Understanding stable levitation of superconductors from
intermediate electromagnetics
A. Badı́a – Majós1, ∗
1
Departamento de Fı́sica de la Materia Condensada–I.C.M.A.,
C.P.S.U.Z., Marı́a de Luna 1, E-50018 Zaragoza, Spain
(Dated: October 19, 2006)
Abstract
Levitation experiments with superconductors in the Meissner state are hindered by a low stability
except for specifically designed configurations. On the contrary, magnetic force experiments with
strongly pinned superconductors and permanent magnets display a high stability, allowing the
demonstration of striking effects, such as lateral or inverted levitation. These facts are explained
by using a minimal theory, in the form of a variational statement for electromagnetic energy related
quantities. Comprehensible illustrations, based on the calculated lines of magnetic field for various
configurations are presented. They provide a qualitative physical understanding of the stability
features. The theory may also be used for quantitative evaluation in terms of relevant material
parameters.
PACS numbers: 74.25.Sv, 74.25.Ha, 41.20.Gz, 02.30.Xx
Submitted to: Am. J. Phys.
1
I.
INTRODUCTION
Levitation experiments based on the repulsion (or attraction) force between permanent
magnets and superconductors are common nowadays. Thus, many of the Introductory
Physics or Materials laboratories in our universities are equipped with a magnetic levitation
kit. Almost everyone is amused and stimulated by the observation of flotating objects. When
the set-up includes a high pinning (or high critical current) superconductor, the possibilities of lifting moderate weights, and displaying lateral or inverted levitation are even more
attractive. Recent developments of vehicles capable of supporting several passengers have
been shown,1 increasing the interest in this topic. As the list of levitation phenomena has
become larger, the difficulties facing a reasonable explanation on how it works or performing
quantitative analysis have also increased. Thus, it is rather simple to understand that a
magnet floating above a superconductor (or viceversa) is related to flux expulsion. One may
even use the standard image technique in magnetostatics in order to perform quantitative estimates. Specialized image models have also been introduced, which allow us to understand
attractive forces.2 However, such approximations are only useful for small displacements of
tiny magnets close to the superconductor and do not include material parameters.
In this work, as an alternative and complementary point of view to previous publications,3
I will give a theoretical framework, and a number of examples which may be of help for
filling the gaps referred above. The presentation is aimed at students who have followed an
intermediate course on Electromagnetics and have some background in Classical Mechanics.
The main concepts involved are: (i) electromagnetic energy, (ii) thermodynamic reversibility
and irreversibility, (iii) Lenz-Faraday’s law of induction, and (iv) the use of variational
principles. For brevity, the superconducting properties will be simply developed in their
simplest form as required. For qualitative purposes, the intuitive picture of lines of force will
be exploited. The main equations of the physical model will then be shown and explained.
Finally, a number of analytical calculations and computational exercises will be both shown
and suggested.
The presentation emphasizes the peculiarities of levitation with the so-called type-I or
type-II superconductors. Each case relates to a way of avoiding the limitations imposed by
Earnshaw’s theorem,4 which restricts levitation in electromagnetic systems.5 Type-I materials allow static levitation as related to diamagnetism. Levitation with type-II materials is
2
a dynamic effect related to magnetic flux conservation and minimum energy losses.
II.
BASIC SUPERCONDUCTIVITY AND VARIATIONAL PRINCIPLES
Superconductivity is a complex phenomenon, whose nature combines electromagnetic,
thermodynamic and quantum effects. However, for our purposes, governing equations may
be built, recalling relatively simple properties. In fact, one may start with principles of
single particle dynamics, and follow a step-by-step process until the proper material law is
reached. Variational statements are aimed.
A.
Type-I superconductors
Let us begin with the definition of the electric current density for conducting media
J~ = N q~v .
N is used for the volume density of the charge carriers, q for their effective charge, and ~v
for their velocity. If the underlying material is such that charges can move without friction,
Newton’s second law gives
N q2 ~
1 ~
dJ~
=
E≡
E.
dt
m
µ0 λ 2
Here λ defines a characteristic length, which one calls London penetration depth. This
equation leads to the property
~
2 dJ
d
~ · J~ = µ0 λ
E
· J~ =
dt
dt
µ0 λ 2 2
J .
2
!
