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F12 Chapter Contents
NAG Library Manual

# NAG Library Chapter IntroductionF12 – Large Scale Eigenproblems

## 1  Scope of the Chapter

This chapter provides routines for computing some eigenvalues and eigenvectors of large-scale (sparse) standard and generalized eigenvalue problems. It provides routines for:
• – solution of symmetric eigenvalue problems;
• – solution of nonsymmetric eigenvalue problems;
• – solution of generalized symmetric-definite eigenvalue problems;
• – solution of generalized nonsymmetric eigenvalue problems;
• – partial singular value decomposition.
Routines are provided for both real and complex data.
The routines in this chapter have all been derived from the ARPACK software suite (see Lehoucq et al. (1998)), a collection of Fortran 77 subroutines designed to solve large scale eigenvalue problems. The interfaces provided in this chapter have been chosen to combine ease of use with the flexibility of the original ARPACK software. The underlying iterative methods and algorithms remain essentially the same as those in ARPACK and are described fully in Lehoucq et al. (1998).
The algorithms used are based upon an algorithmic variant of the Arnoldi process called the Implicitly Restarted Arnoldi Method. For symmetric matrices, this reduces to a variant of the Lanczos process called the Implicitly Restarted Lanczos Method. These variants may be viewed as a synthesis of the Arnoldi/Lanczos process with the Implicitly Shifted $QR$ technique that is suitable for large scale problems. For many standard problems, a matrix factorization is not required. Only the action of the matrix on a vector is needed.

## 2  Background to the Problems

This section is only a brief introduction to the solution of large-scale eigenvalue problems. For a more detailed discussion see, for example, Saad (1992) or Lehoucq (1995) in addition to Lehoucq et al. (1998). The basic factorization techniques and definitions of terms used for the different problem types are given in Section 2 in the F08 Chapter Introduction.

### 2.1  Sparse Matrices and their Storage

A matrix $A$ may be described as sparse if the number of zero elements is so large that it is worthwhile using algorithms which avoid computations involving zero elements.
If $A$ is sparse, and the chosen algorithm requires the matrix coefficients to be stored, a significant saving in storage can often be made by storing only the nonzero elements. A number of different formats may be used to represent sparse matrices economically. These differ according to the amount of storage required, the amount of indirect addressing required for fundamental operations such as matrix-vector products, and their suitability for vector and/or parallel architectures. For a survey of some of these storage formats see Barrett et al. (1994).
Most of the routines in this chapter have been designed to be independent of the matrix storage format. This allows you to choose your own preferred format, or to avoid storing the matrix altogether. Other routines are general purpose, which are easier to use, but are based on fixed storage formats. One such format is currently provided. This is the banded coordinate storage format as used in Chapters F07 and F08 (LAPACK) for storing general banded matrices.

### 2.2  Symmetric Eigenvalue Problems

The symmetric eigenvalue problem is to find the eigenvalues, $\lambda$, and corresponding eigenvectors, $z\ne 0$, such that
 $A z = λ z , A = AT , where  A  is real.$
For the Hermitian eigenvalue problem we have
 $A z = λ z , A = AH , where  A  is complex.$
For both problems the eigenvalues $\lambda$ are real.
The basic task of the symmetric eigenproblem routines is to compute some of the values of $\lambda$ and, optionally, corresponding vectors $z$ for a given matrix $A$. For example, we may wish to obtain the first ten eigenvalues of largest magnitude, of a large sparse matrix $A$.

### 2.3  Generalized Symmetric-definite Eigenvalue Problems

This section is concerned with the solution of the generalized eigenvalue problems $Az=\lambda Bz$, $ABz=\lambda z$, and $BAz=\lambda z$, where $A$ and $B$ are real symmetric or complex Hermitian and $B$ is positive definite. Each of these problems can be reduced to a standard symmetric eigenvalue problem, using a Cholesky factorization of $B$ as either $B=L{L}^{\mathrm{T}}$ or $B={U}^{\mathrm{T}}U$ ($L{L}^{\mathrm{H}}$ or ${U}^{\mathrm{H}}U$ in the Hermitian case).
With $B=L{L}^{\mathrm{T}}$, we have
 $A z = λ B z ⇒ L-1 A L-T LTz = λ LTz .$
Hence the eigenvalues of $Az=\lambda Bz$ are those of $Cy=\lambda y$, where $C$ is the symmetric matrix $C={L}^{-1}A{L}^{-\mathrm{T}}$ and $y={L}^{\mathrm{T}}z$. In the complex, case $C$ is Hermitian with $C={L}^{-1}A{L}^{-\mathrm{H}}$ and $y={L}^{\mathrm{H}}z$.
The basic task of the generalized symmetric eigenproblem routines is to compute some of the values of $\lambda$ and, optionally, corresponding vectors $z$ for a given matrix $A$. For example, we may wish to obtain the first ten eigenvalues of largest magnitude, of a large sparse matrix pair $A$ and $B$.

### 2.4  Nonsymmetric Eigenvalue Problems

The nonsymmetric eigenvalue problem is to find the eigenvalues, $\lambda$, and corresponding eigenvectors, $v\ne 0$, such that
 $A v = λ v .$
More precisely, a vector $v$ as just defined is called a right eigenvector of $A$, and a vector $u\ne 0$ satisfying
 $uT A = λ uT uH A = λ uH when  u  is complex$
is called a left eigenvector of $A$.
A real matrix $A$ may have complex eigenvalues, occurring as complex conjugate pairs.
This problem can be solved via the Schur factorization of $A$, defined in the real case as
 $A = ZT ZT ,$
where $Z$ is an orthogonal matrix and $T$ is an upper quasi-triangular matrix with $1$ by $1$ and $2$ by $2$ diagonal blocks, the $2$ by $2$ blocks corresponding to complex conjugate pairs of eigenvalues of $A$. In the complex case, the Schur factorization is
 $A = Z T ZH ,$
where $Z$ is unitary and $T$ is a complex upper triangular matrix.
The columns of $Z$ are called the Schur vectors. For each $k$ ($1\le k\le n$), the first $k$ columns of $Z$ form an orthonormal basis for the invariant subspace corresponding to the first $k$ eigenvalues on the diagonal of $T$. Because this basis is orthonormal, it is preferable in many applications to compute Schur vectors rather than eigenvectors. It is possible to order the Schur factorization so that any desired set of $k$ eigenvalues occupy the $k$ leading positions on the diagonal of $T$.
The two basic tasks of the nonsymmetric eigenvalue routines are to compute, for a given matrix $A$, some values of $\lambda$ and, if desired, their associated right eigenvectors $v$, and the Schur factorization.

### 2.5  Generalized Nonsymmetric Eigenvalue Problem

The generalized nonsymmetric eigenvalue problem is to find the eigenvalues, $\lambda$, and corresponding eigenvectors, $v\ne 0$, such that
 $A v = λ B v , A B v = λ v , and B A v = λ v .$
More precisely, a vector $v$ as just defined is called a right eigenvector of the matrix pair $\left(A,B\right)$, and a vector $u\ne 0$ satisfying
 $uT A = λ uT B uH A = λ uH B  when  u  is complex$
is called a left eigenvector of the matrix pair $\left(A,B\right)$.

### 2.6  The Singular Value Decomposition

The singular value decomposition (SVD) of an $m$ by $n$ matrix $A$ is given by
 $A=UΣVT, A=UΣVHin the complex case$
where $U$ and $V$ are orthogonal (unitary) and $\Sigma$ is an $m$ by $n$ diagonal matrix with real diagonal elements, ${\sigma }_{i}$, such that
 $σ1≥σ2≥⋯≥σminm,n≥0.$
The ${\sigma }_{i}$ are the singular values of $A$ and the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ columns of $U$ and $V$ are the left and right singular vectors of $A$. The singular values and singular vectors satisfy
 $Avi = σi ui and ATui = σi vi or  AH ui = σi vi so that ATA ui = σi2 ui ( AHA ui = σi2 ui )$
where ${u}_{i}$ and ${v}_{i}$ are the $i$th columns of $U$ and $V$ respectively.
Thus selected singular values and the corresponding right singular vectors may be computed by finding eigenvalues and eigenvectors for the symmetric matrix ${A}^{\mathrm{T}}A$ (or the Hermitian matrix ${A}^{\mathrm{H}}A$ if $A$ is complex).
An alternative approach is to use the relationship
 $0 A AT 0   U V = U V Σ$
and thus compute selected singular values and vectors via the symmetric matrix
 $C = 0 A AT 0   C = 0 A AH 0  if  A  is complex .$
In many applications, one is interested in computing a few (say $k$) of the largest singular values and corresponding vectors. If ${U}_{k}$, ${V}_{k}$ denote the leading $k$ columns of $U$ and $V$ respectively, and if ${\Sigma }_{k}$ denotes the leading principal submatrix of $\Sigma$, then
 $Ak ≡ Uk Σk VTk (or  Uk Σk VHk )$
is the best rank-$k$ approximation to $A$ in both the $2$-norm and the Frobenius norm. Often a very small $k$ will suffice to approximate important features of the original $A$ or to approximately solve least squares problems involving $A$.

### 2.7  Iterative Methods

Iterative methods for the solution of the standard eigenproblem
 $A x = λ x$ (1)
approach the solution through a sequence of approximations until some user-specified termination criterion is met or until some predefined maximum number of iterations has been reached. The number of iterations required for convergence is not generally known in advance, as it depends on the accuracy required, and on the matrix $A$, its sparsity pattern, conditioning and eigenvalue spectrum.

