA.  The examples are typically based upon the solution of a partial differential equation (PDE) using the Finite Difference Method (FDM).  The theory of the FDM is described on [[NumMethodsPDEs]].
+This tutorial walks through a number of example programs which solve equations using linear algebra packages.  These packages include LAPACK, PETSc, PLASMA and MAGMA.  The examples are based upon solutions of partial differential equations (PDEs) using the Finite Difference Method (FDM).  The theory of the FDM is described on [[NumMethodsPDEs]].
 The take-home message for this page is that we--as modellers and natural scientists--should be writing code at and above the level of call to a routine from a linear algebra library and ''we should not write code lower than this level.''  Why not?  Well, our code will be slower (less optimised) and more likely to contain bugs.  We will have wasted our time when we could have been focussing on the higher-level task of doing the science.

@@ Line 434: / Line 434: @@
 \end{bmatrix}
 </math>
 =Solving Parabolic Problems using LAPACK=

@@ Line 543: / Line 543: @@
 <pre>
 make ex1f
-./ex1f
+qsub mpi_submit
 </pre>
 We can again plot a graph of timings given varying N:

@@ Line 437: / Line 437: @@
 =Solving Parabolic Problems using LAPACK=
-Our second example uses a pre-computed LU decomposition when solving a parabolic equation using the Crank-Nicolson scheme.
+In our second example, we'll see 3 different schemes for solving a parabolic problem--forward Euler, backward Euler and Crank-Nicolson.  The forward Euler scheme is explicit and so no system of linear equations is solved.  It is a appealingly simple scheme, but can suffer from unstable solutions under certain conditions.  In both the backward Euler and Crank-Nicolson schemes, we will see the (repeated) use of a pre-computed LU decomposition when solving a parabolic equation.
 <pre>
@@ Line 482: / Line 482: @@
 '''Exercises'''
-* example2 also contains simple programs to compute the forward- and backward-Euler schemes using LAPACK subroutine calls (see [[NumMethodsPDEs]]).  Experiment with values of <math>r=\frac{k}{h^2}</math> for the 3 schemes, so as to highlight the effects of the CFL condition.
+* Experiment with values of <math>r=\frac{k}{h^2}</math> for the 3 schemes, so as to highlight the effects of the CFL condition.
 =Banded Matrices=

@@ Line 197: / Line 197: @@
 * Play around with '''mxm-laplace.f90'''.  Try different domain sizes, and different boundary conditions.
-=Direct Solve using LU Decomposition=
+=Details of the LU Decomposition=
 In practice, the LAPACK routine '''dgesv''' does not perform Gaussian elimination, but rather computes an '''LU decomposition''' and then solves the equation using two substitution steps--one forward and one backward.  This is because for the same computational cost, we can factor '''A''' without knowing about '''b'''.  We can then find '''x''' cheaply, for many different '''b'''.  This is useful when solving parabolic equations, where '''x''' at time-step <math>j</math> forms the basis for '''b''' at time-step <math>j+1</math>.

@@ Line 170: / Line 170: @@
 ./dgesv-example.exe
 </pre>
 Next, take a look at '''simple-laplace.f90'''.  This program solves the 2D laplace equation for a 4x4 domain, as described on [[NumMethodsPDEs]].  This program only differs from the previous one in way in which we initialise the matrix '''A''' and vector '''b'''.  To run the program type:
@@ Line 433: / Line 435: @@
 </math>
-=Crank-Nicolson=
+=Solving Parabolic Problems using LAPACK=
 Our second example uses a pre-computed LU decomposition when solving a parabolic equation using the Crank-Nicolson scheme.

@@ Line 188: / Line 188: @@
 ./mxm-laplace.exe
 </pre>
 =Direct Solve using LU Decomposition=