(1)
If this is included in the Poynting’s theorem, it is apparent that the standard electromagnetic
field energy is augmented by a new form of reversible storage, straightforwardly related to
the kinetics of the moving charges. Thus, neglecting the presence of electrostatic charges,
one has the total energy conservation law
!
Z
d Z
B2
λ2
2
~ dV ≡ dU = 0 ,
dV +
|∇ × B|
3
dt IR 2µ0
dt
VS 2µ0
~ as the unknown variwhere VS denotes the superconducting volume. By using the field B
able, and δ for denoting derivatives respect to it, one may express the equilibrium condition
Z
minimize
IR3
Z
B2
λ2
~ 2 dV
dV +
|∇ × B|
2µ0
VS 2µ0
3
!
⇒
δU
= 0,
~
δB
(2)
in order to obtain the static configuration for fixed sources (boundary conditions for the
fields). Notice that this formulation reflects the flux expulsion (diamagnetism) in superconductors, as one minimizes B 2 . This is the so-called Meissner state. The more conventional
~ + λ2 (∇ × ∇ × B)
~ = 0 (London equation6 ) follows from the null
differential statement B
derivative condition in the previous statement.
B.
Type-II superconductors
For certain superconducting materials, the explanation of many observations requires to
relax the assumption that the electric current flow is completely free of losses. Thus, in the
so-called hard type-II’s only low (undercritical) currents posses such property. As a first
approximation, one may assume the lossless behavior for J ≤ Jc (Jc : critical current for the
material), and a damped regime for J > Jc , quite in the manner of Ohm’s law for normal
metals [here E = ρsc (J − Jc )]. The nature of this phenomenon, and the interpretation of the
intrinsic parameter Jc may be found in Ref.7. In brief, these materials can hold an internal
magnetic flux structure as long as field gradients (J) remain below some threshold. Above
this value, flux is unpinned and the underlying currents flow dissipatively.
Here, I will show that the behavior of these materials also allows a variational model. To
start with, recall that the dynamical equations of a single particle under conservative forces
may be obtained by a minimum action principle
Z
minimize
⇒
L dt
d
dt
∂L
∂ ẋ
!
=
∂L
∂x
(3)
with L = mẋ2 /2 − V the Lagrangian. Notice that, in general, non-conservative forces may
not be treated variationally. In fact, Newton’s second law is equivalent to
d
dt
∂L
∂ ẋ
!
=
∂L
+ Fncons .
∂x
(4)
Nevertheless, if one assumes that Fncons is a viscuos drag force (Fncons = −mγ ẋ) with a
friction constant γ, Eq.(4) may be obtained from
Z
minimize
∆t
⇒
L̂dt
0
d
dt
∂ L̂
∂ ẋ
!
=
∂ L̂
.
∂x
(5)
Here, we have defined L̂ ≡ L + (mγ ẋ2 /2)t, and assume that ∆ẋ ẋ for increments within
the time interval [0, ∆t]. In such case, (5) leads to mẍ = −mγ ẋ − ∂V /∂x as required.
4
E
−Jc
J
Jc
FIG. 1: {E, J} graph (conduction law) for a hard type-II superconductor, according to Bean’s
model. Vertical lines correspond to infinite resisitivity above Jc .
The model (5) is, thus, a quasistationary variational principle, which may be applied in
a time discretized description of the system (redefine [0, ∆t] and iterate). Generalization is
apparent if one recalls that, for the single particle, mγ ẋ2 is just the energy loss per unit time.
For instance, the eddy-current problem in normal (ohmic) metals may be solved iteratively
by the principle
minimize Sn ≡
Z
∆t
0
Z
IR3
L̂ dV dt
(6)
~ · J/2)t
~
with L̂ ≡ B 2 /2µ0 + (E
the modified Lagrangian density for the magnetostatic field.
~ · J~ corresponds to the energy dissipation per unit time and volume. Then, if
Recall that E
~ = ρJ~ and ∇ × B
~ = µ0 J~ within
one assumes ∆E E and uses the stationary relations E
the time interval [0, ∆t], the principle (6) becomes
minimize Fn ≡
Z
IR3
Z
~ 2
(∆B)
ρ∆t
~ 2 dV .
dV +
|∇ × B|
2µ0
2µ
VM ET AL
0
(7)
~ one gets
Eventually, taking variations respect to the variable B
δFn
=0
~
δB
⇔
~
∆B
ρ
~
= −∇ ×
∇×B
∆t
µ0
!
as expected. This is just the time-discretized combination of Faraday’s and Ohm’s laws.