## 3  Recommendations on Choice and Use of Available Routines

### 3.1  Types of Routine Available

The routines available in this chapter divide essentially into three suites of basic reverse communication routines and some general purpose routines for banded systems.
Basic routines are grouped in suites of five, and implement the underlying iterative method. Each suite comprises a setup routine, an options setting routine, a solver routine, a routine to return additional monitoring information and a post-processing routine. The solver routine is independent of the matrix storage format (indeed the matrix need not be stored at all) and the type of preconditioner. It uses reverse communication, i.e., it returns repeatedly to the calling program with the parameter IREVCM set to specified values which require the calling program to carry out a specific task (either to compute a matrix-vector product or to solve the preconditioning equation), to signal the completion of the computation or to allow the calling program to monitor the solution. Reverse communication has the following advantages:
 (i) Maximum flexibility in the representation and storage of sparse matrices. All matrix operations are performed outside the solver routine, thereby avoiding the need for a complicated interface with enough flexibility to cope with all types of storage schemes and sparsity patterns. This also applies to preconditioners. (ii) Enhanced user interaction: you can closely monitor the solution and tidy or immediate termination can be requested. This is useful, for example, when alternative termination criteria are to be employed or in case of failure of the external routines used to perform matrix operations.
At present there are suites of basic routines for real symmetric and nonsymmetric systems, and for complex systems.
General purpose routines call basic routines in order to provide easy-to-use routines for particular sparse matrix storage formats. They are much less flexible than the basic routines, but do not use reverse communication, and may be suitable in many cases.
The structure of this chapter has been designed to cater for as many types of application as possible. If a general purpose routine exists which is suitable for a given application you are recommended to use it. If you then decide you need some additional flexibility it is easy to achieve this by using basic and utility routines which reproduce the algorithm used in the general purpose routine, but allow more access to algorithmic control parameters and monitoring.

### 3.2  Iterative Methods for Real Nonsymmetric and Complex Eigenvalue Problems

The suite of basic routines F12AAF, F12ABF, F12ACF, F12ADF and F12AEF implements the iterative solution of real nonsymmetric eigenvalue problems, finding estimates for a specified spectrum of eigenvalues. These eigenvalue estimates are often referred to as Ritz values and the error bounds obtained are referred to as the Ritz estimates. These routines allow a choice of termination criteria and many other options for specifying the problem type, allow monitoring of the solution process, and can return Ritz estimates of the calculated Ritz values of the problem $A$.
For complex matrices there is an equivalent suite of routines. F12ANF, F12APF, F12AQF, F12ARF and F12ASF are the basic routines which implement corresponding methods used for real nonsymmetric systems. Note that these routines are to be used for both Hermitian and non-Hermitian problems. Occasionally, when using these routines on a complex Hermitian problem, eigenvalues will be returned with small but nonzero imaginary part due to unavoidable round-off errors. These should be ignored unless they are significant with respect to the eigenvalues of largest magnitude that have been computed.
There are general purpose routines for the case where the matrices are known to be banded. In these cases an initialization routine is called first to set up default options, and the problem is solved by a single call to a solver routine. The matrices are supplied, in LAPACK banded-storage format, as arguments to the solver routine. For real general matrices these routines are F12AFF and F12AGF; and for complex matrices the pair is F12ATF and F12AUF. With each pair non-default options can be set, following a call to the initialization routine, using F12ADF for real matrices and F12ARF for complex matrices. For real matrices that can be supplied in the sparse matrix compressed column storage (CCS) format, the driver routine F02EKF is available. This routine uses routines from Chapter F12 in conjunction with direct solver routines from Chapter F11.
There is little computational penalty in using the non-Hermitian complex routines for a Hermitian problem. The only additional cost is to compute eigenvalues of a Hessenberg rather than a tridiagonal matrix. The difference in computational cost should be negligible compared to the overall cost.

### 3.3  Iterative Methods for Real Symmetric Eigenvalue Problems

The suite of basic routines F12FAF, F12FBF, F12FCF, F12FDF and F12FEF implement a Lanczos method for the iterative solution of the real symmetric eigenproblem.
There is a general purpose routine pair for the case where the matrices are known to be banded. In this case an initialization routine, F12FFF, is called first to set up default options, and the problem is solved by a single call to a solver routine, F12FGF. The matrices are supplied, in LAPACK banded-storage format, as arguments to F12FGF. Non-default options can be set, following a call to F12FFF, using F12FDF.

### 3.4  Iterative Methods for Singular Value Decomposition

The partial singular value decomposition, ${A}_{k}$ (as defined in Section 2.6), of an $\left(m×n\right)$ matrix $A$ can be computed efficiently using routines from this chapter. For real matrices, the suite of routines listed in Section 3.3 (for symmetric problems) can be used; for complex matrices, the corresponding suite of routines for complex problems can be used; however, there are no general purpose routines for complex problems.
The driver routine F02WGF is available for computing the partial SVD of real matrices. The matrix is not supplied to F02WGF; rather, a user-defined routine argument provides the results of performing Matrix-vector products.
For both real and complex matrices, you should use the default options (see, for example, the options listed in Section 10 in F12FDF) for problem type (Standard), computational mode (Regular) and spectrum (Largest Magnitude). The operation to be performed on request by the reverse communication routine (e.g., F12FBF) is, for real matrices, to multiply the returned vector by the symmetric matrix ${A}^{\mathrm{T}}A$ if $m\ge n$, or by $A{A}^{\mathrm{T}}$ if $m. For complex matrices, the corresponding Hermitian matrices are ${A}^{\mathrm{H}}A$ and $A{A}^{\mathrm{H}}$.
The right ($m\ge n$) or left ($m) singular vectors are returned by the post-processing routine (e.g., F12FCF). The left (or right) singular vectors can be recovered from the returned singular vectors. Providing the largest singular vectors are not multiple or tightly clustered, there should be no problem in obtaining numerically orthogonal left singular vectors from the computed right singular vectors (or vice versa).
The second example in Section 9 in F12FBF illustrates how the partial singular value decomposition of a real matrix can be performed using the suite of routines for finding some eigenvalues of a real symmetric matrix. In this case $m\ge n$, however, the program is easily amended to perform the same task in the case $m.
Similarly, routines in this chapter may be used to estimate the $2$-norm condition number,
 $K2A = σ1 σn .$
This can be achieved by setting the option Both Ends to get the largest and smallest few singular values, then taking the ratio of largest to smallest computed singular values as your estimate.

### 3.5  Alternative Methods

Other routines for the solution of sparse linear eigenproblems can currently be found in Chapters F02 and F08. In particular, the following routines allow the direct solution of real symmetric systems:
 Band F02SDF and F08UAF (DSBGV) Sparse F02FJF

## 4  General Use of Routines

This section will describe the complete structure of the reverse communication interfaces contained in this chapter. Numerous computational modes are available, including several shift-invert strategies designed to accelerate convergence. Two of the more sophisticated modes will be described in detail. The remaining ones are quite similar in principle, but require slightly different tasks to be performed with the reverse communication interface.
This chapter is structured as follows. The naming conventions used in this chapter, and the data types available are described in Section 4.1, spectral transformations are discussed in Section 4.2. Spectral transformations are usually extremely effective but there are a number of problem dependent issues that determine which one to use. In Section 4.3 we describe the reverse communication interface needed to exercise the various shift-invert options. Each shift-invert option is specified as a computational mode and all of these are summarised in the remaining sections. There is a subsection for each problem type and hence these sections are quite similar and repetitive. Once the basic idea is understood, it is probably best to turn directly to the subsection that describes the problem setting that is most interesting to you.
Perhaps the easiest way to rapidly become acquainted with the modes in this chapter is to run each of the example programs which use the various modes. These may be used as templates and adapted to solve specific problems.

### 4.1  Naming Conventions

Routines for solving nonsymmetric (real and complex) eigenvalue problems have as first letter after the chapter name, the letter ‘A’, e.g., F12ABF; equivalent routines for symmetric eigenvalue problems will have this letter replaced by the letter ‘F’, e.g., F12FBF. For the letter following this, routines for real eigenvalue problems will have letters in the range ‘A to M’ while those for complex eigenvalue problems will have letters correspondingly shifted into the range ‘N to Z’; so, for example, the complex equivalent of F12ADF is F12ARF, while the real symmetric equivalent is F12FDF.
A suite of five routines are named consecutively, e.g., F12AAF, F12ABF, F12ACF, F12ADF and F12AEF. Each general purpose routine has its own initialization routine, but uses the option setting routine from the suite relevant to the problem type. Thus each general purpose routine can be viewed as belonging to a suite of three routines, even though only two routines will be named consecutively. For example, F12ADF, F12AFF and F12AGF represent the suite of routines for solving a banded real symmetric eigenvalue problem.