Now, the application of the previous ideas to superconducting media follows in a natural
way. One has to use a suitable form for the energy loss related to overcritical current
flow. The simplest theory accounting for losses in such conditions was proposed by C. P.
Bean8 within the context of magnetic hysteresis. In terms of a conduction law, Bean’s
model is equivalent to the E(J) relation sketched in Fig.1. Notice that the multivalued
5
graph corresponds to the overdamped limit of the physical properties mentioned before:
(i)non-dissipative current flow is allowed for current densities below a critical value Jc , and
(ii) induced electric fields relate to the current density flow by an infinite slope resistivity.
Obviously, the vertical relationship is just an idealization of real cases in which a very high
slope (ρsc ) appears. In terms of (6), Bean’s law gives place to a quasistationary principle in
which the modified Lagrangian reads L̂ = L + Dt, with the dissipation function
D=


 0 if J < Jc
(8)
.

 ∞ if J > Jc
The variational principle admits a simple treatment of this singular behavior. One may just
minimize the first term in (7) and replace the second one by a cutoff condition J ≤ Jc .
By using a time-discretization in layers of width δt, i.e.: tn = nδt we end up with the
iterative model9
1 Z
~ n+1 − B
~ n |2
|B
minimize
2µ0 IR3
~ n+1 | ≤ µ0 Jc in VS
for |∇ × B
~ n+1 = µ0 J~0,n+1
and ∇ × B
in IR3 \ VS .
(9)
IR3 \VS denotes the region of space outside the superconductor (VS ), where magnetic sources
~ n+1 , a differential
are located. I recall that, owing to the unilateral restriction for ∇ × B
equation statement does not directly follow from the previous model. On the other hand,
the model is clearly linked to the compensation between minimum magnetic flux changes
(this is Lenz-Faraday’s law) and minimum energy losses in the evolution of the system.
III.
LEVITATION IN THE MEISSNER STATE
Diamagnetism is a necessary condition for stability,5 however it is not sufficient. A specific
geometrical configuration is required in order to produce a position of stable equilibrium in
the field of magnetic and gravitational forces. Thus, the radial force on a dipole magnet
over a diamagnetic disk has been shown to be directed outward10 unless the magnet is just
above the center (there, it vanishes). It is for this reason that in his pioneering experiment11
V. Arkadiev used a small permanent magnet which was floated over a concave lead bowl.
Recall that lead is a type-I superconductor.
6
Potential
FIG. 2: Magnetic field lines around a horizontal magnetic dipole over a superconducting tape,
for three positions of the magnet. Induced currents within the tape flow perpendicular to the
plot. Inwards and outwards orientations are marked with two colors. To the right, we plot the
~ S for small horizontal displacements of the magnet
magnetostatic potential energy Um = −m
~ ·B
around each position.
These facts are illustrated in Figs.2 and 3, where we display the structure of magnetic
field lines for a small horizontal magnet on top of a flat sample and on top of a curved
one. It is noticeable that, in the first case, an arbitrarily small horizontal disturbance in the
position of the magnet, irreversibly leads to its fall over the edge. However, as it is apparent
in Fig.3, the concave structure of the superconductor produces a restoring force that pulls
the magnet back to its equilibrium position above the center.
As regards the vertical force for horizontal configurations, the reader may convince himself
that it is repulsive by using the image method limit and considering the two poles of the
magnet. Here, we skip a deeper analysis for shortness.
7
Potential energy
FIG. 3: Same as Fig.2 for a superconducting tape with concave section. Here, the potential has
been calculated for a number of positions around the axis of symmetry.
A.
Minimum energy model.
In Ref.11, Arkadiev remarked that levitation may be understood by the repulsion force
between the real magnet and the corresponding magnetostatic image within the superconductor. Nevertheless, this does not explain the aforementioned facts on stablility. The image
is only simple for the limiting case of a superconducting half-space, becomes a non-trivial
problem of complex variables in the flat disk12 and may require much effort in arbitrary
geometries. Below, we develop a theory including the fewest ingredients (Sec.II) for studying the stabilization process in a general context. Figs.2 and 3 have been obtained as two
specific applications.
We have solved the integral statement (2) by a numerical technique. Some make-up of
the model was of help before tackling it. For our purposes, the so-called bulk approximation
8
(λ → 0) may be used.9 Thus13 we get
µ0 Z
~ 2 dV for ∇ × H
~ = J~0 in IR3 \ VS .
H
(10)
2 IR3
~ = J~0 establishes the magnetic sources.
The condition ∇ × H
We now use vector
minimize
differential calculus to restrict the infinite domain. The problem (10) is equivalent to
minimize

µ Z Z
0
 8π

Z

J~S (~x) · J~S (~x 0 )
0
~ 0 · J~S dV ,
dV
dV
+
A

|~x − ~x 0 |
VS
VS
(11)
~ 0 for the
where, we have used J~S for the current density within the superconductor and A
magnetic source vector potential.
If one can determine a priori (e.g. by symmetry considerations) the current density
streamlines (paths of the charge carriers), a mutual inductance approach may be used.
Eq.(11) becomes
minimize

1 X
2
Ii Mij Ij +
i,j
X
i


Ii A0i  .
(12)
Here the set of unknowns {Ii } stands for the current elements flowing along appropriate
circuits and Mij the problem’s mutual inductance matrix. Notice that, though possibly
in many variables, this is just a quadratic problem that may be solved with moderate
computational effort.
B.
Application
For obtaining Figs.2 and 3 we have considered the magnetic field structure over a long
flat/curved superconducting tape when a parallel dipole line is placed above and horizontally
shifted. Recall that the dipole line vector potential is
~ × ~r
~ 0 = µ0 m
,
A
2π r2
with m
~ the magnetic moment per unit length. On the other hand, the superconducting
current density will flow along parallel infinite straight lines, for which the expressions
µ0
8π
µ0
a2
=
ln
2π (xi − xj )2 + (yi − yj )2
Mii =
Mij
(13)
have been obtained. Above, a stands for the radius of the wires, which are assumed to be
equal, and along the Z axis.
9
IV.
MAGNETIC LEVITATION WITH PINNED SUPERCONDUCTORS
With the advent of high-Tc superconductivity, a number of exotic levitation configurations were reported.14,15 Nowadays, such experiments are routinely reproduced with the
availability of good quality samples at the temperature of liquid nitrogen. As was immediately recognized, the key property behind rigid (stable) levitation is the pinning of magnetic
flux lines within the superconductor. Recall this property for type-IIs, the group to which
high-Tc superconductors belong.
A.
Minimum action model
The stability issues in levitation experiments with type-II superconductors will be discussed within the framework introduced in Sec.II B. Thus, one has to solve the constrained
minimization statement (9).
As was discussed in Sec.III A, minimization is better realized when the problem is mapped
into the finite volume of the superconductor. Proceeding as in that case, we arrive at
(Z Z
minimize


J~n (~x) · J~n+1 (~x 0 ) 
J~ (~x) · J~n+1 (~x 0 )
 n+1
−
2
|~x − ~x 0 |
|~x − ~x 0 |
VS
8π Z ~
~ 0,n · J~n+1
+
A0,n+1 − A
µ0 VS
for
J~n+1 ≤ Jc in VS
and
~ 0,n+1
A
)
given .
(14)
Again, minimization may be further discretized in spatial variables, and we get the mutual
inductance formulation
minimize
X
1X
Ii,n+1 Mij Ij,n+1 −
Ii,n Mij Ij,n+1
2 i,j
i,j
+µ0
X
Ii,n+1 Si (A0,n+1 − A0,n )
i
for
|Ii,n+1 | ≤ Ic
and
~ 0,n+1
A
given .
(15)
This statement has to be solved as follows: one starts from the initial configuration {Ii,1 } and
~ 0,1 , A
~ 0,2 , A
~ 0,3 , . . .
iteratively finds {Ii,2 }, {Ii,3 }, . . . in terms of the desired excitation process A
10
Below we present some applications of the statement (15) related to levitation configurations
using magnets and superconductors. The interested reader is directed to Ref.16 for the
technical details on constrained numerical minimization.
FIG. 4: A small magnet descends towards a cool superconductor, which rejects the magnetic field
by means of critical currents with density ±Jc . Induced current lines are perpendicular to the plot.
Inwards/outwards flow is indicated by two color scales.
B.