### 4.2  Shift and Invert Spectral Transformations

The most general problem that may be solved here is to compute a few selected eigenvalues and corresponding eigenvectors for
 $A x = λ B x , where  A  and  B  are real or complex  n × n  matrices.$ (2)
The shift and invert spectral transformation is used to enhance convergence to a desired portion of the spectrum. If $\left(x,\lambda \right)$ is an eigen-pair for $\left(A,B\right)$ and $\sigma \ne \lambda$ then
 $A - σB -1 B x = ν x , where  ν = 1 λ - σ .$ (3)
This transformation is effective for finding eigenvalues near $\sigma$ since the ${n}_{\nu }$ eigenvalues of $C\equiv {\left(A-\sigma B\right)}^{-1}B$ that are largest in magnitude correspond to the ${n}_{\nu }$ eigenvalues ${\lambda }_{j}$ of the original problem that are nearest to the shift $\sigma$ in absolute value. These transformed eigenvalues of largest magnitude are precisely the eigenvalues that are easy to compute with a Krylov method. (See Barrett et al. (1994)). Once they are found, they may be transformed back to eigenvalues of the original problem. The direct relation is
 $λj = σ + 1 νj$
and the eigenvector ${x}_{j}$ associated with ${\nu }_{j}$ in the transformed problem is also an eigenvector of the original problem corresponding to ${\lambda }_{j}$. Usually the Arnoldi process will rapidly obtain good approximations to the eigenvalues of $C$ of largest magnitude. However, to implement this transformation, you must provide the means to solve linear systems involving $A-\sigma B$ either with a matrix factorization or with an iterative method.
In general, $C$ will be non-Hermitian even if $A$ and $B$ are both Hermitian. However, this is easily remedied. The assumption that $B$ is Hermitian positive definite implies that the bilinear form
 $x,y ≡ xH By$
is an inner product. If $B$ is positive semidefinite and singular, then a semi-inner product results. This is a weighted $B$-inner product and vectors $x$, $y$ are called $B$-orthogonal if $⟨x,y⟩=0$. It is easy to show that if $A$ is Hermitian (self-adjoint) then $C$ is Hermitian self-adjoint with respect to this $B$-inner product (meaning $⟨Cx,y⟩=⟨x,Cy⟩$ for all vectors $x$, $y$). Therefore, symmetry will be preserved if we force the computed basis vectors to be orthogonal in this $B$-inner product. Implementing this $B$-orthogonality requires you to provide a matrix-vector product $Bv$ on request along with each application of $C$. In the following sections we shall discuss some of the more familiar transformations to the standard eigenproblem. However, when $B$ is positive (semi)definite, we recommend using the shift-invert spectral transformation with $B$-inner products if at all possible. This is a far more robust transformation when $B$ is ill-conditioned or singular. With a little extra manipulation (provided automatically in the post-processing routines) the semi-inner product induced by $B$ prevents corruption of the computed basis vectors by roundoff-error associated with the presence of infinite eigenvalues. These very ill-conditioned eigenvalues are generally associated with a singular or highly ill-conditioned $B$. A detailed discussion of this theory may be found in Chapter 4 of Lehoucq et al. (1998).
Shift-invert spectral transformations are very effective and should even be used on standard problems, $B=I$, whenever possible. This is particularly true when interior eigenvalues are sought or when the desired eigenvalues are clustered. Roughly speaking, a set of eigenvalues is clustered if the maximum distance between any two eigenvalues in that set is much smaller than the minimum distance between these eigenvalues and any other eigenvalues of $\left(A,B\right)$.
If you have a generalized problem $B\ne I$, then you must provide a way to solve linear systems with either $A$, $B$ or a linear combination of the two matrices in order to use the reverse communication suites in this chapter. In this case, a sparse direct method should be used to factor the appropriate matrix whenever possible. The resulting factorization may be used repeatedly to solve the required linear systems once it has been obtained. If instead you decide to use an iterative method, the accuracy of the solutions must be commensurate with the convergence tolerance used for the Arnoldi iteration. A slightly more stringent tolerance is needed relative to the desired accuracy of the eigenvalue calculation.
The main drawback with using the shift-invert spectral transformation is that the coefficient matrix $A-\sigma B$ is typically indefinite in the Hermitian case and has zero-valued eigenvalues in the non-Hermitian case. These are often the most difficult situations for iterative methods and also for sparse direct methods.
The decision to use a spectral transformation on a standard eigenvalue problem $B=I$ or to use one of the simple modes is problem dependent. The simple modes have the advantage that you only need to supply a matrix vector product $Av$. However, this approach is usually only successful for problems where extremal non-clustered eigenvalues are sought. In non-Hermitian problems, extremal means eigenvalues near the boundary of the spectrum of $A$. For Hermitian problems, extremal means eigenvalues at the left- or right-hand end points of the spectrum of $A$. The notion of non-clustered (or well separated) is difficult to define without going into considerable detail. A simplistic notion of a well-separated eigenvalue ${\lambda }_{j}$ for a Hermitian problem would be $‖{\lambda }_{i}-{\lambda }_{j}‖>\tau ‖{\lambda }_{n}-{\lambda }_{1}‖$ for all $j\ne i$ with $\tau \gg \epsilon$, where ${\lambda }_{1}$ and ${\lambda }_{n}$ are the smallest and largest algebraically. Unless a matrix vector product is quite difficult to code or extremely expensive computationally, it is probably worth trying to use the simple mode first if you are seeking extremal eigenvalues.
The remainder of this section discusses additional transformations that may be applied to convert a generalized eigenproblem to a standard eigenproblem. These are appropriate when $B$ is well-conditioned (Hermitian or non-Hermitian).

#### 4.2.1  $B$ is Hermitian positive definite

If $B$ is Hermitian positive definite and well-conditioned ($‖B‖‖{B}^{-1}‖$ is of modest size), then computing the Cholesky factorization $B=L{L}^{\mathrm{H}}$ and converting equation (2) to
 $L-1 A L-H y = λy , where  LH x = y$
provides a transformation to a standard eigenvalue problem. In this case, a request for a matrix vector product would be satisfied with the following three steps:
 (i) Solve ${L}^{\mathrm{H}}z=v$ for $z$. (ii) Matrix-vector multiply $z←Az$. (iii) Solve $Lw=z$ for $w$.
Upon convergence, a computed eigenvector $y$ for $\left({L}^{-1}A{L}^{-\mathrm{H}}\right)$ is converted to an eigenvector $x$ of the original problem by solving the triangular system ${L}^{\mathrm{H}}x=y$. This transformation is most appropriate when $A$ is Hermitian, $B$ is Hermitian positive definite and extremal eigenvalues are sought. This is because when $A$ is Hermitian, so is $\left({L}^{-1}A{L}^{-\mathrm{H}}\right)$.
If $A$ is Hermitian positive definite and the smallest eigenvalues are sought, then it would be best to reverse the roles of $A$ and $B$ in the above description and ask for the largest algebraic eigenvalues or those of largest magnitude. Upon convergence, a computed eigenvalue $\stackrel{^}{\lambda }$ would then be converted to an eigenvalue of the original problem by the relation $\lambda ←\frac{1}{\stackrel{^}{\lambda }}$.

#### 4.2.2  $B$ is not Hermitian positive semidefinite

If neither $A$ nor $B$ is Hermitian positive semidefinite, then a direct transformation to standard form is required. One simple way to obtain a direct transformation of equation (2) to a standard eigenvalue problem $Cx=\lambda x$ is to multiply on the left by ${B}^{-1}$ which results in $C={B}^{-1}A$. Of course, you should not perform this transformation explicitly since it will most likely convert a sparse problem into a dense one. If possible, you should obtain a direct factorization of $B$ and when a matrix-vector product involving $C$ is called for, it may be accomplished with the following two steps:
 (i) Matrix-vector multiply $z←Av$. (ii) Solve $Bw=z$ for $w$.
Several problem-dependent issues may modify this strategy. If $B$ is singular or if you are interested in eigenvalues near a point $\sigma$ then you may choose to work with $C\equiv {\left(A-\sigma B\right)}^{-1}B$ but without using the $B$-inner products discussed previously. In this case you will have to transform the converged eigenvalues of $C$ to eigenvalues of the original problem.

### 4.3  Reverse Communication and Shift-invert Modes

The reverse communication interface routine for real nonsymmetric problems is F12ABF; for complex problems is F12APF; and for real symmetric problems is F12FBF. First the reverse communication loop structure will be described and then the details and nuances of the problem setup will be discussed. We use the symbol $\mathrm{OP}$ for the operator that is applied to vectors in the Arnoldi/Lanczos process and $B$ will stand for the matrix to use in the weighted inner product described previously. For the shift-invert spectral transformation mode $\mathrm{OP}$ denotes ${\left(A-\sigma B\right)}^{-1}B$.
The basic idea is to set up a loop that repeatedly call one of F12ABF, F12APF and F12FBF. On each return, you must either apply $\mathrm{OP}$ or $B$ to a specified vector or exit the loop depending upon the value returned in the reverse communication parameter IREVCM.

#### 4.3.1  Shift and invert on a generalized eigenproblem

The example program in Section 9 in F12AEF illustrates the reverse communication loop for F12ABF in shift-invert mode for a generalized nonsymmetric eigenvalue problem. This loop structure will be identical for the symmetric problem calling F12FBF. The loop structure is also identical for the complex arithmetic subroutine F12APF.
In the example, the matrix $B$ is assumed to be symmetric and positive semidefinite. In the loop structure, you will have to supply a routine to obtain a matrix factorization of $\left(A-\sigma B\right)$ that may repeatedly be used to solve linear systems. Moreover, a routine needs to be provided to perform the matrix-vector product $z=Bv$ and a routine is required to solve linear systems of the form $\left(A-\sigma B\right)w=z$ as needed using the previously computed factorization.
When convergence has taken place (indicated by $\mathbf{IREVCM}=5$ and $\mathbf{IFAIL}=0$), the reverse communication loop will be exited. Then, post-processing using the relevant routine from F12ACF, F12AQF and F12FCF must be done to recover the eigenvalues and corresponding eigenvectors of the original problem. When operating in shift-invert mode, the eigenvalue selection option is normally set to Largest Magnitude. The post-processing routine is then used to convert the converged eigenvalues of $\mathrm{OP}$ to eigenvalues of the original problem (2). Also, when $B$ is singular or ill-conditioned, the post-processing routine takes steps to purify the eigenvectors and rid them of numerical corruption from eigenvectors corresponding to near-infinite eigenvalues. These procedures are performed automatically when operating in any one of the computational modes described above and later in this section.
You may wish to construct alternative computational modes using spectral transformations that are not addressed by any of the modes specified in this chapter. The reverse communication interface will easily accommodate these modifications. However, it will most likely be necessary to construct explicit transformations of the eigenvalues of $\mathrm{OP}$ to eigenvalues of the original problem in these situations.