Examples
Most of the properties which are typically displayed in demonstrations with high-Tc
superconductors may be explained within the previous theory. For instance, it is possible
to describe the difference between cooling the superconductor close to the magnet or at a
long distance. One can also predict the appearance of lateral restoring forces, or in general
to calculate what happens when arbitrary displacements of the magnet occur around the
superconductor.
Fig.4 displays the magnetic field configuration which arises when a small magnet descends
towards a cool superconductor. We consider a long superconducting bar (rectangular cross
section 2W ×L) and a magnetic dipole line on top. Notice that the magnetic field penetrates
from the upper side, being excluded in the lower part of the sample. Superconducting
11
FIG. 5: Standard method for levitating a magnet above a hard superconductor. The superconductor is cooled with the magnet close to the surface (bottom-left picture), then the magnet is moved
away and finally put back to its original position.
currents, along the infinite length of the bar, with density J = ±Jc have been induced in
the region penetrated by the field. The combination of induced currents and the dipole line
produces the resultant magnetic field structure.
On the other hand, the repulsion between the magnet and the superconductor follows
~ Orientation of J~
directly from the picture and the elementary force per unit volume J~ × B.
has to be done according to Lenz’s law.
When the superconductor is cooled with the permanent magnet close to the surface, a
new physical picture arises. To start with, the flux structure due to the magnet is frozen
~ = 0 is fulfilled, and no change occurs until
within the sample. The condition J~S = ∇ × H
the magnet is shifted. This corresponds to the condition E = 0 in Fig.1. Then, if one
~ = −µ0 ∂ H/∂t)
~
moves the magnet (see Fig.5), an electric field arises (∇ × E
and according
to the E(J) law, a critical current distribution appears. In Fig.5 we plot the simulation of a
12
Vertical Force
0.4
0.2
descending
0
−0.2
ascending
−0.4
1
2
1.5
Height (y/L)
2.5
FIG. 6: Vertical force as a function of normalized height for the small magnet in Fig.5. Positive
sign corresponds to repulsion, while negative means attraction.
typical process of stable magnetic levitation. The magnet is initially moved away from the
superconductor. Due to Faraday-Lenz’s law, a current distribution appears, which tries to
preserve the frozen flux lines. This induces an attractive force, as related to the paramagnetic
flux line structure, which is observed (the field is compressed towards the superconductor).
However, if the movement of the magnet is reversed (rightmost part of the picture), the new
induced currents flow in opposition to the previous structure. Now, the sample is trying to
avoid field penetration. As a consequence, the interaction force becomes repulsive and one
can observe stable levitation. Fig.6 shows the actual behavior of the levitation force for a
specific process. This quantity has been evaluated from the relation
F~ =
Z
VS
~ 0 × J~S
µ0 H
dV
which combines the magnetic force on moving charges, and Newton’s third law.
One can show that the process illustrated in Fig.5 leads to a highly stable configuration.
Thus, Fig.7 displays the current density and flux line structure that appear when the levitating magnet is laterally displaced. Again, the evolutionary current density distribution
accomodates so as to preserve the flux line structure within the sample, to the highest degree
possible. As a consequence, a deep potential well in the magnetostatic energy (Um ) arises.
We emphasize that the restoring force remains even when the magnet is beyond the edge
of the superconductor. However, a noticeable change in the slope of Um is observed at such
point.
13
Potential Energy
−W
0
W
FIG. 7: Lateral displacement of the small magnet considered in Fig.5. The restoring property is
illustrated by the magnetostatic potential energy. W denotes the superconductor’s half width. Full
symbols are used to indicate the corresponding positions of the magnet in the upper plots
V.
CONCLUDING REMARKS
The main property of type-I superconductors is flux expulsion (diamagnetism). This
phenomenon leads to a vertical repulsion froce, which allows levitation. However, lateral
stability is poor. Thus, magnets have to be levitated over bowl shaped superconductors.
These facts follow from a minimum magnetostatic energy model in a simple fashion. Further
refinements such as the inclusion of finite penetration depth λ are easy to implement.
The basic ingredients underlying the rich phenomenology in levitation experiments with
type-II superconductors are Faraday’s law and a highly non-linear E(J) relation. When
14
these properties are put in the form of a variational principle, the features of field or zero
field cooling, finite size effects, and lateral stability are immediately obtained. As in other
physical systems (air cushions, eddy current based,...) lateral stability with hard type-II superconductors is automatically achieved by a physical tendency to minimize the irreversible
loss of energy. What is unusual in the case of superconductors is that dissipation only occurs
during transitions between metastable states with different magnetic flux structures. This
allows to avoid a continuous supply of energy.
VI.
SUGGESTED PROBLEMS
~ = 0 in (2) leads to B
~ + λ2 (∇ × ∇ × B)
~ = 0 within
(1) Show that the condition δU /δ B
the superconductor. Notes: take the formal derivative of the integrand in the sense of
~ and to (x, y, z) may be interchanged. Integration
a gradient. Derivatives respect to B
may be extended to IR3 by taking λ → ∞ outside the superconductor.
(2) Check that the statement (5) leads to the correct dynamical equations for an object
falling in uniform gravity, against a viscous force, when ∆ẋ ẋ.
(3) Show that the time integration of (6), performed in the sense of averages over the
interval [0, ∆t] leads to the spatial variational principle in (7).
(4) Show that (10) may be transformed into (11) by means of vector calculus manipulations. Hint: introduce the magnetic vector potential, and use the divergence theorem.
(5) Obtain the expressions (13) and write a computer code for solving the matrix problem
in the statement (12). Apply it to different shapes of the superconductor.
Acknowledgements
I am indebted to L. A. Angurel and M. Mora for assistance with levitation experiments
and useful discussions.
∗
Electronic address: anabadia@unizar.es
15
1
J. Wang et al., ”The first man-loading high temperature superconducting Maglev test in the
world”, Physica C 378-381, 809-814 (2002).
2
A. A. Kordyuk, ”Magnetic levitation for hard superconductors”, J. Appl. Phys. 83, 610-612
(1998).
3
E. H. Brandt, ”Rigid levitation and suspension of high-temperature superconductors by magnets”, Am. J. Phys. 58, 43-49 (1990).
4
S. Earnshaw, ”On the nature of the molecular forces which regulate the constitution of the
luminiferous ether”, Trans. Camb. Phil. Soc. 7, 97-112 (1842).
5
M. D. Simon, L. O. Heflinger, ans A. K. Heim, ”Diamagnetically stabilized magnet levitation”,
Am. J. Phys. 69, 702-713 (2001).
6
R. Feynman, R. B. Leighton, and M. Sands, The Feynman lectures on Physics, vol.3, Quantum
Mechanics (Addison-Wesley, Reading, Massachusetts, 1965).
7
M. Tinkham, Introduction to Superconductivity, 2nd ed (McGraw-Hill, New York, 1996).
8
C. P. Bean, ”Magnetization of high-field superconductors”, Rev. Mod. Phys. 36, 31-39 (1964).
9
In most experimental conditions, the kinetic energy of the charge carriers may be neglected due
to the smallness of λ (submicron range), as compared to the scale of the sample (centimeters).
10
L. C. Davis, E. M. Logothetis, and R. E. Soltis, ”Stability of magnets levitated above superconductors”, J. Appl. Phys 64, 4212-4218 (1988).
11
V. Arkadiev, ”A floating magnet”, Nature 160, 330 (1947).
12
L. C. Davis and J. R. Reitz, ”Solution to potential problems near a conducting semi-infinite
sheet or conducting disk”, Am. J. Phys. 39, 1255-1265 (1971).
13
~ = rot×(µ0 H)
~ = µ0 (J~S +J~0 )
In this work, we follow the convention for macroscopic fields: rot×B
~ = div(µ0 H)
~ = 0, i.e.: superconducting carriers are treated as free charges with flow
and divB
density J~S .
14
F. Hellman, E. M. Gyorgy, D. W. Johnson, Jr., H. M. O’Bryan, and R. C. Sherwood, ”Levitation
of a magnet over a flat type-II superconductor”, J. Appl. Phys 63, 447-450 (1988).
15
W. G. Harter, A. M. Hermann, and Z. Z. Sheng, ”Levitation effects involving high Tc thallium
based superconductors”, Appl. Phys. Lett 53, 1119-1121 (1988).
16
A. R. Conn, N. I. M. Gould, Ph. L. Toint, LANCELOT: A Fortran Package for Large-Scale
Nonlinear Optimization, Springer Series in Computational Mathematics (Springer Verlag, Berlin
1992).
16