#### 4.3.2  Using the computational modes

The problem set up is similar for all of the available computational modes. In the previous section, a detailed description of the reverse communication loop for a specific mode (Shift-invert for a Generalized Problem) was given. To use this or any of the other modes listed below, you are strongly urged to modify one of the example programs.
The first thing to decide is whether the problem will require a spectral transformation. If the problem is generalized, $B\ne I$, then a spectral transformation will be required (see Section 4.2). Such a transformation will most likely be needed for a standard problem if the desired eigenvalues are in the interior of the spectrum or if they are clustered at the desired part of the spectrum. Once this decision has been made and $\mathrm{OP}$ has been specified, an efficient means to implement the action of the operator $\mathrm{OP}$ on a vector must be devised. The expense of applying $\mathrm{OP}$ to a vector will of course have direct impact on performance.
Shift-invert spectral transformations may be implemented with or without the use of a weighted $B$-inner product. The relation between the eigenvalues of $\mathrm{OP}$ and the eigenvalues of the original problem must also be understood in order to make the appropriate eigenvalue selection option (e.g., Largest Magnitude) in order to recover eigenvalues of interest for the original problem. You must specify the number of eigenvalues to compute, which eigenvalues are of interest, the number of basis vectors to use, and whether or not the problem is standard or generalized. These items are controlled by setting options via the option setting routine.
Setting the number of eigenvalues NEV and the number of basis vectors NCV (in the setup routine) for optimal performance is very much problem dependent. If possible, it is best to avoid setting NEV in a way that will split clusters of eigenvalues. As a rule of thumb $\mathbf{NCV}\ge 2×\mathbf{NEV}$ is reasonable. There are trade-offs due to the cost of the user-supplied matrix-vector products and the cost of the implicit restart mechanism. If the user-supplied matrix-vector product is relatively cheap, then a smaller value of NCV may lead to more user matrix-vector products and implicit Arnoldi iterations but an overall decrease in computation time. Convergence behaviour can be quite different depending on which of the spectrum options (e.g., Largest Magnitude) is chosen. The Arnoldi process tends to converge most rapidly to extreme points of the spectrum. Implicit restarting can be effective in focusing on and isolating a selected set of eigenvalues near these extremes. In principle, implicit restarting could isolate eigenvalues in the interior, but in practice this is difficult and usually unsuccessful. If you are interested in eigenvalues near a point that is in the interior of the spectrum, a shift-invert strategy is usually required for reasonable convergence.
The integer argument IREVCM is the reverse communication flag that will specify a requested action on return from one of the solver routines F12ABF, F12APF and F12FBF. The options Standard and Generalized specify if this is a standard or generalized eigenvalue problem. The dimension of the problem is specified on the call to the initialization routine only; this value, together with the number of eigenvalues and the dimension of the basis vectors is passed through the communication array. There are a number of spectrum options which specify the eigenvalues to be computed; these options differ depending on whether a Hermitian or non-Hermitian eigenvalue problem is to be solved. For example, the Both Ends is specific to Hermitian (symmetric) problems while the Largest Imaginary is specific to non-Hermitian eigenvalue problems (see Section 10.1 in F12ADF). The specification of problem type will be described separately but the reverse communication interface and loop structure is the same for each type of the basic modes Regular, Regular Inverse, Shifted Inverse (also Shifted Inverse Real and Shifted Inverse Imaginary for real nonsymmetric problems), and for the problem type: Standard or Generalized. There are some additional specialised modes for symmetric problems, Buckling and Cayley, and for real nonsymmetric problems with complex shifts applied in real arithmetic. You are encouraged to examine the documented example programs for these modes.
The Tolerance specifies the accuracy requested. If you wish to supply shifts for implicit restarting then the Supplied Shifts must be selected, otherwise the default Exact Shifts strategy will be used. The Supplied Shifts should only be used when you have a great deal of knowledge about the spectrum and about the implicit restarted Arnoldi method and its underlying theory. The Iteration Limit should be set to the maximum number of implicit restarts allowed. The cost of an implicit restart step (major iteration) is in the order of $4n\left(\mathbf{NCV}-\mathbf{NEV}\right)$ floating point operations for the dense matrix operations and $\mathbf{NCV}-\mathbf{NEV}$ matrix-vector products $w←\mathrm{Av}$ with the matrix $A$.
The choice of computational mode through the option setting routine is very important. The legitimate computational mode options available differ with each problem type and are listed below for each of them.

#### 4.3.3  Computational modes for real symmetric problems

The reverse communication interface subroutine for symmetric eigenvalue problems is F12FBF. The option for selecting the region of the spectrum of interest can be one of those listed in Table 1.
 Largest Magnitude The eigenvalues of greatest magnitude Largest Algebraic The eigenvalues of largest algebraic value (rightmost) Smallest Magnitude The eigenvalues of least magnitude. Smallest Algebraic The eigenvalues of smallest algebraic value (leftmost) Both Ends The eigenvalues from both ends of the algebraic spectrum
Table 1
Eigenvalue spectrum options for symmetric eigenproblems
Table 2 lists the spectral transformation options for symmetric eigenvalue problems together with the specification of $\mathrm{OP}$ and $B$ for each mode and the problem type option setting.
 Problem Type Mode Problem $\mathbf{OP}$ $\mathbit{B}$ Standard Regular $Ax=\mathrm{\lambda x}$ $A$ $I$ Standard Shifted Inverse $Ax=\mathrm{\lambda x}$ ${\left(A-\sigma I\right)}^{-1}$ $I$ Generalized Regular Inverse $Ax=\lambda Bx$ ${B}^{-1}Ax$ $B$ Generalized Shifted Inverse $Ax=\lambda Bx$ ${\left(A-\sigma B\right)}^{-1}B$ $B$ Generalized Buckling $\mathrm{Kx}=\lambda {K}_{G}x$ ${\left(K-\sigma {K}_{G}\right)}^{-1}K$ $K$ Generalized Cayley $Ax=\lambda Bx$ ${\left(A-\sigma B\right)}^{-1}\left(A+\sigma B\right)$ $B$
Table 2
Problem types, computational modes and spectral transformations for symmetric eigenproblems

#### 4.3.4  Computational modes for non-Hermitian problems

When $A$ is a general non-Hermitian matrix and $B$ is Hermitian and positive semidefinite, then the selection of the eigenvalues is controlled by the choice of one of the options in Table 3.
 Largest Magnitude The eigenvalues of greatest magnitude Smallest Magnitude The eigenvalues of least magnitude Largest Real The eigenvalues with largest real part Smallest Real The eigenvalues with smallest real part Largest Imaginary The eigenvalues with largest imaginary part Smallest Imaginary The eigenvalues with smallest imaginary part
Table 3
Eigenvalue spectrum options for real nonsymmetric and complex eigenproblems
Table 4 lists the spectral transformation options for real nonsymmetric eigenvalue problems together with the specification of $\mathrm{OP}$ and $B$ for each mode and the problem type option setting. The equivalent listing for complex non-Hermitian eigenvalue problems is given in Table 5.
 Problem Type Mode Problem $\mathbf{OP}$ $\mathbit{B}$ Standard Regular $Ax=\mathrm{\lambda x}$ $A$ $I$ Standard Shifted Inverse Real $Ax=\mathrm{\lambda x}$ ${\left(A-\sigma I\right)}^{-1}$ $I$ Generalized Regular Inverse $Ax=\lambda Bx$ ${B}^{-1}Ax$ $B$ Generalized Shifted Inverse Real with real $\sigma$ $Ax=\lambda Bx$ ${\left(A-\sigma B\right)}^{-1}B$ $B$ Generalized Shifted Inverse Real with complex $\sigma$ $Ax=\lambda Bx$ $\text{real}\left\{{\left(A-\sigma B\right)}^{-1}B\right\}$ $B$ Generalized Shifted Inverse Imaginarywith complex $\sigma$ $Ax=\lambda Bx$ $\text{imag}\left\{{\left(A-\sigma B\right)}^{-1}B\right\}$ $B$
Table 4
Problem types, computational modes and spectral transformations for real nonsymmetric eigenproblems
Note that there are two shifted inverse modes with complex shifts in Table 4. Since $\sigma$ is complex, these both require the factorization of the matrix $A-\sigma B$ in complex arithmetic even though, in the case of real nonsymmetric problems, both $A$ and $B$ are real. The only advantage of using this option for real nonsymmetric problems instead of using the equivalent suite for complex problems is that all of the internal operations in the Arnoldi process are executed in real arithmetic. This results in a factor of two saving in storage and a factor of four saving in computational cost. There is additional post-processing that is somewhat more complicated than the other modes in order to get the eigenvalues and eigenvectors of the original problem. These modes are only recommended if storage is extremely critical.
 Problem Type Mode Problem $\mathbf{OP}$ $\mathbit{B}$ Standard Regular $Ax=\mathrm{\lambda x}$ $A$ $I$ Standard Shifted Inverse $Ax=\mathrm{\lambda x}$ ${\left(A-\sigma I\right)}^{-1}$ $I$ Generalized Regular Inverse $Ax=\lambda Bx$ ${B}^{-1}Ax$ $B$ Generalized Shifted Inverse $Ax=\lambda Bx$ ${\left(A-\sigma B\right)}^{-1}B$ $B$
Table 5
Problem types, computational modes and spectral transformations for complex non-Hermitian eigenproblems

#### 4.3.5  Post processing

On the final successful return from a reverse communication routine, the corresponding post-processing routine must be called to get eigenvalues of the original problem and the corresponding eigenvectors if desired. In the case of Shifted Inverse modes for Generalized problems, there are some subtleties to recovering eigenvectors when $B$ is ill-conditioned. This process is called eigenvector purification. It prevents eigenvectors from being corrupted with noise due to the presence of eigenvectors corresponding to near infinite eigenvalues. These operations are completely transparent to you. There is negligible additional cost to obtain eigenvectors. An orthonormal (Arnoldi/Lanczos) basis is always computed. The approximate eigenvalues of the original problem are returned in ascending algebraic order. The option relevant to this routine is Vectors which may be set to values that determine whether only eigenvalues are desired or whether corresponding eigenvectors and/or Schur vectors are required. The value of the shift $\sigma$ used in spectral transformations must be passed to the post-processing routine through the appropriately named argument(s). The eigenvectors returned are normalized to have unit length with respect to the semi-inner product that was used. Thus, if $B=I$ then they will have unit length in the standard-norm. In general, a computed eigenvector $x$ will satisfy ${x}^{\mathrm{H}}Bx=1$.

#### 4.3.6  Solution monitoring and printing

The option setting routine for each suite allows the setting of three options that control solution printing and the monitoring of the iterative and post-processing stages. These three options are: Advisory, Monitoring and Print Level. By default, no solution monitoring or printing is performed. The Advisory option controls where solution details are printed; the Monitoring option controls where monitoring details are to be printed and is mainly used for debugging purposes; the Print Level option controls the amount of detail to be printed, see individual option setting routine documents for specifications of each print level. The value passed to Advisory and Monitoring can be the same, but it is recommended that the two sets of information be kept separate. Note that the monitoring information can become very voluminous for the highest settings of Print Level.

## 5  Functionality Index

 Standard or generalized eigenvalue problems for complex matrices,
 banded matrices,
 initialize problem and method F12ATF
 selected eigenvalues, eigenvectors and/or Schur vectors F12AUF
 general matrices,
 initialize problem and method F12ANF
 option setting F12ARF
 reverse communication implicitly restarted Arnoldi method F12APF
 reverse communication monitoring F12ASF
 selected eigenvalues, eigenvectors and/or Schur vectors of original problem F12AQF
 Standard or generalized eigenvalue problems for real nonsymmetric matrices,
 banded matrices,
 initialize problem and method F12AFF
 selected eigenvalues, eigenvectors and/or Schur vectors F12AGF
 general matrices,
 initialize problem and method F12AAF
 reverse communication implicitly restarted Arnoldi method F12ABF
 reverse communication monitoring F12AEF
 selected eigenvalues, eigenvectors and/or Schur vectors of original problem F12ACF
 Standard or generalized eigenvalue problems for real symmetric matrices,
 banded matrices,
 initialize problem and method F12FFF
 selected eigenvalues, eigenvectors and/or Schur vectors F12FGF
 general matrices,
 initialize problem and method F12FAF
 option setting F12FDF
 reverse communication implicitly restarted Arnoldi(Lanczos) method F12FBF
 reverse communication monitoring F12FEF
 selected eigenvalues and eigenvectors and/or Schur vectors of original problem F12FCF

None.

None.

## 8  References

Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia
Lehoucq R B (1995) Analysis and implementation of an implicitly restarted iteration PhD Thesis Rice University, Houston, Texas
Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821
Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
Saad Y (1992) Numerical Methods for Large Eigenvalue Problems Manchester University Press, Manchester, UK

F12 Chapter Contents

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# Microphone

##  History

The word "microphone" (Greek mikros "small" and phone "voice" or "sound") originally referred to a mechanical hearing aid for small sounds.<ref>[1]</ref>

The invention of a practical microphone was crucial to the early development of the telephone system. Several early inventors built primitive microphones (then called transmitters) prior to Alexander Bell, but the first commercially practical microphone was the carbon microphone conceived in October, 1876 by Thomas Edison. Many early developments in microphone design took place at Bell Laboratories. See also Timeline of the telephone. The main basic designs still popular are of American, British or Russian origin.

##  Principle of operation

Image:Oktava319.jpg
An Oktava condenser microphone.

A microphone is a device made to capture waves in air, water or hard material and translate it to an electrical signal. The most common method is via a thin membrane producing some proportional electrical signal. Most microphones in use today for audio use electromagnetic generation (dynamic microphones), capacitance change (condenser microphones) or piezoelectric generation to produce the signal from mechanical vibration.

##  Microphone varieties

###  Condenser or capacitor microphones

Image:Oktava319-internal.jpg
Inside the Oktava 319 condenser microphone.

Technology
In a condenser microphone, also known as a capacitor microphone, the diaphragm acts as one plate of a capacitor, and the vibrations produce changes in the distance between the plates. Since the plates are biased with a fixed charge (Q), the voltage maintained across the capacitor plates changes with the vibrations in the air, according to the capacitance equation:

$Q = C \cdot V$

where Q = charge in coulombs, C = capacitance in farads and V = potential difference in volts. The capacitance of the plates is inversely proportional to the distance between them for a parallel-plate capacitor. (See capacitance for details.)

The charge across the capacitor is not maintained perfectly constant. As the capacitance changes, the charge across the capacitor changes to make the voltage drop across the capacitor equal to the bias voltage. However, the rate of this change is kept slow by using a series resistor of a very high value (of the order of 10 MΩ). Note that the time constant of a RC circuit equals the product of the resistance and capacitance.

Within the time-frame of the capacitance change (on the order of 100 μs), the charge thus appears practically constant and the voltage across the capacitor adjusts itself instantaneously to reflect the change in capacitance. Within the time-frame of the capacitance change, the voltage across the capacitor never really catches up to the bias voltage. The voltage difference between the bias and the capacitor is seen across the series resistor. The voltage across the resistor is amplified and reproduced to audio or recording. Condenser microphones are elongation receivers.

Usage
Condenser microphones range from cheap to expensive. They generally produce a high-quality audio signal and are now the popular choice in laboratory and studio recording applications. They require a power source, provided either from microphone inputs as phantom power or from a small battery. Professional microphones often sport an external power supply for reasons of quality perception. Power is necessary for establishing the capacitor plate voltage, and is also needed for internal amplification of the signal to a useful output level. Condenser microphones are also available with two diaphragms, the signals from which can be electrically connected such as to provide a range of polar patterns (see below), such as cardioid, omnidirectional and figure eight. It is also possible to vary the pattern smoothly with some microphones, for example the Røde NT2000.<ref>http://www.rodemic.com/?pagename=Products&product=NT2000</ref>

####  Electret condenser microphones

Main article: Electret microphone

An electret microphone is a relatively new type of capacitor microphone invented at Bell laboratories in 1962 by Gerhard Sessler and Jim West<ref>"Electret Microphone Turns 40"</ref>, and often simply called an electret microphone. An electret is a dielectric material that has been permanently electrically charged or polarized. The name comes from electrostatic and magnet; a static charge is embedded in an electret by alignment of the static charges in the material, much the way a magnet is made by aligning the magnetic domains in a piece of iron. They are used in many applications, from high-quality recording and lavalier use to built-in microphones in small sound recording devices and telephones. Though electret mics were once low-cost and considered low quality, the best ones can now rival capacitor mics in every respect (apart from low noise) and can even have the long-term stability and ultra-flat response needed for a measuring microphone. Unlike other capacitor microphones, they require no polarizing voltage, but normally contain an integrated preamplifier which does require power (often incorrectly called polarizing power or bias). This preamp is frequently phantom powered in sound reinforcement and studio applications. While few electret microphones rival the best DC-polarized units in terms of noise level, this is not due to any inherent limitation of the electret. Rather, mass production techniques needed to produce electrets cheaply don't lend themselves to the precision needed to produce the highest quality microphones.

###  Dynamic microphones

Dynamic microphone works via electromagnetic induction. Two types can be distinguished: the plunger coil microphone and the ribbon microphone. Dynamic microphones are velocity receivers.

####  Plunger coil microphones

Image:SM57&Beta57.JPG
The Shure SM57 and Beta 57A dynamic microphones

Technology
A small movable induction coil, positioned in the magnetic field of a permanent magnet, is attached to the diaphragm. When sound enters through the windscreen of the microphone, the sound wave moves the diaphragm. When the diaphragm vibrates, the coil moves in the magnetic field, producing a varying current in the coil through electromagnetic induction. The frequency content of the generated signal is proportional to the perceived frequency. So a 1KHz sine wave would generate an identical frequency output from the microphone. A single dynamic membrane will not respond linearly to all audio frequencies. Some microphones for this reason utilize multiple membranes for the different parts of the audio spectrum and then combine the resulting signals. Combining the multiple signals correctly is difficult and designs that do this are rare and tend to be expensive. There are on the other hand several designs that are more specifically aimed towards isolated parts of the audio spectrum. AKG D112 is for example designed for bass content rather than treble. In audio engineering several kinds of microphones are often used at the same time to get the best result.

The dynamic principle is exactly the same as in a loudspeaker, only reversed.

Usage
Dynamic microphones are robust, relatively inexpensive, and resistant to moisture, and for this reason they are widely used on-stage by singers.

####  Ribbon microphones

Technology
In ribbon microphones a thin, usually corrugated metal ribbon is suspended in a magnetic field. The ribbon is electrically connected to the microphone's output, and its vibration within the magnetic field generates the electrical signal. Ribbon microphones are similar to dynamic microphones in the sense that both produce sound by means of magnetic induction. Basic ribbon microphones detect sound in a bidirectional (also called figure-eight) pattern because the ribbon, which is open to sound both front and back, responds to the pressure gradient rather than the sound pressure. Though the symmetrical front and rear pickup can be a nuisance in normal stereo recording, the high side rejection can be used to advantage by positioning a ribbon mic horizontally, for example above cymbals, so that the rear lobe picks up only sound from the cymbals. Other directional patterns are produced by enclosing one side of the ribbon in an acoustic trap or baffle, allowing sound to reach only one side. Ribbon mics give very high quality sound reproduction, and were once valued for this reason, but a good low-frequency response can be obtained only if the ribbon is suspended very loosely, and this makes them fragile. Protective wind screens can reduce the danger of damaging the ribbon, but will somewhat reduce the bass response at large miking distances.

Ribbon microphones don't require phantom power; in fact, this voltage can damage these microphones.

###  Carbon microphones

Main article: Carbon microphone

Technology
A carbon microphone, formerly used in telephone handsets, is a capsule containing carbon granules pressed between two metal plates. A voltage is applied across the metal plates, causing a small current to flow through the carbon. One of the plates, the diaphragm, vibrates in sympathy with incident sound waves, applying a varying pressure to the carbon. The changing pressure deforms the granules, causing the contact area between each pair of adjacent granules to change, and this causes the electrical resistance of the mass of granules to change. The changes in resistance cause a corresponding change in the voltage across the two plates, and hence in the current flowing through the microphone, producing the electrical signal. Carbon microphones were once commonly used in telephones; they have extremely low-quality sound reproduction and a very limited frequency response range, but are very robust devices.

Unlike other microphone types, the carbon microphone can also be used as a type of amplifier, using a small amount of sound energy to produce a larger amount of electrical energy. Carbon microphones found use as early telephone repeaters, making long distance phone calls possible in the era before vacuum tubes. These repeaters worked by mechanically coupling a magnetic telephone receiver to a carbon microphone: the faint signal from the receiver was transferred to the microphone, with a resulting stronger electrical signal to send down the line. (One illustration of this amplifier effect was the oscillation caused by feedback, resulting in an audible squeal from the old "candlestick" telephone if its earphone was placed near the carbon microphone.)

###  Piezo microphones

Technology
A piezo microphone uses the phenomenon of piezoelectricity — the ability of some materials to produce a voltage when subjected to pressure — to convert vibrations into an electrical signal. An example of this is Rochelle salt (potassium sodium tartrate), which is a piezoelectric crystal that works as a transducer, both as a microphone and as a slimline loudspeaker component.

Usage
Piezo transducers are often used as contact microphones to amplify sound from musical instruments for live performance, or to record sounds in unusual environments (underwater, for instance). Saddle mounted pickups in acoustic guitars are piezos and are mechanically connected to the strings through the saddle. This is not to be confused with magnetic coil pickups commonly visible on typical electric guitars that use steel strings. Instruments that use non-metallic strings cannot use magnetic pickups, but can use piezo transducers which pickup sound vibrations mechanically. Small acoustic microphones have also been used, but are less practical because they also pickup ambient noise and feedback. Some instruments feature hybrid combinations of the piezo and magnetic technologies. John Petrucci of Dream Theater has a signature series of typical electric guitars which also sports a piezo pickup in the saddle. A saddle is where the strings are mounted to the guitar on the opposite end of where the head and tuning pegs are.

###  Laser microphones

Technology
A laser microphone is an exotic application of laser technology. It consists of a laser beam that must be reflected off a glass window or another rigid surface that vibrates in sympathy with nearby sounds. This device essentially turns any vibrating surface near the source of sound into a microphone. It does this by measuring the distance between itself and the surface extremely accurately; the tiny fluctuations in this distance become the electrical signal of the sounds picked up. Usage
Laser microphones are new, very rare and expensive, and are most commonly portrayed in the movies as spying devices.

####  Liquid Microphones

Technology
Early versions did not produce intelligible speech, Alexander Graham Bell innovated a set of improvements in acidity and diaphram movement. Bell’s liquid transmitter consisted of a metal cup filled with a mixture of sulfuric acid and water. A sound wave causes the diaphragm to move thereby forcing the a brass tube to move up and down in the liquid. The electrical resistance between the wire and the cup is inversely proportional to the amount of wire submerged. Elisha Gray patented a version using a needle instead of the brass tube. Other minor variantions and/or improvements were made to the liquid microphone by Majoranna,Chambers,Vanni and Sykes Elisha Gray and one version was even patented by Reginald Fessenden in 1903.

Usage
These were the first microphones and are utterly obsolete now. It was with this arrangement that the famous first phone conversation between Bell and Watson took place. The liquid variable resistance was not practical as a commercial application. Other inventors soon devised superior devices.

### Speakers as microphones

Usage
A loudspeaker is the exact opposite of a microphone, since it's a transducer that turns an electrical signal into sound waves. However, because a conventional speaker is constructed much like a dynamic microphone (with a diaphragm, coil and magnet), speakers can actually work "in reverse" as microphones. The result, though, is a microphone with poor quality, limited frequency response (particularly at the high end), and poor sensitivity.

In practical use, speakers are sometimes used as microphones in such applications as intercoms or walkie-talkies, where high quality and sensitivity are not needed. However, there is at least one other novel application of this principle; using a medium-size woofer placed closely in front of a "kick" (bass drum) in a drum set to act as a microphone. This has been commercialized with the Yamaha "Subkick".[2]

If you were to connect a pair of any kind of ordinary headphones to a mic or line input, you would be able to record a stereo track. For example to a Minidisc or iPod. It might not be high fidelity but it does work to a certain degree. With the new accessiorie mic input modules for iPod 5th generation you could fold a regular small sized pair of earphones over your earlobes and get a proper stereo recording. Since the membranes are tiny in small headphones sensitivity would be usable.

##  Capsule design and directivity

The shape of the microphone defines its directivity. Inner elements are of major importance and concerns the structural shape of the capsule, outer elements may be the interference tube.

A pressure gradient microphone is a microphone in which both sides of the diaphragm are exposed to the incident sound and the microphone is therefore responsive to the pressure differential (gradient) between the two sides of the membrane. Sound incident parallel to the plane of the diaphragm produces no pressure differential, giving pressure-gradient microphones their characteristic figure-eight directional patterns.

The capsule of a pressure microphone however is closed on one side, which results in an omnidirectional pattern.

##  Microphone Polar Patterns

Common polar patterns for microphones
(microphone facing up in diagram)
Image:Omnipattern.svg Image:Cardioidpattern.svg Image:Hypercardioidpattern.svg Image:Bidirectionalpattern.svg Image:Shotgunpattern.svg
Omnidirectional Cardioid Hypercardioid Bi-directional Shotgun

A microphone's directionality or polar pattern indicates how sensitive it is to sounds arriving at different angles about its central axis. The above polar patterns represent the locus of points that produce the same signal level output in the microphone if a given sound pressure level is generated from that point. How the physical body of the microphone is oriented relative to the diagrams depends on the microphone design. For large-membrane microphones such as in the Oktava (pictured above), the upward direction in the polar diagram is perpendicular to the microphone body. For dynamic microphones such as the Shure (also pictured above), it extends from the axis of the microphone. Some microphone designs combine several principles in creating the desired polar pattern. This ranges from shielding (meaning diffraction/dissipation/absorption) by the housing itself to electronically combining dual membranes.

An omnidirectional microphone's response is generally considered to be a perfect sphere in three dimensions. In the real world, this is not the case. As with directional microphones, the polar pattern for an "omnidirectional" microphone is a function of frequency. The body of the microphone is not infinitely small and, as a consequence, it tends to get in its own way with respect to sounds arriving from the rear, causing a slight flattening of the polar response. This flattening increases as the diameter of the microphone (assuming it's cylindrical) reaches the wavelength of the frequency in question. Therefore, the smallest diameter microphone will give the best omnidirectional characteristics at high frequencies. The wavelength of sound at 10 kHz is about an inch (2.5 cm) so the smallest measuring microphones are often 1/4" (6 mm) in diameter, which practically eliminates directionality even up to the highest frequencies. Omnidirectional microphones, unlike cardioids, do not employ resonant cavities as delays, and so can be considered the "purest" mikes in terms of low coloration; they add very little to the original sound. Being pressure-sensitive they can also have a very flat low-frequency response down to 20 Hz or below. Pressure-sensitive mikes also respond much less to wind noise than directional (velocity sensitive) mikes.

A unidirectional microphone is sensitive to sounds from only one direction. The diagram above illustrates a number of these patterns, with the microphone capsule being represented as a red dot. The mike faces upwards in each diagram. The sound intensity for a particular frequency is plotted for angles radially from 0 to 360°. (Professional diagrams show these scales and include multiple plots at different frequencies. These diagrams just provide an overview of the typical shapes and their names.)

The most common unidirectional mike is a cardioid microphone, so named because the sensitivity pattern is heart-shaped (see cardioid). A hyper-cardioid is similar but with a tighter area of front sensitivity and a tiny lobe of rear sensitivity. These two patterns are commonly used as vocal or speech mikes, since they are good at rejecting sounds from other directions. Because they employ internal cavities to provide front-back delay, directional mikes tend to have more coloration than omnis, and they also suffer from low-frequency roll-off. These problems are overcome to a large extent by careful design, but only the best cardioids can begin to approach the performance of a tiny low-cost omni in terms of absolute accuracy. This is not always recognised, but is the price paid for directionality, often needed to exclude ambient reverberation wherever very close placement is impossible.

Figure 8 or bi-directional mikes receive sound from both the front and back of the element. Most ribbon microphones are of this pattern.

Shotgun microphones are the most highly directional. They have small lobes of sensitivity to the left, right, and rear but are significantly more sensitive to the front. This results from placing the element inside a tube with slots cut along the side; wave-cancellation eliminates most of the off-axis noise. Shotgun microphones are commonly used on TV and film sets, and for location recording of wildlife.

An omnidirectional microphone is a pressure transducer; the output voltage is proportional to the air pressure at a given time.

On the other hand, a figure-8 pattern is a pressure gradient transducer; the output voltage is proportional to the difference in pressure on the front and on the back side. A sound wave arriving from the back will lead to a signal with a polarity opposite to that of an identical sound wave from the front. Moreover, shorter wavelengths (higher frequencies) are picked up more effectively than lower frequencies.

A cardioid microphone is effectively a superposition of an omnidirectional and a figure-8 microphone; for sound waves coming from the back, the negative signal from the figure-8 cancels the positive signal from the omnidirectional element, whereas for sound waves coming from the front, the two add to each other. A hypercardioid microphone is similar, but with a slightly larger figure-8 contribution.

Since pressure gradient transducer microphones are directional(partially), their frequency response is dependent on the distance to the sound source. This is known as the proximity effect, a bass boost at distances of a few centimeters. A phenomenon related to the physics of wave and particle propagation. Low frequency audio does not exhibit the same properties as high frequency audio.

##  Design concerning practical application

A lavalier microphone is made for hands-free operation. These small microphones are worn on the body and held in place either with a lanyard worn around the neck or a clip fastened to clothing. The cord may be hidden by clothes and either run to an RF transmitter in a pocket or clipped to a belt (for mobile use), or run directly to the mixer (for stationary applications).

A wireless microphone is one which does not use a cable. It usually transmits its signal using a small FM radio transmitter to a nearby receiver connected to the sound system, but it can also use infrared light if the transmitter and receiver are within sight of each other.

A contact microphone is designed to pick up vibrations directly from a solid surface or object, as opposed to sound vibrations carried through air. One use for this is to detect sounds of a very low level, such as those from small objects or insects. The microphone commonly consists of a magnetic (moving coil) transducer, contact plate and contact pin. The contact plate is placed against the object from which vibrations are to be picked up; the contact pin transfers these vibrations to the coil of the transducer. Contact microphones have been used to pick up the sound of a snail's heartbeat and the footsteps of ants. A portable version of this microphone has recently been developed.

A throat microphone is a variant of the contact microphone, used to pick up speech directly from the throat, around which it is strapped. This allows the device to be used in areas with ambient sounds that would otherwise make the speaker inaudible.

A parabolic microphone uses a parabolic reflector to collect and focus sound waves onto a microphone receiver, in much the same way that a parabolic antenna (e.g. satellite dish) does with radio waves. Typical uses of this microphone, which has unusually focused front sensitivity and can pick up sounds from many meters away, include nature recording, outdoor sporting events, eavesdropping, law enforcement, and even espionage. Parabolic microphones are not typically used for standard recording applications, because they tend to have poor low-frequency response as a side effect of their design.

##  Connectivity

### Connectors

The most common connectors used by microphones are:

• Male XLR connector on professional microphones
• ¼ inch mono phone plug (UK "jack plug") on less expensive consumer microphones
• 3.5 mm mono mini phone plug on very inexpensive and computer microphones

Some microphones use other connectors, such as TRS, 5-pin XLR, or stereo mini phone plug on some stereo microphones. Some lavaliers have a proprietary connector to connect them to their transmitter. Since 2005, professional-quality microphones with USB connections have begun to appear, designed for direct recording into computer-based software studios.

###  Impedance matching

Microphones have an electrical characteristic called impedance, measured in ohms (Ω) that depends on the design. Low impedance is considered under 600 Ω. Medium impedance is considered between 600 Ω and 10 kΩ. High impedance is above 10 kΩ. Most professional microphones are low impedance, about 200 Ω. Less expensive models have an impedance of at least 600 Ω. Low-impedance microphones are preferred over high impedance on long-run cables for two reasons: one is that using a high-impedance mike with a long cable is likely to result in loss of high frequency signal; the other is that long high-impedance cables tend to pick up more hum (and possibly radio-frequency interference (RFI) as well).

To get the best sound, the impedances of the microphone and the equipment to which it is connected must match. There are transformers (called matching transformers) that adapt impedances, such as DI units. In general, any XLR microphone can be connected to any mixer with XLR inputs, and any plug microphone can be connected to any jack plug that is marked as a microphone input, but it can't be connected to a line input.

##  Measurements and specifications

Image:Oktava319vsshuresm58.png
A comparison of the far field on-axis frequency response of the Oktava 319 and the Shure SM58

Because of differences in their construction, microphones have their own characteristic responses to sound. This difference in response produces non-uniform phase and frequency responses. In addition, mics are not uniformly sensitive to sound pressure, and can accept differing levels without distorting. Although for scientific applications microphones with a more uniform response are desirable, this is often not the case for music recording, as the non-uniform response of a microphone can produce a desirable coloration of the sound. There is an international standard for microphone specifications (IEC 60268-4), but very few manufacturers adhere to it.

A frequency response diagram plots the microphone sensitivity in decibels over a range of frequencies (typically at least 0–20 kHz), generally for perfectly on-axis sound (sound arriving at 0° to the capsule). Frequency response may be less informatively stated textually like so: "30 Hz–16 kHz ±3 dB". This is interpreted as a (mostly) linear plot between the stated frequencies, with variations in amplitude of no more than plus or minus 3 dB. However, one cannot determine from this information how smooth the variations are, nor in what parts of the spectrum they occur. Note that commonly-made statements such as "20 Hz–20 kHz" are meaningless without a decibel measure.

The self-noise or equivalent noise level is the sound level that creates the same output voltage as the inherent noise of the microphone. This represents the lowest point of the microphone's dynamic range, and is particularly important should you wish to record sounds that are quiet. The measure is often stated in dB(A), which is the equivalent loudness of the noise on a decibel scale frequency-weighted for how the ear hears, for example: "15 dBA SPL" (SPL means sound pressure level relative to 20 micropascals). The lower the number the better. Some microphone manufacturers state the noise level using ITU-R 468 noise weighting, which more accurately represents the way we hear noise, but gives a figure some 11 to 14 dB higher. A quiet microphone will measure typically 20 dBA SPL or 32 dB SPL 468-weighted.

The maximum SPL (sound pressure level) the microphone can accept is measured for particular values of total harmonic distortion (THD), typically 1%. This is generally inaudible, so one can safely use the mic at this level without harming the recording. Example: "142 dB SPL peak (<1% THD)". The higher the value, the better.

The clipping level is perhaps a better indicator of maximum useable level as the 1% THD figure usually quoted under max SPL is really a very mild level of distortion, quite inaudible especially on brief high peaks. Harmonic distortion from microphones is usually of low-order (mostly third harmonic) type, and hence not very audible even at 3-5%. Clipping, on the other hand, usually caused by the diaphragm reaching its absolute displacement limit (or by the preamplifier), will produce a very harsh sound on peaks, and should be avoided if at all possible. For some mikes the clipping level may be much higher than the max SPL.

The dynamic range of a mike is the difference in SPL between the noise floor and the maximum SPL. If stated on its own, for example "120 dB", it conveys significantly less information than having the self-noise and maximum SPL figures individually.

Sensitivity indicates how well the mike converts acoustic pressure to output voltage. A high sensitivity mic creates more voltage and so will need less amplification at the mixer or recording device. This is a practical concern but not directly an indication of the mic's quality, and in fact the term sensitivity is something of a misnomer, 'transduction gain' being perhaps more meaningful, (or just "output level") because true sensitivity will generally be set by the noise floor, and too much "sensitivity" in terms of output level will compromise the clipping level. There are two common measures. The (preferred) international standard is made in millivolts per pascal at 1 kHz. A higher value indicates greater sensitivity. The older American method is referred to a 1 V/Pa standard and measured in plain decibels, resulting in a negative value. Again, a higher value indicates greater sensitivity, so −60  dB is more sensitive than −70 dB.

##  Measurement microphones

Some microphones are intended for use as standard measuring microphones for the testing of speakers and checking noise levels etc. These are calibrated transducers and will usually be supplied with a calibration certificate stating absolute sensitivity against frequency.

###  Microphone calibration techniques

####  Pistonphone apparatus

A pistonphone is an acoustical calibrator (sound source) using a closed coupler to generate a precise sound pressure for the calibration of instrumentation microphones. The principle relies on a piston mechanically driven to move at a specified rate on a fixed volume of air to which the microphone under test is exposed. The air is assumed to be compressed adiabatically and the SPL in the chamber can be calculated from PV = const. The pistonphone method only works at low frequencies, but it can be accurate and yields an easily calculable sound pressure level. The standard test frequency is usually around 250 Hz.

####  Reciprocal method

This method relies on the reciprocity of one or more microphones in a group of 3 to be calibrated. It can still be used when only one of the microphones is reciprocal (exhibits equal response when used as a microphone or as a loudspeaker).

##  Microphone techniques

There exist a number of well-developed microphone techniques used for miking musical, film, or voice sources. Choice of technique depends on a number of factors, including:

• The collection of extraneous noise. This can be a concern, especially in amplified performances, where audio feedback can be a significant problem. Alternatively, it can be a desired outcome, in situations where ambient noise is useful (hall reverberation, audience reaction).
• Choice of a signal type: Mono, stereo or multi-channel.
• Type of sound-source: Acoustic instruments produce a very different sound than electric instruments, which are again different from the human voice.
• Situational circumstances: Sometimes a microphone should not be visible, or having a microphone nearby is not appropriate. In scenes for a movie the microphone may be held above the pictureframe, just out of sight. In this way there is always a certain distance between the actor and the microphone.
• Processing: If the signal is destined to be heavily processed, or "mixed down", a different type of input may be required.

###  Basic techniques

There are several classes of microphone placement for recording and amplification.

• In close miking, a directional microphone is placed relatively close to an instrument or sound source. This serves to eliminate extraneous noise, including room reverberation, and is commonly used when attempting to record a number of separate instruments while keeping the signals separate, or when trying to avoid feedback in an amplified performance. This technique was first used by Les Paul on his version of "The Tennessee Waltz" with Mary Ford.[citation needed]
• In ambient or distant miking, a microphone — typically a sensitive one — is placed at some distance from the sound source. The goal of this technique is to get a broader, natural mix of the sound source or sources, along with ambient sound, including reverberation from the room or hall.

###  Stereo recording techniques

There are two features of sound that the human brain uses to place objects in the stereo sound-field between the loudspeakers. These are the relative level (or loudness) difference between the two channels Δ L, and the time-delay difference in arrival times for the same sound in each channel Δ t. The "interaural" signals (binaural ILD and ITD) at the ears are not the stereo microphone signals which are coming from the loudspeakers, and are called "interchannel" signals (Δ L and Δ t). These signals are normally not mixed. Loudspeaker signals are different from the sound arriving at the ear. See the section "Binaural recording for earphones".

####  Conventional stereo recording for loudspeakers

The following microphone techniques can be used to capture the live "soundstage":

• The X-Y technique involves the coincident placement of two directional (cardioid) microphones. When two directional microphones are placed coincidentally, typically at a 90° angle (or greater) to each other (typically with each microphone pointing to a side of the soundstage), a stereo effect is achieved simply through intensity differences between the sound entering each microphone. Due to the lack of time-of-arrival stereo information, the stereo effect in X-Y recordings has less ambience. The main advantage is that the signal is mono-compatible, i.e., the signal is suitable for playback on non-stereo devices such as AM radio. If two bi-directional (figure 8) microphones are used instead of cardioid microphones, this technique is known as a Blumlein pair . Angles for microphones are: 90° - bidirectional, 131° - cardioid, 105° - hypercardioid, 115° - supercardioid.
• The Middle-Side (M-S) array technique is a special case of X-Y and uses a directional cardioid or an omnidirectional pressure microphone (M - middle microphone) and a bidirectional (figure-8) microphone (S - side microphone), placed at a 90° angle to each other with the directional microphone facing the soundstage. The outputs of these microphones are mixed in such a way as to generate sum and difference signals between the outputs. The S signal is added to the M for one channel, and is subtracted (by reversing phase and adding) to generate the other channel. M-S has two advantages: when the stereo signal is combined into mono, the signal from the S microphone cancels out entirely, leaving only the mono recording from the directional M microphone; additionally, M-S recordings can be "remixed" after recording to alter or even remove the stereo spread. The M-S technique with an omnidirectional M microphone is equivalent to X-Y with two cardioids at a 180° angle.
• Near-coincident recording is a variant of the X-Y technique and incorporates interchannel time delay by placing the microphones several inches apart. The ORTF stereo technique of the Office de Radiodiffusion Télévision Française (Radio France), calls for a pair of cardioid microphones placed 17 cm apart at an angle of 110°. In the NOS stereo technique of the Nederlandse Omroep Stichting (Holland Radio), the angle is 90° and the distance is 30 cm. The choice between one and the other depends on the recording angle of the microphone system, not on the distance to and the width of the sound source. This technique leads to a realistic stereo effect and has reasonable mono-compatibility. These interchannel signals have nothing to do with interaural signals which come only from artificial head recordings. Even the spacing of 17 cm has nothing to do with human ear distance. The ORTF and NOS engineers did not think in those terms, because this microphone system was developed for a set of stereo loudspeakers, not for earphones.
• The A-B technique uses two omnidirectional microphones at a moderate distance from each other (20 centimeters up to a few meters). Stereo information consists of large time-of-arrival distances and some sound level differences. With excessively large distances, the stereo image can be perceived as somewhat unnatural, as if the left and right channel are independent sound sources without an even spread from left to right. A-B recordings are not so good for mono playback because the time-of-arrival differences can lead to certain frequency components being canceled out and others being amplified, the so-called comb-filtering effect, but the stereo sound can be really convincing. If wide A-B is used for large orchestras, the center can be filled with another microphone. Then one gets the famous "Decca tree", which has brought us many good sounding recordings.
• The Blumlein shuffler technique uses two microphones spaced around 20 cm (head width), and these are usually, but not necessarily, omnidirectional. A special "Blumlein shuffler" circuit integrates the difference signal, before matrixing it to produce an output in which phase (time delay) information has been converted to amplitude difference. This is a purist technique for providing true stereo from binaural capture, permitting omnidirectional microphones to be used (with their low coloration and flat low-frequency response) for true stereo. It has been little used, probably because of the lack of commercial shufflers. While offering very realistic stereo, it can emphasise low frequencies picked up from the sides unless the shuffler incorporates rolloff in the difference path. A central baffle, in the form of a foam disc suspended between the microphones, provides level separation above 2 kHz where the shuffling has to be phased out.
• The Baffled Omnidirectional technique uses a pair of near-coincident omnidirectional microphones with an absorptive baffle between them and is closely related to binaural technique. Stereo information consists primarily of time-of-arrival differences between the microphones and intensity differences from the baffle. The Jecklin Disk, described by the Swiss radio technician Juerg Jecklin, uses of a 30 cm flat circular sound absorbing baffle arranged vertically with the faces perpendicular to the sound source. Pressure microphones are placed 16.5 cm apart, directly left and right of the disk's center. The KFM Sphere, described by Guenther Theile, consists of two pressure microphones mounted on opposite sides of a 20 cm sphere. The microphones are mounted flush with the surface and arranged with the 0-axis perpendicular to the sound source.

###  Surround microphone techniques

• The Double MS Technique was developed by Chris Wittig and Neil Muncy, and uses a front-facing mid-side microphone pair of direct sound pickup and a rear MS pair facing away from the front. The rear pair is placed at or just beyond the critical distance of the room where the reverberant sound level equals the direct sound level. The matrixed outputs feed front-left, front-right, rear-left, and rear-right speakers.
• The Surround Ambience Microphone Array was developed by Gunther Theile of the Institute for Rundfunktechnik (IRT). Four cardioid microphones are placed 90 degrees to each other and 21 to 25cm apart. No center channel is described.
• The Spider Microphone Array uses a special mike mount with five arms that radiate out from a center point, like a star. At the end of each arm is a condenser microphone aiming outward from the center. Two examples: The Microtech Gefell INA 5 uses five M930 mics in shock mounts. In the SPL Atmos 5.1/ASM 5 Surround Recording System, five Brauner condenser mikes feed a five channel mixing console, which adjusts the mic polar patterns and offers panning, bass management, and surround monitoring. SPL's Web site is www.spl-usa.com Both systems use the Ideal Cardioid Arrangement (ICA 5, ITU-775 specification), developed by Volker Henkels and Ulf Herrmann.


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# Closer form for $\int_0^\infty\frac{(\arctan{x})^2\log^2({1+x^2})}{x^2}dx$

I Would like to know the value of this integral. $$\int_0^\infty\frac{(\arctan{x})^2\log^2({1+x^2})}{x^2}dx$$ I think $$I=\frac{a}{b}(\pi^3\ln2)+\frac{c}{d}(\pi\ln^32)+\frac{e}{f}(\pi\ln^22)+\frac{g}{h}(\pi\ln2)+\frac{i}{j}(\pi^3)+\frac{k}{m}\zeta({3})??$$ Where a,b,c,d...are integers Thanks.

• Where did this horrible, integral-creature come from? – graydad Feb 17 '15 at 22:42
• W|A does not give me a value. But I think I've seen something similar in the past. I would start to tackle it using the series expansion of $\ln (1+x)$. I also think that complex analysis will come in handy here. Especially contour integration. – Tolaso Feb 17 '15 at 22:49

It has a closed form a bit different from what you conjectured: $$I=\frac{4\pi}3\ln^32+\frac{2\pi^3}3\ln2+\frac\pi2\zeta(3)$$
Let $\displaystyle\small\gamma=\lim_{R\to\infty}[-R,R]\cup Re^{i[0,\pi]}$. Observe that $$\small\oint_\gamma\frac{\ln^4(1-iz)}{z^2}dz=\frac{1}{8}\int^\infty_0\frac{\ln^4(1+x^2)}{x^2}dx-3\int^\infty_0\frac{\ln^2(1+x^2)\arctan^2{x}}{x^2}dx+2\int^\infty_0\frac{\arctan^4{x}}{x^2}dx=0$$ since
• The integral over the arc vanishes as $\small\mathcal{O}\left(\dfrac{\ln^4{R}}{R}\right)$.
• The singularity at $\small 0$ is removable.